More Math talk

Election Amendment 11/13/2024

Admitting that this proposal has a probabilty of ever happening that is close to zero, here are my thoughts of amending the US Constitution. First, keep the election to US Senate the same with the following changes: No one can be elected to the Senate for more than one six-year term and give Puerto Rica and D.C. statehood with 2 senators each. Puerto Rica is more populous than 19 of our states! The District of Columbia is more populous than Wyoming and Vermont. Second, make the House of Representatives have a 4-year term with a one term limit. Have each half of the house elected on alternate even number years. Notice this plan eliminates incumbents running for re-election in either the House or the Senate! Third, make the election of the President and Vice President by popular vote every 8 years on odd number years. After 4 years the President and Vice President swap positions. No President or Vice President can be re-elected for a second term. Thus, no incumbent can run for these offices. Furthermore, the Presidential elections would occur on years different from the Senate and House elections. I will have more comment on these plans in future entries in the Math Teacher’s Journal.

Sunday, November 10, 2024

Edward Burr Van Vlek wrote a paper on Continued Fractions. One of Van Vlek’s doctoral students was Hubert Stanley Wall, who wrote one of the most informative and widely circulated books on the theory of Continued Fractions. One of Dr. Wall’s graduate, doctoral students was Jack B Brown. Dr. Brown was my major professor at Auburn.

Thursday, November 7, 2024

I am reading the book “The Irrationals” by Julian Havil. In this book Havil states that the golden mean is the most irrational of all irrational numbers. The perspective that points to this conclusion is that the golden mean is the most difficult of all numbers to approximate by rational numbers. I am trying to wrap my brain about the logic and procedures that seem to make this so.

Monday, November 4, 2024

Both the numerator and denominator sequences in the November 1 journal entry are close to being strictly increasing – though the decimal representations for the fractions are not.

Friday, November 1, 2024

The continued fraction, quadratic equation and rational number sequences investigations that were discussed in my “Math Teacher Journal” are the type of mathematics that keep generating new questions. Again I want to look at the sequence of rational numbers -3/2, -6/1, 3/4, -12/11, -33/10, 30/13, -39/56, -168/73, 219/22, -66/263, … In the October 23, 2024 entry, we looked at this sequence in decimal form and its appearance was quite chaotic. But there are two other sequences generated here. The sequence of numerators -3, -6, 3, -12, -33, 30, -39,… and the sequence of denominators 2,1,4,11,10,13,56, … are both intriguing to me. I have some question about the nature of these sequences.

Wednesday, October 30, 2024

“beauty can be perceived but not explained.” Arthur Cayley. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws. At the age of 17 Cayley began residence at Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. The cause of the Analytical Society had now triumphed, and the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects that had been suggested by reading the Mécanique analytique of Joseph Louis Lagrange and some of the works of Laplace. He took great interest in the movement for the university education of women. At Cambridge the first women’s colleges were Girton and Newnham. In the early days of Girton College he gave direct help in teaching, and for some years he was chairman of the council of Newnham College, in the progress of which he took the keenest interest to the last. Cayley was also an accomplished lawyer, but gave up this profession to be a full time mathematician.

Monday, October 28, 2024

“.. it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician.” a quote from Karl Weierstrass, the great German mathematician. I doubt that I will ever be remembered as a great mathematician or a great poet. But, I have tried my hand at both math and poetry and at least it may be said that I have a great love for both. I was watching a movie the other day when lines from the poem “The Children’s Hour” by Henry Wadsworth Longfellow were quoted by the characters. The poem’s opening lines are: “Between the dark and the daylight, When the night is beginning to lower,” Ah, many a mathematician has been, metaphorically, in such a time or place! But the words that catch my mind’s eye are: “They almost devour me with kisses, Their arms about me entwine, Till I think of the Bishop of Bingen In his Mouse-Tower on the Rhine!” These words led me to discover the horrible legend associated with this mouse tower … and cause me to ponder why Longfellow would use this reference in a poem seemingly written for parents to read to their children. I have suspected for some time that Longfellow sometimes had a deeper, darker side. The poem goes on to end with some beautiful lines, I have you fast in my fortress,
  And will not let you depart,
But put you down into the dungeon
  In the round-tower of my heart.

And there will I keep you forever,
  Yes, forever and a day,
Till the walls shall crumble to ruin,
  And moulder in dust away!

Friday, October 25, 2024

Several weeks ago, I was in New York City, visiting a friend. We took an Uber over to Manhatton to visit Grant’s Tomb and the Riverside Church. These two sights are adjacent to each other within a great view of the Hudson River. Grant’s Tomb is impressive, It’s managed and run by the National Parks Service and you can visit it free. The Riverside Church is a great church and a magnificent example of architecture. Its original pastor was Harry Emerson Fosdick, one of the great protestant preachers of the 20th century. We took the elevator up to where the carillon is housed and took over 100 steps to the top of the 391 foot church tower, where the view of Manhatton was spectacular. I got to ring one “dong” on the largest “tuned” bell in the world.

Wednesday, October 23, 2024

Let’s take another look at the sequence listed in Math Teachers Journal last Saturday and Thursday, 10/12, 10/17: -3/2, -6, 3/4, -12/11, etc. Below are the 1st 24 numbers as decimals (1,-1.500); (2, -6.000); (3, 0.750); (4, -1.0909); (5, -3.3000); (6, 2.3077); (7, -0.6964); (8, -2.3017); (9, 9.9545); (10, -0.259); (11, -1.7152); (12, -10.5343); (13, 0.2525); (14, -1.2758); (15,-4.1423); (16,1.4003); (17, -0.8823); (18, -2.6840); (19, 5.3860);(20, -0.4698); (21, -1.9608); (22, -75.0506) (23, 0.0406); (24,-1.4702); … This does look chaotic, but there seems to be a pattern to the chaos. The first iteration is -1.5, then the results go further down to -6, then up to a positive 0.750. This pattern of down, further down, up to positive, down, further down, up to positive continues proving some order to the chaos. But wait! Look at iterations 10, 11, 12 and 20, 21, 22. In both of these sequences we get 3 negatives in a row! And the 22nd number seems to be an anomaly with a relatively big magnitude negative number, -75.0506. I wrote a simple computer program to number look for more large magnitude “outliers” and found some. the 831st number is 2915.37! Other numbers with surprising magnitudes are (1002, 444.13587); (1492, 303.383) and (1665, 1457.237). I suspect as we follow this sequence toward infinitely many iterations there is no limit to how large the magnitudes might get to be. I changed my program to run until it found a result bigger than 10 thousand It ran for quite a while (it was a graphing calculator program) until it stopped on the twelve thousand, eight hundred, and 21st number which gave a result of 17,958.32!

Monday, October 21, 2024

The sequence discussed in the October 17 journal entry and also in the October 12 journal entry can be calculated from the quadratic equation x^2 + 2x + 3 = 0, and translating the quadratic into a continued fraction. (see the procedure shown in October 12 journal entry). This sequence -3/2, -6, 3/4, -12/11, etc. does not converge, in fact looks quite chaotic. Any quadratic equation, using the method referenced, can be changed to a continued fraction and then a sequence. If the quadratic has real roots, the sequence will converge. If the sequence has non-real complex roots, the sequence will not converge and appear to be chaotic. Let’s take a look at the equation x^2 – 1x – 2 = 0. This has real roots (zeros) at x=2 and x=-1. The continued fraction can be formed by changing the equation to the form, x(x-1) = 2, then x = 2/(-1 + x). So, we get the sequence -2, -2/3, -6/5, -10/11, -22/21, -42/43, … converging to the real root -1.

Thursday, October 17, 2024

In Saturday’s Journal entry we generated the following sequence: -3/2, -6/1, 3/4, -12/11, -33/10, 30/13, … etc. There is a, somewhat hidden, simple recursion formula which for any fraction in the sequence will calculate the next numerator and denominator. Multiply the denominator by negative three to get the next numerator. Add two times the denominator to the numerator to get the next denominator. If this makes the next fraction a negative ratio, place the negative sign with the numerator. For example, to get the next fraction after -12/11, multiply eleven by negative three giving -33. Add twice 11 to -12 giving 10. This makes the next fraction -33/10. Using the same rules, to find the fraction following 30/13, negative three times 13 is -39 and twice 13 plus 30 is 56. So, the next fraction is -39/56.

Saturday, October 12, 2024

If we choose a quadratic equation with complex roots, weird stuff starts happening! For example, the equation x^2 + Ax – B =0 (from my October 8 journal entry) and letting A=2 and B=-3, we get the equation x^2 + 2x + 3 = 0 which has no real roots. However, a continued fraction can be constructed. Moving 3 to the right of the equals sign and factoring out x on the left, we get x(2 + x) = -3. then dividing by (2 + x) gives x = -3/(2 + x). With successive substitution for x (as in October 8 discussion) we get the infinite continued fraction x = -3/(2 + -3/(2 + -3/(2 + -3/(…)))). This generates the following sequence -3/2, -6, 3/4, -12/11, -33/10, 30/13, -39/56, -168/73, … which doesn’t seem to be converging at all, and in fact doesn’t. It is not a simple matter even to find a recursive pattern. Though in a future journal entry, I will show a fairly simple one.

Tuesday, October 8, 2024

The quadratic equation, x^2 + Ax – B = 0 can be rewritten as an infinite continued fraction as follows. If we move B to the right side of the equals sign and factor x out of the terms remaining on the left of the equals sign we get x(x + A) = B. Dividing this equation by (x + A) we get the equation x = B/(A + x). Replacing the x on the right side of the equation with B/(A + x), we get x = B/(A + B/(A + x). And if the substitution is repeated ad infinitum we get, x = B/(A + B/(A + B/(A + B/(A + B/( …)))) and the solution to the quadratic equation can be found as the limit of sequence formed if A and B are replaced with numerical values.

Wednesday, October 2, 2024

Here is recursive rule I found for 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, etc. Take the preceding fraction, multiply its denominator by 4 to get the next fraction’s numerator. Subtract the preceding numerator from this numerator to get the denominator, Reduce if possible. For example to get the fraction after 12/7, multiply 7 by 4 to get 28, subtract 12 to get 16 – this gives 28/16 which reduces to 7/4.

Sunday, September 29, 2024

“From the rocking of the cradle, to the rolling of the hearse.” Kris Kristofferson R.I.P. I will reveal my answer of a recursion formula for the sequence in 9/27/2024 in a few days. (If old age or dementia doesn’t stifle my brain cells too much.)

Friday, September 27, 2024

1/1, 4/3, 3/2, 8/5, 5/3, 12/7, 7/4, 16/9, 9/5, … 25/13, … etc. approaching a limit of? This sequence has to do with the continued fraction discussed in my journal entry of 9/23/2024. And a recursion pattern can be found using an analysis similar to what we did to find a recursion pattern in the sequence we examined in journal entry 8/22/2024. By this we mean there is a formula to get the next number in the sequence after the number preceding it. For example, a try might be “add 9 to the numerator and 5 to the denominator”. This works for finding 16/9 after 7/4, but I suspect it works nowhere else. A better try might be “add 2 to twice the numerator and add 1 to twice the denominator.” This will also get 16/9 after 7/4 and furthermore works for 1/1, 3/2, 5/3 and many others … but not for all numbers in the sequence. Is there ONE formula that will work for every number in the sequence?

Wednesday, September 25, 2024

“It is the consequence of humanity. We are all formed with frailty and error, let us pardon reciprocally each other’s folly – that is the first law of nature.” Voltaire

Monday, September 23, 2024

I have been doing some more napkin scribbling on a favorite topic. Continued fractions give so many chances to look at number patterns. Looking at the infinite continued fraction x = A/(A + A/(A + A/(A + A/(… Since at “infinity” x contains itself we can rewrite the expression as x = A/(A + x). Multiplying by (A + x) and setting the expression equal to zero we get x^2 +Ax – A = 0. Using the plus case of the quadratic formula we get the solution that x = (-A + sqrt(A^2 +4A))/2. If we let A equal 1, this solution is (-1 + sqrt(5))/2 which is the Golden Mean”. If we replace A with the next positive integers 2,3,4,5,6,7,8 we start a recognizable pattern: (-2 + sqrt(12))/2; (-3 + sqrt(21))/2; (-4 + sqrt(32))/2; (-5 + sqrt(45))/2; (-6 + sqrt(60))/2; (-7 +sqrt(77))/2; (-8 + sqrt(96))/2…

Notice that the numerator of this expression is (-N + sqrt((N+2)^2 – 4). This value approaches 2 from below as N gets very large. For example, -100 + sqrt(102^2 – 4) = 1.98039 and -1000 + sqrt(1002^2 – 4) = 1.998004. So, the denominator two makes the limit (as N gets very large) of (-N + sqrt((N+2)^2 -4)/2 equal to 1. The conclusion is that the continued fraction A/(A + A/(A + A/(A + … for positive numbers, A, can never get larger than 1.

Tuesday, September 17, 2024

I think of Toffler’s prediction of “Future Shock” as I ponder what is presently occurring in higher education, robotics and artificial intelligence. Events, inventions and societal progress are moving at us at the speed of overwhelming bafflement. Education seems to be moving toward utilitarianism. Robotics and A.I. are progressing in attempts to augment (or substitute for) human capability. So, in a sense, humans are becoming more like machines and machines are becoming more like humans. Let’s propose the following question for thought. Will humans replace machines, or will machines replace humans? If this question troubles you, get a good book and read it and deeply think about what it is saying. Perhaps this is one activity in which we humans will continue to excel.

Friday, September 13, 2024

Ah yes, today is Friday the 13th. Triskaidekaphobia … There are numerous possible origins of the superstitions around his date. 13 people attending the “Last Supper of Jesus”, Norse mythology has the 13th guest at the feast of Valhalla was Loki, etc., are two of many. There was a calendar proposal to eliminate Friday the 13th’s by going to 13 months of 28 days which gives 364 days – adding New Year’s Day as a special “non-weekday” Of course, this would make an entire month unlucky!

Sunday, September 8, 2024

This morning at the Methodist Church which I sometimes attend, I was motivated to ponder the difference between “inspired writing” and “writing that inspires”. My mathematical mindset led me to the “hypothesis” that this is NOT an “if and only if” situation. Perhaps a defining difference could be that “inspired writing” is given that status when a sufficient number of people with generally accepted authority formally approve the designation, while “writing that inspires” simply inspires me or some other reader.

Wednesday, September 4, 2024

A wise person once said, “we should base our actions on our convictions, not on convenience.” This idea floats to the top of my thoughts, when I ponder that much of the testing questions in the U.S. are designed to be convenient to score, rather than any strenuous desire to measure how much critical thinking a student is performing. I believe that a very good education reform would be to “outlaw” multiple choice testing. Multiple choice testing tells what a student “marks” rather than what a student knows. Here are a couple of math questions that can reveal how a student thinks.

(1) Describe 3 different rectangles (i.e. different widths and lengths) that have area 108 square inches but have different perimeters. What rectangle, with area 108 square inches, would have the smallest possible perimeter? What rectangle with the same area would have the largest possible perimeter? (Now isn’t this more thought provoking than – What is the area of a rectangle with width 9 inches and length 12 inches?)

(2) Design 3 different polynomial functions F(x), G(x) and H(x) so that F(1)=G(1)=H(1) =5 and F(2)=G(2)=H(2) = 7, but F(3), G(3), and H(3) have 3 different values.

I concede that such questions would take longer to grade, but how much fun it would be to see how creative students can be when given a chance! for some answers see 8/20/2024

Monday, September 2, 2024

The function F(x) = 4 – 4/x has been in my investigations for several postings. But, I have tweaked this function for some fascinating results. In fact, my initial studies in this area of using this type of recursion to generate sequences, were using: 1 -1/x; 2 – 2/x; and 3 – 3/x. These 3 functions do not generate sequences that converge to limits. These 3 functions are “mysteriously” cyclical.

Saturday, August 31, 2024

In Thursday’s journal comments, I mentioned that the function F(x) = 4 – 4/x had an initial value that would crash the function in the second recursion. If you begin with x=1, then F(1) = 4 – 4/1 = 0, but F(0) = 4 – 4/0 is undefined. If you reverse the rules discussed in previous journal entries and work backwards from x=1, you will get the following sequence: 16/9, 7/4, 12/7, 5/3, 8/5, 3/2, 4/3, 1, giving next an undefined value. This shows that 16/9 will eventually crash the sequence and in doing so will not give a sequence converging to 2. In fact, every number in the sequence listed above will do the same. Furthermore, the given sequence can be continually calculated in reverse to find infinitely many other initial values that will crash the sequence. Also, it appears that this “backwards sequence” converges to 2, but approaches 2 from numbers less than 2.

Thursday, August 29, 2024

Good math questions are those that are not only interesting but are questions that provoke and lead to other questions. Whether questions lead to “useful” discoveries has some importance – but to me, a math nerd, as long as questions lead to fun results, they are of primary importance. For example, let us stay with the function F(x) = 4 – 4/x and use a different initial value. Starting with x=7 and using recurson we get the sequence 7/1, 24/7, 17/6. 44/17, 27/11, 64/17, 37/16, 84/37, 47/21, 104/47, … Does the same rule work for finding the next numerator and denominator? Does the sequence approach the same limit? The answer to both of these questions is yes. This brings us to another interesting question. Does every possible initial value generate the same rules for the sequence and produce a sequence with limit equal to 2? The answer is no! Initial value x=1 crashes F(x) = 4 – 4/x on the second recursive iteration. There are other initial values that can crash the recursion!

Wednesday, August 28, 2024

In my journal entry Monday, I made the observation that the limit of the sequence 3/1, 8/3, 5/2, 12/5, … gave evidence of being 2. Let’s go after this with some creative algebra. The formula of the function that generated this function was F(x) = 4 – 4/x. Since we were using recursion, each output of the function became the next input. And as we observed, as the sequence progressed, each output got closer to 2 and the resulting next output (using this for an input) got even closer to 2. After some length of the sequence, the output and the input were “extremely” close to the same number. Another way to say this is that x and F(x) were nearly the same number. So F(x) = 4 – 4/x becomes x = 4 – 4/x. This gives us an equation that we can solve. If we multiply x = 4 -4/x through by x we get x^2 = 4x -4. Setting this quadratic equal to zero we get x^2 – 4x +4 =0 which factors into (x-2)(x-2) = 0 The solution is 2. In other words, if input x ever equals output F(x), both values must be 2. Or, 2 must be the limit of the sequence.

Monday, August 26, 2024

For the last two entries in my journal, I have been working with the sequence 3/1, 8/3, 5/2, 12/5, 7/3, … It is of interest to examine whether this sequence is approaching a limit. It seems to be approaching two. 7/3 = 2.33333… 11/5 = 2.2 28/13 =2.1538 and somewhat farther out the sequence 43/21 = 2.04761904….

 Sunday, August 25, 2024

Thursday, 8/22/24, I was looking at the sequence generated by the Function F(x)= 4 – 4/x using recursion and an initial value of 3/1. 3/1, 8/3, 5/2, 12/5, 7/3, 16/7, 9/4, 20/9, 11/5, 24/11, 13/6, 28/13, … It turns that each numerator becomes the next fraction’s denominator. This is easy to see in some examples, like going from 7/3 to 16/7. To see the pattern more fully we need to discover the pattern of finding the next numerator. It seems that if we subtract the denominator from the numerator and multiply this difference by 4 we get the next fraction’s numerator. for example, 7 – 3 =4, 4 time 4 is 16. At first, this may seem wrong since the next fraction is 9/4. But 16 – 7 is 9 and 4 times 9 is 36 which will make (by the rule) 36/16 reducing to 9/4. So, we can state the complete rule for the pattern for getting the next fraction in the sequence as follows: First subtract denominator from the numerator and multiply this difference by 4 to get the next numerator. Second, use the current numerator for the next denominator. Finally, reduce the fraction if possible. For example, take the number 52/25 which occurs later in the sequence. 52 – 25 is 27, so the next numerator is 4 times 27 = 108. The next denominator is 52. Finally, 108/52 reduces to 27/13.

Thursday, August 22, 2024

I have been investigating the sequence generated by F(x)= 4 – 4/x. Starting with x=3, F(3)= 8/3. If we run this result through the same function we get F(8/3) = 4 – 4/(8/3) = 5/2. Continuing this procedure F(5/2) = 12/5. Listing the first several terms of this recursion : 3/1, 8/3, 5/2, 12/5, 7/3, 16/7, 9/4, 20/9, … To find a pattern for this recursion we can notice that that for each pair (3/1, 8/3); (5/2. 12/5); (7/3, 16/7) the first numerator = the second denominator.

Tuesday, August 20, 2024

Here are some answers to some questions posed in the journal entry 9/4/2024. The smallest possible perimeter for a rectangle with area 108 square inches is length = sqrt(108) and width sqrt(108). There is no maximum perimeter since you can choose an increasingly smaller width approaching zero and a length = 108/width. For the 3 functions, F(x) = 2x+3 +(x-1)(x-2)(3), G(x) = 2x+3+(x-1)(x-2)(4) and for the H function, H(x) = 2x+3 +(x-1)(x-2)(5).

Wednesday, November 22, 2023

If the formulas: For Positive integers: If N is even then Next N=N/2; If N is odd then Next N=3N-1 For Negative integers: If N is even then Next N=N/2; If N is odd then Next N=3N+1 the results will be one of three possibilities. A. the sequence will hit a power of two and then have a two number cycle, for example 64, 32,16,8,4,2,1,2,1,2,1,2,1, … B. the sequence will hit the number five and then have a 5 number cycle. Example 27, 80, 40, 20,10 5, 14, 7, 20, 10, 5, 14, 7, 20, 10, 5, … etc or C. the sequence will hit the number seventeen and then cycles for 18 steps. Example 136, 68, 34, 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34,17, … etc.

Saturday, November 18, 2023

Ulam Sequence (see previous entry) will work for negative numbers the same way if you consider the absolute values of the negatives. But to get the sequence to look like the positive results, use 3N-1 for odd negatives. For example if you start with -19 and follow the two formulas, the result is:

-19, -58, -29, -88, -44, -22, -11, -34 etc (compare to the list for 29 in previous entry)

Monday, November 13, 2023

I have been working with a recursion formula that is adjusted from Ulam’s sequence.  Ulam’s sequence is a two rule procedure that can start with any Natural Number, N, and proceeds as follows.  If N is an even integer then the next number in the sequence is N/2.  If N is an odd integer then the next number is 3N+1.  For example, if N = 19, the sequence proceeds 19, 58. 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.  If N=128, the sequence is 128, 64, 32, 16, 8, 4, 2, 1.  The conjecture is that every Ulam sequence will get to 1.  This has not been proven and has turned out to be one of the most difficult and most famous unsolved problems in number theory.

I have been looking at changing the odd number rule to 3N – 1.  With this change, starting with 19 we get the sequence 19, 56, 28,14, 7, 20, 10, 5, 14, 7, 20, 10, 5, 14, 7 …  and the sequence continues to cycle with the lowest number ever reached is 7.

For the purpose of this discussion, we call this sequence UlamTwo.  Here are some questions which have interested me.

(1)  Does UlamTwo ever “march off” to infinity?

(2)  is there a maximum length to a UlamTwo cycle?

(3)  What is the largest possible minimum number of a UlamTwo cycle?

 

Saturday, October 14, 2023

I have constructed (discovered) an interesting function, which perhaps could be considered “pathological”, that maps the real numbers onto the Whole Numbers W={0.1.2.3.4.5.6….}.  Obviously, this function cannot be one-to-one.  It has a property similar to the “strongly Darboux ” property in that it maps every interval of the real numbers onto the Whole Numbers.  The function F(x) is described as follows:

If x is a positive rational number N+A/B where N is a whole number and A/B is a proper fraction where A/B is in lowest terms, then F(x) = A+B.  For example F(25+9/37) =46 and F(4) = F(4+0/1) =1.

If x is a negative rational number -N – A/B where N is a whole number and A/B is in lowest terms then F(x)=A+B.  For example F(-8 – 4/13) = 17.

If x is an irrational number and d1d2d3d4d5d6d7d8 … is the decimal part of x then F(x) is defined as follows:

If S = the sum(d2 – d4 +d6 -d8 + … )  converges then F(x) =Absolute value of S.  Otherwise F(x) = 0

For example if x = .361941105090n0n0n0n0n0n0n…. where the odd placed decimals are the digits of pi then F(x) = Abs(6-9+1-0+0-0+0…) =2

…………………………………………..

Sunday, April 23, 2023

If you continue the patterns, each sequence of fractions will converge.  We have described the recursion patterns of some of these in previous posts.

(A)  2/3; 7/5; 17/12; 41/29; 99/70 … 

(B)  2/1; 5/3; 7/4; 19/11; 26/15; 71/41 ”’ 

(C)  5/2; 13/7; 41/20; 121/61; 365/182 …

(D)  3/1; 2/1; 7/3; 11/5; 9/4/ 29/13; 47/21; 38/17 …

(E)  7/2; 19/9; 73/28; 241/101; 847/342; 2899/1189 … 

There are quite a few interesting patterns showing in these lists (if you are a math nerd like me)  I may write about some of the patterns in subsequent posts.  But for now, I will tell you that sequence (A) converges to sqrt(2).  Sequence (B) converges to sqrt(3), (C) to sqrt(4), (D) to sqrt(5) and (E) to sqrt(6)

Friday, April 21, 2023

The sequence mentioned yesterday: 2/1; 5/3; 7/4; 19/11; etc. can be extended by noticing the pattern.  The next denominator is always the sum of the numerator and denominator for the fraction you are looking at.  For example the next denominator after 19/11 will be 19+11=30.  The next numerator can be calculated by adding the numerator and 3 times the denominator which gives  19+3*11=52.  So the next fraction is 52/30 or 26/15.  To get the next fraction after 265/153 … denominator is 265+153=418 and the next numerator is 265+3*153=724 so the fraction is 724/418 or 362/209 giving 1.732057416 accurate to 5 decimal places to the sqrt(3).

The pattern of the formulas seems to be hinted at if we look at the sequence approaching the sqrt(2) examined in the April 9, 2023 post: 3/2; 7/5; 17/12; 41/29; etc where each next numerator is calculated by numerator plus 2 times the denominator.  Can you guess my prediction for the sequence of fractions converging to the sqrt(4)

Thursday, April 20, 2023

Here is another sequence of fractions which has caught my interest:  2/1; 5/3; 7/4; 19/11; 26/15; 71/41; 97/56; 265/153 …  The pattern here is more complicated than the pattern for the list in previous posts.  This list seems to be approaching the square root of 3.   for example 71/41 = 1.7317; 97/56 = 1.7321428 and 265/153 = 1.732026144 compared to sqrt(3) = 1.732050807…..  I am fairly confident from my investigations (doodling on napkins) that there is a patterned sequence to be found that will converge to any positive integer’s square root.  Also, there is a pattern to the patterns!

Tuesday, April 11, 2023

Where does the list: 3/2, 7/5, 17/12, 41/29 … (see previous two posts) come from?

This list of fractions come from the infinite continued fraction started by the following sequence of expressions:  First expression 1 + 1/2 =3/2;  2nd expression 1 + 1/(2+1/2) = 7/5 and keep extending the denominators the 3rd expression is 1 + 1/(2+1/(2+1/2)) = 17/12 and we will do one more (this is easier with pencil and paper because of the awkward notation) 4th expression 1+(2+1/(2+1/(2+1/2))) =41/29. If you continue this out to infinitely many terms some slick but elegant algebra will confirm that the limit of this continued fraction is the square root of two.

Sunday, April 9, 2023

Referring back to the previous post, (4/7/2023) I wonder who was able to predict the next number in the sequence,  3/2, 7/5, 17/12, 41/29, 99/70, 239/ 169, ??      The next number is 577/408 You can extend the list by using a variation of “double recursion”.  But before showing that let us look at the decimal values of the fractions.  1.5; 1.4; 1.416666666; 1.413793;  1.41428571; 1.414201183 and 577/408 is equal to 1.41421568627….  If you are a number fan, you may see that this sequence seems to be approaching the square root of 2 = 1.41421356237…..  which is. in fact, the situation.  I will get to some math logic showing this in a future post.  But back to the “double recursion”  ….

To find the next fraction in the sequence from any fraction in the list: add the numerator and denominator to get the next denominator.  To get the next numerator, add the numerator to twice the denominator.  For example the fraction after 17/12 will have a denominator of 17+12 = 29 and the numerator will be 2 times 12 plus 17 = 24+17 =41.  So the next fraction is 41/29.  More about this in the next post.

Friday, April 7, 2023

Surprise!  I am still alive and well after an extended absence from my website, mathonanapkin.com

Perhaps I can get back to the habit of making regular posts to Math Teacher’s Journal.  I was a substitute teacher this past week and during a break of limited supervision (the students were very well behaved!) I managed to come up with a bit of math worthy of scribbling on a napkin.  Here it will be presented as a “puzzle teaser” with a minimum of hints.  In subsequent posts, more hints and background sources and the answer will be posted.  Or at least that is my intention.  Consider the sequence of fractions shown below:

3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ??

Look for a pattern and make a guess at the next fraction.  If no one gets it then we will award a “fabulous prize” to the closest wrong answer.  Or give a shout out to the most original analysis of the question using “Whacko Logic” … my fall back plan on all problems that baffle me.

Sunday, October 3, 2021  

It turns out that F(x) = 2 – 2/x is a “4 cycle” function.  ( see Sept 18 journal entry)  For example if we start with the number 7: F(7)=12/7; F(12/7)=5/6; F(5/6)=-2/5; and F(-2/5)=7  Note that we have to restrict the domain to avoid division by zero; our starting input can not be x=0, x=1 or x=1/2  It turns out that the other function mention in the journal entry on September 15,  F(x) = 3 – 3/x, is a “cycle 6” function.  I will say more about that and perhaps a general observation in a future journal explanation. 

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Wednesday, September 15, 2021  Between December 8 and December 19 of 2017 (see journal entries for those days) I worked hard at getting a “cycle three” function.  Lately while working on another problem, I stumbled upon an incredibly simple cycle three function!  The function is F(x) = 1 – 1/x  this works for any real number except zero or one.  example:  let x=5,  F(5) =1- 1/5 = 4/5;  next F(4/5) =1- 1/(4/5) = 1 – 5/4 = -1/4;  Finally F(-1/4) = 1- 1/(-1/4) = 5

These 3 results cycle back to the original input F(5) = 4/5; F(4/5) = -1/4 and F(-1/4) = 5 an infinite loop! 

Similarly observe for F(x) = 1 – 1/x ;  F(12) = 11/12; F(11/12) = -1/11 and F(-1/11) =12 and so on for any input other than zero or one, which crash for division by zero.  This motivated me to look at the following functions:

F(x) = 2 – 2/x         F(x) = 3 – 3/x   and F(x) = 4 – 4/x    You can try these for yourself.  I will comment in a future post. 

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Friday, July 23, 2021   I am reading Chernow’s biography of U.S. Grant and it is excellent!  I was surprised to learn that President/General Grant was proficient enough in mathematics to have been offered a teaching job at West Point and also several colleges.

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Sunday, May 9, 2021 (Mother’s Day)

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Friday, May 7, 2021

I chanced upon a charming, old black and white movie starring Joel McCrea as a post civil war preacher in a small town.  To my surprise I recognized a young Amanda Blake (Kitty in her Gunsmoke days) playing the part of the town’s school teacher.  It’s fun for me to recognize actors in older films before they reach their high level of stardom.

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Monday, May 3, 2021

It has been over a year since i got to be a substitute teacher at Madison Academic Magnet H.S.  Today I got to work with the French classes.  Below is the French A.P. class

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March 11, 2021

I am reading a well done and an interesting book on Math History, The Mathematical Century, The 30 Greatest Problems of the Last Hundred Years.  The author is Piergiorgio Pdifreddi and there s a foreword by Freeman Dyson.

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February, 27, 2021  I read an wonderful article by Kimberle Williams Crenshaw in the January/February issue of The New Republic.  Her writing brought into clear focus the extent of the denial possessed by so many of my white brothers and sisters.  I sent Dr. Crenshaw an email expressing my appreciation.  She responded with a kind thank you for my email. 

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Sunday, January 31, 2021  To follow up on my journal entries of this January, I will comment of the other prime factor of 2021 which is 47.  

the reciprocal of 47, 1/47, has a 46 digit block that repeats. 

1/47 = .0212765957446808510638297872340425531914893617….

This pattern has 4 occurrences of each digit except 0,3,6,9 which occur five times each.  This makes the mean average of the block = 4.5 and thus makes the decimal fraction of 1/47 a “Mean Digit Decimal”.

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Saturday, January 30, 2021  More on “Mean Digital Decimals”  The prime number 43 is a factor of 2021 (our current year)  The reciprocal 1/43 has a 21 digit block that repeats 1/43 = .023255813953488372093… This block contains 2 zeros, 1 one, 3 twos, 5 threes, 1 four, 3 fives, no sixes, 1 seven, 3 eights, and 2 nines.  The mean average of these 21 digits is 90/21 which is about 4.286.  This is not a “Mean Digit Decimal”. 

If all proper fractions with denominator 43 are examined 21 out of the 42 have this same digital pattern though starting differently, eg 15/43 = .348837209302325581395… The other 21 proper fractions have the same repeating block of 21 digits as the fraction 2/43 = .046511627906976744186…   This block contains 2 zeros, 3 ones, 1 two, no threes, 3 fours, 1 five, 4 sixes, 3 sevens, 1 eight and 2 nines.  The mean average of the digits in this pattern is 99/21 which is about 4.714  If these two blocks are considered together the average of the combined two blocks is 189/42 = 4.5  So I perhaps we could call the set of proper fractions with denominator 43 “kind of, sort of” Mean Digit Decimals.  We can also notice that the combined two blocks of digits contain four occurrences of all the digits except  3 and 6 which show up five times each, which also average 4.5. 

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Thursday, January 28, 2021  “Mean Digital Decimals”  Although decimal fractions that contain each of the ten digits an equal number of occurrences are plentiful, they are more easily artificially constructed than found.   For example N=.01234567890123456789… is obviously such a decimal fraction.  Such decimal fractions are defined as simply normal “Intuitively, a number being simply normal means that no digit occurs more frequently than any other.”  For purposes of my own investigation, I define a term “Mean Digital Decimal” If we average the ten digits we get (0+1+2+3+4+5+6+7+8+9)/10 = 4.5 So the arithmetic mean of the ten digits is 4.5 It is easy to find decimal fractions that have repeating blocks of digits that have an arithmetic mean of 4.5.  My favorite is 1/7 = .142857142857… with mean 27/6 = 4.5  A simpler example is 1/11 = .09090909… with mean = 4.5  So both 1/7 and 1/11 are “Mean Digital Decimals.  If we get the product of 1/7 and 1/11 we get some spectacularly interesting patterns all of which are “Mean Digital Decimals”  For example 1/77 = .012987012987  and 76/77 = .987012987012…  Not that 3,4,5,6 are left out of these patterns and that the first 3 digits and last 3 digits are sequential.  It turns out all proper fractions with denominator 77 are “MDD”  Observe 2/77 = .025974025974… or  21/77 = .27272727…  Or the fraction 33/77 =.428571428571 …  These patterns and patterns of other fractions fascinate me.  In a future entries  I want to go back to my 1/1/2021 post and look at the factors of 2021 which are 43 and 47 and see what properties the patterns of their reciprocals possess.

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Friday, January 1, 2021  This year 2021 CE is not a prime number.  But it almost is.  2021 factors into 43 times 47.  The reciprocals of these two factors have wild decimal patterns.  1/43 =0.023255813954388372093… this 21 digit pattern repeats.  1/47 has a 46 digit pattern that repeats.  This causes me to wonder what the digit patter for 1/2021 would be?

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Monday, December 21, 2020  One of my dearest friends and fellow Barbershop Quartet singer has passed away.  Barry Geise was a Navy veteran, an educator and one of the finest Christian Gentleman that I have ever known.  Barry “the bass”  R.I.P.

Monday, December 14, 2020  I can now adjust the numbers given in my December 6 entry.  President Elect Biden won the popular vote by over 7 million votes, which averages out to more than 140,000 votes per state.

Sunday, December 6, 2020   Whatever your politics, this 2020 Presidential Election shows the mathematical foolishness of the Electoral college System.  President Elect Joe Biden won by over 6 million votes in the popular vote count.  This averages out to more than 120,000 votes per state.  Yet if you switch less than a total 25,000 votes in Wisconsin, Arizona and Georgia it is likely President Trump would have been re-elected and still behind in the popular vote count by over six million votes.  Other nations around the world continue to scratch their heads at the apparent vulnerability to political mischief available in our system.

Saturday, July 11, 2020  My entry for July 6 was in error in that the mapping was not one-to-one.  But I think I have a fix.  Using a base 9 notation for the Natural Numbers and arranging all finite subsets, as before, in order from smallest integer to largest then the mapping can be defined as follows.  Put the smallest integer after a decimal point then place a digit 9 at the end of the first integer.  Continue with the remaining integers in the subset, each followed by the digit 9 and terminating the created decimal after the last 9.  Example {5,73,204, 8888} maps to the base ten decimal 0.59739204988889.  This will map all finite subsets to a set of rational numbers between zero and one.  This is also a one-to-one mapping and is sufficient to show that that the set of all finite subsets of the Natural Numbers 1.2.3.4.5…. is not more numerous than the rational numbers and is therefore countable.

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Monday, July 6, 2020  If all finite subsets of the Natural Numbers: 1,2,3, … are listed in order from smallest to largest then a mapping from the subsets into the rational numbers can be done as follows.  Using a base ten format the subset {1,5,11,555} can map to the decimal 0.1050110555.  The set with natural numbers {A,B,C} would be mapped to the decimal 0.A0B0C   Such a mapping would be one-to-one and no more numerous that the rational numbers between zero and one, which is a countably infinite set.

Monday, July 1, 2019  I have been thinking about trying to map a countably infinite set onto the set of all its finite subsets.  For example it seems to me that the natural numbers 1,2,3,4,5,… can be mapped with a one-to-one correspondence to all the finite subsets of the natural numbers.

Thursday, August 29, 2019  I have taken somewhat a vacation from typing any entries into this Journal.  I have recently read “A Mathematician’s Lament” by Paul Lockhart.  I can recommend this short book to any mathematics teacher.

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July 20, 2019,  I have been reading some biographical material about George Boole.  He was quite a remarkable and interesting man.  His work was considerably ahead of his time!

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Monday, June 3, 2019  While in Mobile, AL on this day, I went to the Crescent Theater on Dauphin Street and watched the movie “Rocketman”.  I thought the actor playing Elton John was superb.

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June 1, 2019  At a reunion with some old church friends, I experienced that some tried to persuade me that Christianity was growing faster than Islam.  My intuition was that this notion is bogus, and it appears to be.  With any research using several reputable sources the data is overwhelming – though not unanimous that Islam is growing much faster and at current rates will overtake Christianity.  By birth rate alone, Islam has the advantage.  I remain hopeful that free thinkers and reason will eventually win out over all religions,  But sometimes I get discouraged!

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Monday, April 29, 2019   I was a substitute teacher today with the French classes.  The students were patient with the retired math guy.  I did share with them the names of some of the great French Mathematicians.

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Thursday, April 11, 2019  It’s been a long gap between entries in this journal!  My brain has been somewhat on a math vacation, although I have returned to my interest in examining real valued functions that map every interval onto the real numbers.

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Monday, March 18, 2019  Woodin’s Axiom?

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Friday, March 15, 2019  A shopper goes into a clothing store where the sign reads, “Shirts 10 percent off”.  The shopper finds three shirts he likes and takes them to the clerk where the shopper is stunned to see the clerk mark off 30 percent!  Which of the following should the shopper do? ( A.)  Say nothing to correct the sales clerk’s mistake and walk out with this huge bargain! (B.)  Patiently explain to the clerk that he has made an error.  Or (C.) Go get seven more shirts! 

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Wednesday, March 13, 2019  How do you place 8 sugar cubes in three cups of coffee so that each cup has an odd number of sugar cubes?  Solution:  Place one cube in the first cup, one cube in the second cup and six cubes in the third cup.  Everyone should know that putting six cubes in a cup of coffee is odd!

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Sunday, March 10, 2019  Einstein, Emily Noether, Paul Halmos, and Gorge David Birkhoff are some notable mathematicians or physicists who were born in March.  Along with Keith Devlin, NPR’s math guy.

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Friday, March 8, 2019  The book “Euler’s Gem” shows how to use the formula V-E+F=2 to prove that there are exactly 5 regular polyhedrons.

These are the: tetrahedron, cube, octahedron, dodecahedron and icosahedron.

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Wednesday, March 6, 2019  I am reading a book entitled “Euler’s Gem” by David S. Richeson.  It is the history of the formula for the edges, faces and vertices of polyhedrons.  The formula V-E+F=2 first proved by Euler.

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Monday, March 4, 2019  Daylight Savings time is coming between Saturday night and Sunday.  I read where Maine is considering to getting in the “Atlantic Time Zone” which is the next zone east of Eastern.  I seem to remember there was a time zone some where that shifted their clocks just 30 minutes.  Newfoundland in Canada I believe is three and a half hours off of UTC, (Coordinated Universal Time)

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Saturday, March 2, 2019 pi Day is coming soon.  March 14, or 3/14  We used to have a contest to see who could recite the most digits of the decimal representation

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Thursday, February 28  February is the perfect month in more than one way.  28 is the first perfect number past 6.  It’s the only month where there is exactly four weeks (most years)  Valentine’s day is on the 14th day exactly half way thru.  The heart can be drawn with parts of two Archimedean spirals.

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Wednesday, February 27, 2017

Monday, February 25, 2019   OK!  I admit mathematicians can be described as peculiar people.  How many outside of mathematicians get excited about decimal digit patterns?  But perhaps no more that anyone beset by a huge amount of curiosity.  There is some much in this universe and not enough time to explore and enjoy it.  The mathematical universe, in a sense, is the biggest of all universes.  But there are so many (even to my little mind) interesting fields of study.  I believe that there are no boring subjects only shallow, easily bored people.  I have a scientist friend who loves to research lizards and the effect different environments cause.  A physicist acquaintance has been involved trying to conduct experiments to get a few more decimals in the accuracy of the gravitational constant.  I have been amazed at the skill of a Cherokee rock carver… at my daughter’s skill at learning new software… a Bible scholar who has learned to read Greek, Hebrew and Coptic to the extent that he can translate non canonical books… A high school friend who like to crawl through Tennessee caves

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Saturday, February 23, 2019  Back to decimal digit patterns, see Friday February 15 entry.  I made the mistake of trying to count the non trivial 8 digit repeating patterns and filled up my napkin and got a headache.  For example:  .4311431143114311 is an 8 digit pattern but is more simply a 4 digit pattern.  But 5522883355228833 can’t be reduced to a smaller pattern. I can’t seem to get a handle on all the ways 8 digit patterns can reduce. 

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Thursday, February 21, 2019  The fact that math is so useful in science, technology, measurements, data analysis, engineering and so many other fields is both a blessing and a curse.  Sometimes its usefulness encourages the education institutions to make the study of math solely for utilitarian purposes.  Mathematics is also and perhaps primarily an art form with supreme abstract beauty.  I heard a renown mathematician once say, “Mathematicians do not study math because it is useful, but rather they study it because it is fun.”  Often the initial study is for the abstract logical beauty, then much later the tremendous applications are discovered.  For example: binary arithmetic, Boolean algebra, and even Hamiltonion quaternions.

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Tuesday, February 19, 2019  I visited with my tax man – CPA today and my compliments to his suggestions on quarterly payments.  I was very close to breaking even… which is what should be a taxpayer’s plan.

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Sunday, February 17, 2019  It’s been a rainy February.  I suspect the computers at N.O.A.A. and other meteorological labs have been crunching all kinds of numbers about rainfall patterns and predictions and flood stages and snow accumulations.

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Friday, February 15, 2019  I have done a bit more thinking about repeating digit patterns and which are irreducible i.e. nontrivial.  For example the six digit pattern for 1/7 = .142857142857142857 is not reducible.  An example of a reducible six digit pattern is 17/111 = .153153153153…which is more simply a 3 digit pattern.  So trying to count only the irreducible repeating patterns in six digit blocks or 12 digit blocks or a highly factorable composite number is a bit tricky.

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Wednesday, February 13, 2019  After some more scribbling, I am thinking that there are 9900 different “non trivial” 4 digit patterns. (see the previous discussion of February 10)  If we start with ten thousand possible arrangements of 4 digits we can toss out 100 trivial patterns.  There are ten one digit patterns from 0000 to 9999.  There are 90 two digit patterns of the form ABAB.  There are 45 ways to choose two digits from 0123456789, an example being 5757.  But we need to consider that reversing each block pattern such as 7575 we arrive at 90 different 4 digit blocks of this type.  This gives us the 100 trivial examples.

Sunday, February 10, 2019  Repeating decimal patterns continue to interest me.  Some recent doodling on a napkin concerned an investigation of 4 digit patterns.  An example of a 4 digit pattern is the decimal generated by the fraction 36/101 = 0.35643564356435643564…..  The questioned that came to mind was, “How many different 4 digit patterns are there?”  The simple answer to the question is 10 thousand since that is how many numbers ther are between 0000 and 9999.  But this list contains many “trivial” patterns such as 4444 which is more simply a one digit pattern.  Similarly a 4 digit pattern like 2/11 = .1818181818181818… is more simply described as a two digit pattern.  The revised question (a bit more difficult) is, “How many non-trivial 4 digit patterns are there?”

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Friday, February 8, 2019  Science and reason are not the perfect tools for understanding the universe and human relations but they are the best tools we have.  Sometimes when we push science as far as current knowledge allows, we make a rational and reasonable solution to use some other method … but even then we are being rationalists.  Einstein said, “One thing I have learned in a long life: that all our science, measured against reality is primitive and childlike – and yet it is the most precious thing we have.”

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Wednesday, February 6, 2019 In a discussion in the church choir room before the start of rehearsal, some of the singers seemed to learn for the first time that I was a retired high school math teacher.  We discussed the connection between math and music.  I shared my opinion that band and choral students were often my best math students.

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Monday, February 4, 2019  During 17 years of teaching first year algebra at North Side Junior High School, I came to the belief that two skills were very important and indeed were litmus tests for predicting a students success in algebra.  One was a skill at adding and subtracting negative numbers.  The other was the ability to understand and perform addition and subtraction with fractions with different denominators.  I planned and developed some drills for improving these two skills, with some (though not total) success.  I remember with a smile what happened during a day of practice and drill with negative numbers.  After several repetitions of a drill and during another try a whisper floated to the front of the classroom.  A girl whispered, “I don’t understand this negative shit!”  She had been producing supportive evidence of her assertion.

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Friday, February 1, 2019  February was a month of birthdays for some significant mathematicians.  Galileo and Copernicus, Dirichlet, G.H. Hardy and Alfred North Whitehead all were born in February.

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P.C. Scott, Feb 4. 2019

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Tuesday, January 29, 2019,  I am half way through my third book of the year.  It is “The Innocent Man” by John Grisham.  I am also reading a collection of short stories by Lawrence Block, “Resume Speed”.  

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Thursday, January 24, 2019  I’ve had a slow start reading books this year.  I read over 100 in 2017 and over 70 last year.  I have been reading my favorite magazines, though perhaps I am taking one too many. 

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Monday, January 21, 2019  The total lunar eclipse was a subtle but wondrous sight!  I was moved by the beauty of the event.  The math and science behind the effect is spectacularly beautiful in its own abstract way.

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Friday, January 18, 2019  I was a substitute teacher today at Madison Academic High School … for Spanish classes.  My knowledge at this subject is a number than easily rounds to zero.  It gave me thoughts on the inadequacy of the long term subs for the striking teachers in California.  But it was good to be around the young people again.  They were kind to this senior citizen, retired mathematician.

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Tuesday, January 15, 2019  I have been scribbling on napkins trying to extend answers to the “2019 puzzle”  I have all the numbers from 1 to 40 except 37.  The most difficult for me was 31 which I have shown below:

.2 ^(-(0!+1)) +(sqrt(9))! =31  I can get 37 if I change the order of 2,0,1,9 to 9,2,0,1  as shown below:

(sqrt(9))!^2 +0+1 = 37

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Friday, January 11, 2019  Yesterday my wife Cathy and I had lunch at Baudo’s Restaurant.  The Baudo family opened the very first Pizza take-out in the mid sixties and later became Jackson Tennessee’s finest Italian Restaurant.  It will be closing after 50* years on January 19.  I will be sad to see it go.  We went again with Cathy’s daughter and granddaughter tonite but there was at least an hour wait so we gave up to make room for others.

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Tuesday, January 8, 2019  I taught high school math for 41 years.  The most students that I taught in any one year was probably 175.  That year (late 1970’s) I taught 5 classes of algebra one, each with 35 students.  From 1973-1974 until 1989-1990 I don’t think that I ever had fewer than 125 students.  For these years at North Side Junior High School, it is a safe minimum estimate to say that I taught 130 times 17 = 2200 plus students.For the rest of my high school teaching career (12 years at JCM High School, one year in Hopkinsville KY and 11 at Madison Academic Magnet High School it is a safe estimate to say I averaged teaching 110 students per year.  This give a total of 24 times 110 = 2600 plus students.  If I count the number I taught at two years at Auburn University and 2 years moonlighting at the local community college, then a total of 5000 students is a reasonable guess.

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Sunday, January 6, 2019  Today is Epiphany … “On the 12th day of Christmas”.  Besides the religious and historical significance of Epiphany or “Twelfth Night”, is the song that has motivated some seasonal (perhaps silly) math problems.  For example.  What is the total number of gifts the “true love” gave?  This question depends on whether a partridge, etc, is given again on subsequent days.  I will not celebrate the problem by giving answers here.  There is plenty of discussion available on the internet.

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Thursday 01/03/2019  Did Monday’s discussion fascinate you?  One of my friends intimated that the info made the new year prematurely tiresome. Oh Well!

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Monday 12/31/2018  Last day of the year!  2019 is an interesting number, as are all numbers to a math nerd like myself.  2019 is not prime since it is the product of 3 and 673 which are primes.  The reciprocal of 2019 is a repeating decimal.  There are 224 digits in the block that repeats.  The decimal for 1/673 also has 224 digits and begins and ends as shown:

1/673 = 0014858841010411887072808………573551263

If you multiply this decimal by 1/3=.3333333333333…. it is “obvious” that the result 1/2019 will also have the 224 digit repeating block.  Knowledge of this will not necessarily make you a wealthy person, but will provide a source of somnambulant conversation.

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Thursday, 12/27/2018  There have been some who have suggested switching to a 13 month calendar, each with 28 days … letting the 365th day be called “leap day” or “New Year’s Day”.  Since 13 times 28 plus 1 = 365 there would be obvious advantages to this scheme.  Of course there also would be disadvantages.

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Tuesday, 12/25/2018  Merry Christmas!  The closest thing to a book on math that I received was a book on Norse mythology.  Several days of the week are named after Norse Gods. Tuesday, Wednesday, Thursday, Friday.

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Sunday, December 23, 2018 Christmas on Tuesday in 2018 means that Christmas will be on Wednesday in 2019, since 365 divided by 7 has a remainder of one.  Thursday will be skipped in 2020 because of the extra day in February. The sequence for the next few years, using 1 for Sunday is as follows beginning with 2019: 4,6,7,1,2,4,5,6,7,1

Friday, December 21, 2018  I was invited to the Madison Academic H.S. faculty Christmas luncheon.  It was good to visit with old friends among the faculty and staff.

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Sunday, December 16, 2018  The choir at First Methodist Church in Jackson, TN did their Christmas special music this morning.  We had a 25 piece orchestra accompanying us.  I managed to sing with the rest of the bass singers without making too many mistakes. 

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Saturday, December 15, 2018  We are back from a one day trip to Mobile, AL which is a favorite place for a short get-away.  While there we watched the movie, “The Green Book” which I highly recommend.

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Friday, December 14, 2018  The first day of winter and the winter solstice  is coming up very soon.  The full moon in December is called the “cold moon” or the “long night moon”.  This year the solstice and the cold moon are less than 24 hours apart which is relatively rare.

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Wednesday, December 12, 2018  I remember being taught that it takes one calorie to raise the temperature of one gram of water one Celsius degree.  I have tried to find more information about this (and am still searching) but am curiously skeptical.  Can we assume that this is a linear relation?  For example:  does it take the same energy to raise a gram of water from 5 to 6 degrees Celsius as it does to raise the same gram from 55 to 56 degrees Celsius?

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Monday, December 10, 2018 One of our favorite getaway places is Mobile, AL.  The trip to Mobile is a straight route south on U.S. Highway 45.  Last Saturday on the return trip I watched the trip set and did some mental math.  I noticed that when we passed 8 miles on the mile markers my trip set showed 8.2 miles.  With a bit of thought for this ratio of 40/41, I can figure the actual miles traveled on a long trip.  For example, if my trip set shows a distance of 410 miles I can subtract 1/41 of this distance and get the more accurate figure of 400 miles.  Of course a person should not become so fascinated by the mathematics of travel that driving becomes a hazardous menace to others!

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Friday, December 7, 2018  On the back of a sheet listing our church choir schedule I worked out the probability of arranging the letters AABBCDEF so that no two identical letters are adjacent.  I came up with 4/7.  This extends the sequence of probabilities (see 12/2/2018 journal note) to:

1/3; 2/5; 7/15; 32/63; 4/7 

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Sunday, December 2, 2018  After working with extending the discussions on 11/24 and 11/28, I have calculated results for arranging the letters AABBCD and AABBCDE.  Previous results for counting ways to list the letters so that no two adjacent letters are the same are:  8 out of 24 for AABB; 48 out of 120 for AABBC.

For AABBCD I have counted 336 out of 720 ways so that no two adjacent letters are the same. and 2560 out of 5040 ways to arrange AABBCDE so that no two adjacent letters are the same.

If you reduce these ratios you get a sequence  1/3; 2/5; 7/15; and 32/63.  This has a ghostlike appearance of a pattern.  It seems that you have to get to AABBCDE before you break 50 percent.  It was easier for me to get these counts with a “back door” approach.  That is to say, count the ways that identical letters could touch and subtract this from the total number of possible arrangements.

Wednesday, November 28, 2018  To follow up on the November 24 journal discussion, I will add the letter C to the two A’s and 2 B’s;  There are 120 ways to place 5 objects in a linear order ( 5 factorial)  Let’s consider how many of these 120 ways have the two A’s adjacent:

AAXXX; XAAXX; XXAAX; and XXXAA these are the 4 possible patterns using X to represent the other 3 letters (the two B’s and the letter C) For each of these four patterns there are 6 ways the other 3 letters can be permuted. Note we are considering “different’ ways rather than “different appearing” ways.   (You can demonstrate this by using the numbering plan we used in the 11/24 journal entry and assigning the number 5 for the letter C.) This gives us 4 times 6 or 24 ways the two letter A’s can be adjacent.  Then if you permute the two letter A’s that gives you 2 times 24 or 48 ways that two A’s can be adjacent. Similar logic gives you 48 ways the two B’s can be adjacent.

Now how many ways can the two A’s and the two B’s be adjacent in the same list?  Let’s look at the possible patterns:   AABBC; AACBB; CAABB; BBAAC; BBCAA; and CBBAA  Each of these six patterns can be permuted 4 ways which gives us 24 ways.  These 24 ways would be included in each of the preceding lists of 48 and we shouldn’t count them twice.  It follows that the number of ways that a list could have two A’s and/or two B’s adjacent would be 48+48-24 = 72 ways.  This leaves 120-72=48 ways to arrange the five letters so that no two of the same letters are adjacent. 

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Saturday, November 24, 2018  The last time I was substituting for a teacher at Madison Academic H.S. a math problem came to my attention that has held my interest.  Since it initially baffled me, I simplified it to a similar problem to help me get started.  The simpler problem is as follows:

If you have 2 A’s and 2 B’s how many different ways can you write them in a single row.  First I looked at how many different appearing rows:  AABB; BBAA; ABBA; BAAB; ABAB; BABA  there are six of these.  We note in passing that there are 2 of these ways where neither the A’s nor the B’s are touching.  But if we consider how many “different”  ways instead of how many “different appearing” ways there are 24 ways to arrange 4 letters (4 factorial)  You would notice this if you used 2 different colors for the A’s and B’s.  I’ll illustrate this below by using 1 and 2 for A and 3 and 4 for B.

1234; 1243; 1324; 1342; 1423; 1432; 2134; 2143; 2314; 2341; 2413; 2431

3124; 3142; 3214; 3241; 3412; 3421; 4123; 4132; 4213; 4231; 4312; 4321

Note that 8 out of the 24 ways have neither the 1 and 2 touching nor the 3 and 4 touching.  So it seems that either approach to this puzzle gives a ratio of 2/6 = 8/24 = 1/3  So that if you write this at random the chances of having neither A’s or B’s touching is 1 out of 3.

Now the follow up to this question is what happens when you add a letter C?

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Monday, November 19, 2018  I just finished reading “Street Lawyer” by John Grisham.  It reminded me of the plight of the homeless in America’s cities and how low rent situations are taken away from them by business decisions to make huge profits.  Not all the homeless are there because of their own faulty choices.

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Friday, November 16, 2018  I was a substitute for a European History class today at Madison Academic Magnet High School.  At my age I am a bit of History.  I was alive when the last Civil War veteran died.  I was also around to remember when the last veterans from the Spanish American War and World War I passed away.  Soon we will lose the last living veteran of W.W. II.  Below is a photo taken today in class.

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Wednesday, November 14, 2018  A November snow is a rarity in West Tennessee, but we got one today!

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Wednesday, November 7, 2018  Why is the 11th month named November?  The math is wrong.  It took me years to realize the misnomer.  The calendar is a storehouse of historical oddities.

November was originally the 9th month of early versions of the Roman calendar and consisted of 30 days. It became the 11th month of the year with a length of 29 days when the months of January and February were added. During the Julian calendar reform, a day was added to November making it 30 days long again.

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Sunday, November 4, 2018  It has been almost exactly 19 years since my Mother Mae passed away.  She was always proud of my work as a public school mathematics teacher.

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Wednesday, October 31, 2018  Halloween!  The weather was not ideal but the heavy rains and thunderstorms held off.  Never-the-less we had not trick or treat visitors.  We seldom do have any costumed kids on Halloween at our Jack’s Creek home.

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Wednesday, October 24, 2018  I picked up again my short book on Galois Theory today.  It is slow reading but I still would like to get a better grasp of the subject.

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Friday, October 19, 2018  I spent some time today at Gulf Shores, AL.  The weather was grand and the sight and sound of the waves rolling in to the white sand was soothing.  There were no visible signs of damage from Hurrican Michael.

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Friday, October 12, 2018  We drove south to Pontotoc,MS to visit the”Bodock Festival”  One of my favorite artists was there displaying some of his oil paintings. I purchased a couple of Roy Mobley’s nature scenes.

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Thursday, October 11, 2018  I drove north to Union City, TN to have lunch with my daughter Carol.  The trip throught Kenton, TN didn’t present any sightings of white squirrels.  West Tennessee has a good supply of white squirrels … I see them in Jackson fairly often.  Perhaps Kenton doesn’t deserve any special recognition for the critters.

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Monday, October 8, 2018  Columbus Day,  This seems to be a holiday that generates little excitement.  I think holidays on Fridays are more fun.  Of course my perspective has changed since I retired from full time teaching.

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Friday, October 5, 2018  The summer heat is lingering and the grass still is green and growing.

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Monday, October 1, 2018  I subbed for a Spanish I and Spanish II class today.  The teacher left a good plan and the students were very polite and cooperative with this retired math teacher.

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Saturday, September 29, 2018  It was an enjoyable week getting to be he substitute teacher for the A.P. Calculus classes at Madison Academic High School.  We derived the formulas for the derivatives of arcsecant and arccosecant and also the exponential and natural log functions.  I pondered once again which seems easier … to derive the natural log derivative first then use that to get the derivative of the exponential- or vice versa.  In either case it seems that one must ultimately go back to the least upper bound property of the real numbers.

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Wednesday, September 26, 2018  There is news that perhaps a break through on proving the Riemann Hypothesis  is happening.  It is the greatest of the unsolved problems of mathematics.

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Monday, September 24, 2018  This starts a week of being a substitute teacher at Madison Academic Magnet H.S. for the Calculus – Statistics classes.  I have had much less experience trying to teach stat but I am at home with probability which is what the students are working with.

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Friday, September 21, 2018  An anonymous quote: “When arguing with a fool, make sure that he is not doing the same thing.”

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Tuesday, September 18, 2018  I managed to get in touch with artist Roy Mobley.  I have several of his water color paintings and enjoy his work with nature scenes.  I found out where and when he is next showing some of his work.  October 12, 13 in Pontotoc, MS

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Friday, September 14, 2018 Tonight we attended a concert of A.J. Croce, son of the late Jim Croce.  A.J. is a fabulous piano player and the most talented musician that I have ever experienced in a live setting.

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Tuesday, September 11, 2018  This will always be a special date in U.S. History

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Saturday, September 8, 2018  September in West Tennessee is a grand time of year.  The leaves of the trees start a very subtle change.  The nights are beginning to have “porch sitting” weather.  The morning sun comes later and the air is just a bit crisper.

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Wednesday, September 5, 2018  Yesterday I was a substitute teacher at Madison Academic High School.  The subject was French.  About all I know about French is some familiarity with a number of great French mathematicians: Descartes, Fermat, Cauchy, Galois, Hermite, etc.  Despite my ignorance of the subject, the day went well and the students were polite and cooperative.

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Monday, September 3, 2018  The high school where I worked before my retirement from full time teaching was Madison Academic Magnet High School.  In 2012 we started a program called “Reading Scholars” which is still ongoing at Madison.  One of the promoting features of the program has been the creation of bookmarks which are given to all of our student participants.  I have attached a pdf document showing all five of our bookmark creations.  Last year we had a contest among students to come up with a slogan for our newest bookmark.  The winning submission was from student Reagan Vintzel … his bookmark is the one with the cloud design.

BookMarks

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Friday, August 31, 2018  I have long been interested in the mathematics involved with decimal representations of real numbers.  Trying to find a real number that contains all ten digits in its decimal representation may seem a tough task at first.  But such examples are surprisingly abundant.  I have recently posted a proof of “how abundant” in the EXAMPLE section of this web site.

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Wednesday, August 29, 2018  We had our judging panels listen to six student presentations for the “Reading Scholar” program at Madison Academic Magnet High School today.  One of the students gave an interesting presentation on a book about the God of the Old Testament.  One of my interests is theology (see Dust Currents and the Last Baptist).  The student held up will on some of my tough questions.

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Monday, August 27, 2018  The simplest fraction that as a decimal contains all ten digits is 1/17.  The simplest fraction that contains all ten digits in order is not very simple.  The fraction 1/81 contains all but one of the ten digits.  The decimal for 1/81 is .012345679012345679012345679…  The messy fraction 123456789/9999999999 will give the decimal .01234567890123456789… and it can be reduced somewhat by dividing the numerator and denominator by nine.  But 9 seems to be the only common factor of numerator and denominator.

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Friday, August 24, 2018  A recent news show had a guest who had done research on the incidence of mental illness among artists and other types of genius.  The study got into the left brain, right brain analysis.  I am suspicious of the results that mental illness is more prevalent among “gifted people”.  I tend to believe that the samples studied are skewed.  Gifted people seem to be studied more than the average sample of the population.

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Tuesday, August 21, 2018  Yesterday I was the substitute teacher in the calculus classes at Madison Academic Magnet High School.  It’s early in the course and the students work working with limits, using algebraical, numerical and graphical techniques.  There was no work with delta, epsilon proofs… although hey had studied the limit theorems.

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Saturday, August 18, 2018  We drove back from Mobile, Alabama to our home in West Tennessee today.  We were back just in time to visit a reunion of the Casey Jones Barbershop Chorus.  It was a joy to get to sing with the fellows after a lay-off of several months.  The singing group started back in 1979 and has been performing for 39 years.

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Tuesday, August 14, 2018  Referring back to the question asked last Thursday, the 2 smallest whole numbers that are the sum of 2 squares in THREE different ways are:

325= 1 + 18^2;  325= 6^2 + 17^2; 325= 10^2 + 15^2  and……….

425= 5^2 + 20^2;  425= 8^2 + 19^2;  425= 13^2 + 16^2

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Saturday, August 11, 2018  I stayed outside last night and early this morning to observe the Perseus meteor shower.  I only saw 3 before I gave up and went to bed.  I may try again Sunday night.

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Thursday, August 9, 2018  One question in arithmetic that I have doodled with for many years is the search for whole numbers that are the sums of squares in two different ways.   For example 65= 16+49 also 65=1+64.  A smaller example (but perhaps more trivial) is 25=0+25 and 25=9+16.  The next few  numbers with this property are 85, 125, and 130.  This leads to may other questions.  One I have followed is a challenge … but only needs the tools of basic arithmetic.  Are there numbers that are the sum of two squares in THREE different ways?

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Tuesday, August 7, 2018  This may be my last year to stay on the substitute teacher list.  I have already accepted and been scheduled for 5 days this year.  This will be my 5th year of retirement from full time teaching.  The students have been very kind to me when I appear as their sub.  They are looking at me as sort of a grandfather figure.

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Saturday, August 4, 2018  The math of tuning pitches while singing Is quite interesting.  If you assume concert A is 440 vibrations per second, then inverting the ratio of a major sixth you arrive at middle C being 264 vibrations per second.  If you use 264 for middle C then D (a whole step using a 9/8 ratio) is 297 vibrations per second.  But in the key of D, A is a major 5th.  Using the 3/2 ratio of a major 5th A would be 445.5 vibrations per second.  This means, along with other things, that A above middle C is different in the key of C from what it is in the key of D.  This sort of disconcerting situation was one of the motivations for tuning instruments with “equal temper”.   However, when singing without instruments you can be mathematically justified in assuming that G sharp and A flat are different pitches.  But your hearing needs to be better than mine to discern the difference.

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Wednesday, August 1, 2018  After a visit with my investment advisor I pondered the extremely complicated mathematics of economics of bonds and the stock market.  It calls to mind the poetry of Oliver W. Holmes:

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Saturday, July 28, 2018  I have been reading some philosophical essays of David Hume where he comments on the history of monotheism and polytheism.  This subject motivated the ghost of an idea of applying mathematical progression to theology.  If monotheism is more logical and “civilized” than polytheism, then what shall we say regarding the sequence of polytheism to monotheism to atheism?

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Wednesday, July 25, 2018  If 3 points are selected at random inside of a circle what is the probability that the points form the vertices of an obtuse triangle?  I once tried an experiment in my math classroom using a circle on the floor marked off with masking tape.  The students dropped 3 white beans and decided whether the beans formed an obtuse triangle.  The repeated the drop until 100 drops were made.  Obtuse triangles were formed considerably more than 50 percent of the time.  This question is similar to one of Lewis Carrol’s “Pillow Problems” about which there is some disagreement on what the theoretical probability is.

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Sunday, July 22, 2018   I sang in the service choir at the Methodist Church this morning.  I have been singing in church choirs for the last 52 years.  In the July 10 journal note I mentioned visiting the man who first enrolled me in a church choir.

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Thursday, July 19, 2018  The general answer to the question of Tuesday’s pondered question in terms of length and width is (1/4)times (L+W-sqrt(L^2 + W^2)) that is to say subtract the distance of the diagonal from the sum of the length and width and then divide by four.  Mow this measured strip around the original rectangular plot and you are half finished.

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Tuesday, July 17, 2018  West Tennessee has been having tropical weather since early June.  It is quite an effort to keep up with mowing the grass.  But as sweat pours off my face and my white shirt gets soaked, I ponder math questions … such as… If you are mowing a rectangular plot that measures 30 yards by 20 yards how much of a rectangular strip has to be mowed around the outer part to accomplish cutting half of the plot?  The mathematics fades in an out as I battle heat exhaustion and covet the glass of iced tea in my cooler.

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Friday, July 13, 2018  There seemed to be no bad luck today for me.  I am still doodling around on the algebraic number problem without much progress.  I finished reading “Hitman” by Lawrence Block today.  It is an unusual bit of fiction.

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Tuesday, July 10, 2018  We returned from Louisville, KY yesterday after visiting a couple who had a very positive influence on my life over 50 years ago.  He was my first choir director.  We had a wonderful visit and hope to go back again.  On the drive back from Louisville to our home in Jack’s Creek, we avoided the Interstate highways as much as possible  and enjoyed views of the Kentucky and Tennessee country side and light traffic.

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Saturday, July 7, 2018  I drove to Athens, Alabama to attend the celebration of my sister’s wedding anniversary.  She and her husband have been married 50 years.  It was a 260 mile round trip by the time I returned home to Jack’s Creek.  Almost any trip I take gets me to thinking about geography, geology and the mathematics of the distances and times that are involved.  Between Savannah, TN and Florence. AL I passed though some interesting appearing hills that are on the edge of the “Highland Rim”

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Wednesday, July 4, 2018  Why don’t we change July 4 to the first Saturday in July?  I know it would no longer be on July the 4th (most years).  But this Wednesday celebration is really awkward.  Just call it “Independence Day”.  Most colonists did not learn of the Declaration on the 4th anyway.  If you do not like this idea, then why not take the whole week off for whichever week the 4th falls on!  This idea should get some votes.

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Wednesday, June 27, 2018  Today is my sister’s birthday.  I remembered to send a card.  Hope it arrived on time. Her age is a number with exactly 12 divisors, including 1 and the number itself.  But she is not 2^11 years old!  (See January 17, 2018 journal entry)

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Monday, June 25, 2018  I am not making much progress on constructing polynomials that demonstrate “addition closure” of addition with algebraic numbers.  Perhaps I should leave it alone for a time.

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Friday, June 22, 2018  Summer is officially here.  The grass in my yard is still growing full speed.  I have done some scribbling on a problem involving the “closure with respect to addition” of algebraic numbers.  Suppose R1 is a real root of x^5 + 3x – 1 and R2 is a real root of x^3+x – 3.  How can an integer polynomial be constructed so that one of its roots is (R1 + R2).

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Monday, June 18, 2018 My daughter Carol telephoned me yesterday and wished me Happy Fathers Day.  We enjoyed having lunch together last Thursday.

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Friday, June 15, 2018  I am reading a book by Willy Ley entitled “Exotic Zoology”.  Dr. Ley was known best for his work with rocketry, propulsion and the possibility of space travel.  He was born in Germany and managed to leave before Hitler came to power.  I was privileged to hear Ley speak in 1968 a year before he passed.  He spoke at Union University on U.F.O.’s or as he called them “Unexplained Observations” or U.O.’s

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Monday, June11, 2018  Last week at Magazine Mountain in Arkansas, I watched Hang Gliders launch from an altitude of about 2400 feet above sea level and climb using updrafts to well over 4000 feet.  I was fascinated at seeing the skill of the flyers.  Some stayed aloft over an hour.  Some I never saw land and possibly glided many miles away.

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Friday, June 8, 2018  I just returned from an enjoyable trip to Magazine Mountain in Arkansas.  Yesterday I hiked to the Signal Hill summit which at an altitude of 2735 feet is the highest elevation in Arkansas.  We stayed at Mount Magazine Lodge which is a wondrously beautiful place overlooking the Petit Jean Valley and vistas of over 50 miles into the Ouachita Mountains.

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Sunday, June 3, 2018  I sang with the choir at First United Methodist Church in Jackson, TN.  A big part of my life has been related to singing in church choirs and It has only been recently that I have been out of choral singing.  It is good to get back in the bass section of a church choir.

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Friday, June 1, 2018  There was an article in a publication of the M.A.A. about the “Mobius Function” that I found interesting.  It seems that I should have heard of this function before because of its relation to prime number theory.  Just one of the many gaps in my knowledge of number theory. August Ferdinand Mobius (1790 – 1868) was an interesting person as well as a superb mathematician.  I had heard of the “mobius strip” very early in my teaching career, but not the function.

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Tuesday, May 29, 2018  The polynomial I found that has sqrt(2) + sqrt(3)  +sqrt(3) for a zero is an eighth degree polynomial.  I suspect that it would take a sixteenth degree polynomial to successfully construct a polynomial that had a zero that was the sum of four square roots.  Do we sense a pattern?

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Friday, May 25, 2018   In reference to Friday, May 18 journal remarks, X^4 – 10X^2 + 1 has a zero of sqrt(2) + sqrt(3).  Finding a polynomial with zero sqrt(2) + sqrt(3) + sqrt(5) is messier, but I have found one.

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Wednesday, May 23, 2018  The recent NFL decision to punish players who do not “properly respect the U.S. flag” seems to tread dangerously on hard won court decisions!  The most important U.S. Supreme Court legal victory won by the Jehovah’s Witnesses was in the case West Virginia State Board of Education vs. Barnette (1943), in which the court ruled that school children could not be forced to pledge allegiance to or salute the U.S. flag. The Barnette decision overturned an earlier case, Minersville School District vs. Gobitis (1940), in which the court had held that Witnesses could be forced against their will to pay homage to the flag.  I think that the best way to honor the flag is to respect the freedoms that we enjoy in the U.S.  If we say we respect the flag but don’t allow citizens to express sincere beliefs we are like the hypocrites described in MATTHEW 6: 5,6

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Friday, May 18, 2018  I scribbled some on a napkin today and found a polynomial that had a zero of sqrt(2) + sqrt(3).  This got me to trying to find a polynomial with a zero 0f sqrt(2) + sqrt(3) + sqrt(5).  Of course the fact that the Algebraic numbers are closed with respect to addition guarantees such a polynomial exists… but constructing a polynomial still remains a challenge.

Wednesday, May 16, 2018  Getting back to factoring X^4 + 4, of course the rookie, absurd answer is:

X^4  + 4  =  4(X + 1)  Aint it  a grand try for a student wanting a simple, lazy way out!  But the solution takes some outside the box, creative algebra.

X^4 + 4

X^4 + 4X^2 + 4 – 4X^2

(X^2 +2)^2 – (2X)^2

(X^2 + 2 + 2X)(X^2 +2 – 2X)     a brilliant give and take!

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Friday, May 11, 2018  I survive the week but was worn to a frazzle!  I share with the students one of the trickiest factoring problems I ever encountered in high school algebra.  We looked at how to factor X^4 + 4

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Tuesday, May 8, 2018  What a busy day!  I am substituting all week for a math teacher at Madison Academic H.S.  The students have been great and receptive … not easy to do this late in the school year.  After school I took my car to the shop for some minor repair work.  Tuesday night I attended a meeting of our local Barbershop Singing group.

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Tuesday, May 1, 2018 I have discovered a writer of mysteries that I am enjoying very much and am quite impressed with his style.  Lawrence Block is in his 80’s now and seems to be mostly retired from writing.  But his novels about a retired policeman, recovering alcoholic detective are jewels.  Matt Scudder, Block’s protagonist, is the lead character played by Liam Neeson in the movie “A Walk Among the Tombstones”.  The movie is good but the series of detective mystery novels is outstanding.  I am trying to read them in order and have read the first four.  My recent cataract surgery has enabled me to read without eye glasses.

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Saturday, April 28, 2018   I lost a dear outdoor pet companion this morning.  About five years ago, a passing car tossed out a little calico kitten that was determined to make our property her home.  She was one terrific mouser and an affectionate little friend who followed me around as I performed outdoor chores.  We named her “Little Bit” and took her to the vet and had her spayed.  She was the most agile and acrobatically gifted cat I had ever seen.  She could run up trees, fence posts and brick walls and run back down them head first (never seen another cat do that)  She had to stay outside … I am extremely allergic to cat dander.  She greeted us every morning and often left us a dead mouse, mole or even a dead rat.   When we sat on the porch, she would come see us and climb up in our laps. But even with her amazing reflexes and speed, she made one attempt too many to cross the road.    I miss you Little Bit.  R.I.P.

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Monday, April 23, 2018  Sunday we drove back from a reunion of my first cousins that was held just northeast of Atlanta.  As a life long resident of western Tennessee, I am fascinated by the hills and low mountains that are absent from my home region.  Between Birmingham and Atlanta on Interstate 20 we passed the “Talledega Mountains” which seem to be the last bit of the Appalachian Chain as they dwindle away in Alabama just above the last of the Piedmont.  From previous trips I am aware of an area that looks very much like a glacier moraine to the south of the area of Mount Cheaha, Alabama’s highest peak at 2407 ft..  It was pointed out to me years ago as I traveled from Auburn, AL with my friend to his home in Oxford, AL.  Northwest of Birmingham on highway 157 we crossed what I call Battleground ridge and a spectacularly beautiful view of the plain to the east.  Then between Muscle Shoals and the Mississippi line on US 72 I see some ridges and hills that belong to some sort of geological formation that gives up at Woodall mountain (only 800 ft) in NE Mississippi though perhaps it connects to the hills of the Highland Rim area just east of Savannah, TN.  I ponder what movement of the earth’s crust made all these wrinkles … and I sort of wish western TN had more wrinkles.

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Wednesday, April 18, 2018  After being “burned out” on math for some weeks, I picked up my pencil and dabbled and doodled with subsets of the rational numbers, Q.  I came up with a subset, M, that is everywhere dense on the real number line that is sort of 1/3 of Q. If I add one to every number in M I get a set M1 that is disjoint from M and since the plus one map makes it topologically equivalent to M, M1 is dense on the real number line.  If 2 is added to every number in M then I get set M2 which is disjoint from M and M1.  Furthermore M+M1+M2= Q so I have managed to separate Q into dense, topologically equivalent sets.  This “discovery” will not make me wealthy but I find it interesting.

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Thursday, April 12, 2018  Tomorrow is one of two Friday 13th that we will have this year.  The other will be in July.  Yesterday I had cataract surgery on my right eye.  Everything seemed to go smoothly.  I can already tell that colors seem brighter with this eye.  My other eye will get fixed in two weeks.

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Sunday, April 8, 2018  I am getting a bit lazy with my teacher’s journal and haven’t been doing much with mathematics the last few days.  Maybe I need a dose of some sort of Spring Tonic.  I am not inclined to try the old fashioned remedy of sulfur and molasses.

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Easter Sunday, April 1, 2018  I got curious about when the last time Easter occurred on April Fool’s Day.  It has not happened before in my lifetime!  In 1945 we had Valentine’s Day occur on Ash Wednesday and Easter on April Fool’s Day.  Both of those are eyebrow lifting coincidences.

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Sunday, March 25, 2018  I have not made much progress with the pentagram problem.  But in line with my usual problem solving techniques, I have found three very clever ways to get sums of more than 4 units!  I need some new napkins.

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Friday, March 23, 2018  I have been scribbling on napkins and on backs of envelopes on the problem of finding the smallest possible sum of line segments connecting the vertex points of a regular pentagon that measures one unit on a side.  There has not been any impressive progress so far.  The best I have done is 4 units, which to quote Sherlock (and Watson) is absurdly simple.

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Wednesday, March 21, 2018  I checked out “A Brief History of Time” a day before Hawking died.  An eerie coincidence.  I am thoroughly enjoying the book.  I also checked out a book on Complex Variables.  So many interesting topics in math and science and at 68 years old – I hear the clock ticking!

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Monday, March 19, 2018  Back on March 7, Madison Academic Magnet H.S. had the “Reading Scholars” presentations.  All of the presenters did an excellent job, especially in the question and answer session.  The members on the judging panels asked some challenging questions.  Below is a photograph of the winners, with myself being the old guy in the background.

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Saturday, March 17, 2018  St Patrick’s Day.  I have posted the “least possible sum” sketch and work in the EXAMPLE section.  See March 11 journal comments

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Thursday, March 15, 2018  Yesterday, I celebrated “Pi Day” by driving down to my favorite getaway city, Mobile, Alabama.  I had two people send me texts wishing me Happy Pi Day. Today we attended the I-Max theater at the Gulf Coast Science Museum and watched a great science film, “Nature’s Wonderland, Galapagos”  I challenge anyone to see this film and then go see evolution denier Ken Ham’s “Ark Encounter” and “Creation Museum” in Kentucky and try to believe Ham’s nonsensical phony science.

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Tuesday, March 13, 2018  The answer to the question asked in the March 7 journal entry is that the probability is about 1.5 percent or exactly 5!/6^5.  I enjoyed getting to substitute for the calculus and pre-calc classes at Madison Academic H.S.  The students are talented and sharp minded.  It is days like this that cause me to miss teaching!

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Sunday, March 11, 2018  The least possible sum for Friday’s question is 2.732… or 1 + Sqrt(3).  The surprise answer uses 5 line segments.  I’ll attach the sketch in the EXAMPLES section when I get around to it.

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Friday, March 9,  Another question for which I have been scribbling work on a napkin – Consider the four points on the corners of a unit square: (0,0); (1,0); (1,1); and (0,1).  Of all the ways to connect these four points with straight line segments, which way will have the least possible sum of segment lengths?

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Wednesday, March 7, 2018  Probability is not one of my strong points but I really enjoy short, simple sounding puzzles.  The question that crossed my mind recently while I was scribbling on a napkin is as follows.  If a single die is rolled six times, what is the probability that you get six different numbers?

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Monday, March 5, 2018  This is a follow up to the discussion about Cantor in the previous (March 3) journal entry.  Cantor’s injective function map from the unit square to the unit interval showed that the set of real ordered points in the plane was not more numerous than the real number line.  But there are infinitely points in the line that are not covered.  Since all terminating decimals have two different representations, one or the other must be left out.  For example .25 = .25000000000… but also .25 = .249999999999…  so in the mapping one or the other would be left out.  If the function F described in the previous journal entry was such that F(1/3, 1/4) = .3235303030303030… then the F inverse value of .32353939393939393939393939… would be undefined.

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Saturday, March 3, 2018  I read a bit of history about Georg Cantor’s work with infinity  and cardinal numbers of infinite sets.  Cantor showed that the set of real numbers has an infinite cardinal larger than the infinite cardinal of the set of all integers.  But Cantor was surprised than the 2 dimensional plane is not a larger order of infinity than the real number line.  He demonstrated this with an injective map from the plane to the real number line.  Using decimal notation for ordered pairs in the plane, he mapped the ordered pair digits to a real number by alternating digits.  For example (.333333333…, .25000000…) maps to .3235303030303030….  I have a great fondness (as you will notice) for using decimal notation to demonstrate unusual functions!

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Wednesday, February 28, 2018  I filed my Federal Income Taxes today and for the 2017 year I came close to breaking even with my tax bill and the amount withheld.  I am both amused and annoyed at those who think that the “rich” pay a bigger percent than the “poor”.  This is only true if one uses tunnel vision and only looks at Federal income tax.  But if you consider social security tax, sales tax, property tax (which renters pay for the landlords), gasoline tax, license plate tax, telephone tax, state lottery tickets, hotel room tax, hunting and fishing license taxes, state and national park fees, business licenses, automobile inspections, etc. etc etc.  … the “poor” get to keep much less of each dollar earned than the “rich”

Tuesday, February 27, 2018  I am still reading a biography about the great Irish mathematician William Rowan Hamilton.  Hamilton was one of the successors of  Bishop James Ussher as the head of the Irish astronomic observatory at Dunsink.  Ussher is known for using the Bible to calculate the age of the earth to be 6000 years old.  It is noted that even Sir Isaac Newton was reluctant to believe the earth could be older than 60,000 years.  While these perspectives can be partially excused because of the limits of known science in the past, no scientist can be taken seriously who believes such nonsense now.  It is deplorable that some religious people still hang on to such foolishness!  If young people hear such anti-scientific rants from pulpits, the are likely to later reject religion and throw the baby out with the bath water.

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Sunday, February 25, 2018  We just returned from a short trip to Mobile, AL.  While there we visited the independent theater, The Crescent, on Dauphin street.  We watched “Three Billboards Outside Ebbing Missouri.”  It’s an interesting film that has 7 nominations for an Oscar.

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Thursday, February 22, 2018  More 2018 solutions:

44 = Sqrt((.2^-0!)!)^(-.1+Sqrt(Sqrt(8))    this gives 44.0978

45=(.2^-0!)*(1+8)

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Tuesday, February 20, 2018  Below are a few more solutions to the 2018 puzzle.  See the January 5 journal entry for my rules.  I have been doing this type puzzle since 1976 with my math classes.  The year 2000 presented special difficulties!  I thought that this was an original idea with me  Recently I have discovered other teachers have had similar ideas … though not with exactly the same rules.  Perhaps some of these teachers were former students (I taught about 5000 students in my 41 year career)  The solutions for some of the numbers between 40 and 50 are difficult to find as you will see as I continue to post some of my best efforts.

41= Sqrt((2+0!)!!)^Sqrt(Sqrt(Sqrt(Sqrt(-1+8))))  Obviously a tough reach. equal to 41.06079

42= ((2+0!)!)*(-1+8)

43= -2 +Sqrt(Sqrt((0!/.1)!)) + Sqrt(Sqrt(8))  A REALLY tough one for me, equal to 43.327  I hope for a closer answer

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Monday, February 19, 2018  I am reading a biography of the great Irish mathematician William Rowan Hamilton.  I got to visit Trinity College a few years ago in Dublin, Ireland and was a bit surprised that our guide, though excellent had not heard of Hamilton who was perhaps Ireland’s greatest mathematician and scientist.

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Saturday, February 17, 2018  When I look back at my childhood and youth, sometimes I wonder what helped me develop my mathematical skills.  Not that I am a great mathematician … I consider myself a “good” mathematician.  There are mathematicians who make me look very slow witted at math.  Then there are those rare math genius who make these look slow witted.  So I am at least two levels of talent from the very top.  Two experiences in my youth helped develop some math skill.  One was the countless games of checkers my father (a superb checker player) played with me.  These taught me much about “if – then” and lengthened my attention span.  Another experience was the games I invented for myself … such as tossing up rocks and batting them with a broom handle and mentally calculating the percentage of the rocks landing certain distances away in the pasture.  Besides the growth of mental math skills, I think this activity nearly depleted portions of our gravel driveway.

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Thursday, February 15, 2018   I have been on a busy track of reading mystery stories this year.  One author that really keeps my interest is C.J. Box.  His settings in Wyoming have made me want to plan a trip to visit that state.

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Tuesday, February 13, 2018  A follow up question from yesterday’s entry.  Noting that 0!=1 and 1!=1, are there other real numbers X so that X!=1?

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Monday, February 12, 2018  In my younger days I was troubled by the fact that zero factorial was equal to one.  But if you think of factorial only from the perspective of permutations it makes sense.  For example there are six ways to put A,B,C in order (3 objects, 3 factorial) and 24 ways to put A,B,C,D in order which is 4 factorial.  In a quite logical sense there is only one order to arrange zero letters.  But mathematicians have found a very creative way to define the factorial function continuously.  K factorial is defined as the area under the curve of Y=(X^K)*(e^(-X)) from zero to infinity.  Using the improper integral from this definition we get zero factorial is one.  This function gives all the correct values for N factorial where N is a positive integer,  It also gives intriguing values such as 1/2 factorial is equal to sqrt(pi)/2.  And my contribution, from a bunch of trial and error today is that 100=4.892518712 factorial.  I am also surprised that TI graphing calculators will use the ! symbol to calculate the factorials for 1/2, 3/2, 5/2, 7/2 etc.  But will not for other fractions or decimals.  It is surprising what a good calculator can teach you if you use it as an experimental tool instead of an answer machine.

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Sunday, February 11, 2018  If I can quote the agnostic Bertrand Russell then why not quote the very religious Leonhard Euler

For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.

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Friday, February 9, 2018

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

In all affairs it’s a healthy thing now and then to hang a question mark on the things you have long taken for granted.

The biggest cause of trouble in the world today is that the stupid people are so sure about things and the intelligent folks are so full of doubts.

Quotes from Bertrand Russell … He had a way of nailing the truth!

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Wednesday, February 7, 2019  A few more answers on the 2018 puzzle:

36= (2+0)*18

37= sqrt(2) + sqrt(sqrt((0!/.1)!)) – 8  this one has been very tough and what I have here is 37.06

38= 2*(0!+18)

39=sqrt(20)^sqrt((sqrt(1+8)!)  even tougher for me and equal to 39.213  I’ll try to get closer

40=(.2^-0!)*(1*8)

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Monday, February 5, 2018  During forty plus years of working in a high school math classroom, I was persuaded that the real importance of tests and quizzes was not to assess what the students had learned.  The true value of “assessments” was their importance as learning tools.  If a student can take a math test and not learn something in the process, then the test will be a great waste of time.  I believe that this is one of the many problems of the high stakes testing foolishness that plagues public schools in the U.S.  These inane multiple choice tests show only what students “mark” … not what students know or learn or how they think!  Teachers get little or no feedback on how a student demonstrated thinking or application skills.  More often than not, teachers and students never learn which specific problems are answered correctly or answered incorrectly.  Most math teachers relish the times when students have a “Oh Yeah! I get it!!” experience.  Opportunities for these moments on high stakes, multiple choice assessments are somewhere between rare and nonexistent.  Most of the questions on these tests seem to me to be mind numbing, time pressured, “one size fits all”, regurgitations of the memorized obvious!  I may rant on this topic again in the future, but next time I won’t hold back!

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Saturday, February 3, 2018  I have used guess and test to find a close fit to Euler’s “inexplicable function” discussed in the Feb. 2 journal entry.  My best guess is F2(x)= ln(1+0.78x).  For example F(6)=1+1/2+1/3+1/4+1/5+1/6 which equals 49/20 or 2.45.  Using ln(1+0.78x) where x=6 gives 2.458.  Other examples are below.  Euler used an Integral formula which gives exact results.  There is no closed form expression that works for F(x)=1+1/2+1/3+…+1/x, but the formula I give here ln(1+0.78x) is a very close approximation.

F(20)=3.6   F2(20)=3.60004

F(75)=4.90135   F2(75)=4.90156

and here is one more entry in the 2018 puzzle

35=(.2^-0!)*(-1+8)

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Friday, February 2, 2018  The book about L. Euler that I have been perusing is full of beautiful mathematical jewels!  There is a section on what Euler called inexplicable functions… functions that are neither algebraic or transcendental.  One function that caught my eye was F(x)= 1 + 1/2 + 1/3 + 1/4 + … 1/x, where x is not necessarily a positive integer. Euler defined the function to be continuous.  His formula agrees with all the positive integer values of the harmonic series and fills in the in between values in a continuous differentiable curve.  For example F(3) = 11/6; F(4) = 25/12 but F(3.5) = approximately 1.966086591.

32=(2+0!+1)*8

33=.2^(-0!-1)+8

34=2*(-0!+18)

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Wednesday, January 31, 2018

29= 20+1+8

30= (.2^-0!)*(Sqrt(1+8))!

31= see January 22

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Monday, January 29, 2018  I gave a short presentation at the Madison Academic Magnet High School today for the math club – Mu Alpha Theta.  I shared some of L. Euler’s work with 1/9 + 1/99 + 1/999 + 1/9999 + …  There is a photo of the club’s Vice President and me in the gallery under “Rookie Stuff”

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Saturday, January 27, 2018  John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”

Blaise Pascal: “It is not certain that everything is uncertain.”

27= (2+0!)*(1+8)

28= 20 + (1*8)

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Friday, January 26, 2018  Surviving three school days as a substitute teacher for some Spanish classes was not so difficult.  The students were cooperative and kind to this old man … about the age of their grandparents.

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Thursday, January 25, 2018  Next Wednesday we will have a blue moon and a lunar eclipse.  It’s been 150 years sinve that last happened.  It is also a “supermoon”.

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Wednesday, January 24, 2018  I survived as a substitute for a Spanish teacher.  I only know about 20 words of Spanish and can barely make a sentence.  That is embarrassing!  I am proud to see that students are now taking 2 or 3 years of foreign language.  I know English, a bit of Latin, one computer language and mathematics.

26= 2 + (0! + Sqrt(1+8))!

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Tuesday, January 23, 2018  Though I have retired after 40 plus years of teaching mathematics, I occasionally go back to my last school and substitute.  This week will be a new experience for me as I will be a substitute for a Spanish class.  My normal situation is to fill in for an absent math teacher.  In this case the teacher is a former colleague and he will have the student tasks all arranged.  I may learn some new language skills!

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Monday, January 22, 2018  The first roadblock, at least for me, in working the 2018 puzzle is an attempt to use the digits 2,0,1,8 in that order and math symbols to get a result of exactly 31.  I have found expressions for one through 30 with exact values.  Most of these results have been given in my journal entries.  Getting exactly 31 has eluded me so far.  But below is a way to get “very close” to 31.

31= (2+0!)/.1 + a bunch of nested square roots around 8               (12 square roots will get 3 place accuracy)          You can look at the examples section of this site to see some work on changing 8 into one.   Below are a few more solutions:

23= 20 + Sqrt(1+8)

24= (2+0!)*(1*8)

25=Sqrt(Sqrt(((2+0!)!-1)^8)   (another tough one!)

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Sunday, January 21, 2018  Here are a few more solutions to the 2018 puzzle.  See the January 5 journal entry.

21=20 + 1^8

22= -2 + (0! +sqrt(1+8))!      (this was a tough one!)

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Saturday, January 20, 2018 There is just the smallest bit of snow lingering around our house.  In some ways snow acts as a blanket to hold heat in the ground and also as a reflector of sunlight to keep radiant energy from warming the ground.  I must be getting old because I neither tossed snowballs or mad a snowman during this winter episode.  We probably received a total of about 5 inches of snow.

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Thursday, January 18, 2018  Here are some more thoughts on the series 1/9 + 1/99 + 1/999 + 1/9999 … =?  This series converges to a sum, call it S.  Using my trusty TI-84 calculator, the sum of the first 7 terms is given below:

S is approximately equal to 0.1223242432 the great Swiss mathematician L. Euler noticed that the Nth digit of this decimal is equal to the number of divisors of N.  For example the 6th digit is 4 and 6 has divisors 1,2,3 and 6.  The 9th digit is 3 and 9 has divisors1,3 and 9.  At some point this digital pattern breaks down … where?

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Wednesday, January 17, 2018  Sometimes students in a math class can ask questions that can lead to an interesting (and to a math nerd like myself, a fascinating) bit of further research.  Once in an algebra class a student asked, “Mr. Scott, are there any whole numbers that have exactly 100 factors?”  I replied at once that 2 raised to the 99th power had exactly 100 factors from 2^0 on up to 2^99.  Another student replied with a comment that 2 raised to the 99th power was a very large number … are there smaller numbers that have exactly 100 factors?  I gave a common response that I didn’t know, but I would think about it.

We made it an extra credit project.  Extra points would be given to the student who by the end of the week had come up with a smaller number with exactly 100 factors.  The result turned out that points were awarded to the student with the best wrong answer.  One student who couldn’t find a smaller number than 2^99 turned in a “proof” that there could not be a smaller number since 2 is the smallest prime.  It was the best effort we got for this question by Friday.

The question lingered in my curiosity and occasionally I scribbled on some napkins with a few minor bits of progress.  Then one night I woke up with the answer.  “I’ve got it!”  My awakened wife mumbled, “what are you talking about?”  I replied that I had found the smallest possible number with exactly 100 factors.  She made a comment that indicated that she did not share my enthusiasm with this discovery.

The smallest number with exactly 100 factors is 45,360.  How I got this answer is another story.  But what surprises me is that over 30 years have passed since my discovery and the question becomes relevant to L. Euler’s question (see the January 15 journal entry) about the series 1/9 + 1/99 + 1/999 + 1/9999 + …  I may write more about this later.

17= -2 + 0! +18

18= 2*0 + 18

19= 2^0 + 18

20=2 + 0 + 18

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Monday, January 15, 2018  One of my favorite topics in mathematics is converging number series.  I found an interesting series question while looking at a book about some of L, Euler’s work.  It involved the infinite sum:

1/9 + 1/99 + 1/999 + 1/9999 + … = ?

Most calculus students have encountered the series 1/10 + 1/100 + 1/1000 + 1/10000 + … = 1/9  Euler took this simpler series and made some amazing discoveries about the series with nines.  Of course an obvious conclusion is that the series with nines must sum to a number larger than 1/9.

Here are a few more results of the 2018 puzzle. (see January 5th journal entry)

14=(2+0!)! + 1*8

15=(2+0!)! +1+8

16= -2+0 +18

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Saturday, January 13, 2018  I am reading a book about WW II rescues in Greenland, “Frozen in Time” by Mitchell Zuckoff.  In describing Greenland’s brutally cold climate it mentioned that there is only one temperature where Farenheight and Celsius  agree – that being 40 degrees below zero.  It makes me shiver to contemplate that this is a common temp in Greenland!  I am thinking that it would be a good first year algebra assignment to solve equations to show that this temp is the same in both systems.

10= 2 + 0 + 1*8

11=2 + 0 + 1 + 8

12=2 + 0! + 1 + 8  (remembering that zero factorial is one)

13=-2 – 0! + 18

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Thursday, January 11, 2018  During the last very cold weather when West Tennessee dipped into the low teens,  I had to daily toss out the ice in my outdoor cat’s water bowl and refill it so the cat could have a drink.  Noticing how quickly the water froze brought to mind the old saying “hot water will freeze quicker than cold”.  If you take this as an absolutely true axiom, perhaps you should try the following experiment.  In very cold freezing weather, set out 100 identical empty bowls.  Fill the first with hot water, freshly raised to the boiling point in your tea kettle.  Wait until this water cools significantly – say ten minutes.  Fill the second bowl with similarly freshly boiled water.  After ten more minutes fill the 3rd with freshly boiled water,  Continue this procedure until the 100th bowl is filled.  Now recall our axiom and predict which bowl freezes first.  By the way, our outdoor cat has a nice enclosure with a heating pad so she can stay nice and comfortable even in frigid weather.

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Tuesday, January 9, 2018  “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.”
David_Hilbert

Not ignorance, but ignorance of ignorance, is the death of knowledge. Alfred North Whitehead

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Sunday, January 7, 2018  I again picked up a book on L. Euler and as usual am thrilled and intrigued by many of his results.  Euler must have had the math gods whispering in his ear!

7= 2*0 – 1 + 8

8=2*0 + 1*8

9=2+0-1+8

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Friday, January 5, 2018  West Tennessee should have air temperatures creeping above freezing today.  Our outdoor cat will enjoy a good hunting day.  We have built a small enclosure with a heating pad that keeps her somewhat comfortable during the brutally cold nights so the cold is not a life threatening situation.  There are times she indicates wanting to come in the house, but she enjoys her freedom.  My allergy prevents an indoor cat.

Here are a few more solutions to the 2018 puzzle:

4 = 2^0 + Sqrt(1+8)

5= 2+0 + Sqrt(1+8)

6= (2+0)*Sqrt(1+8)

I still claim that we can go all the way to 100.  For those who want to play along, the rules are as follows.  The digits of the year must be kept in order.  Then add any grouping or operational symbols found on a calculator to get the desired result.

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Wednesday, January 3, 2018   It is difficult for me to part with some of my paperback books.  The late John D. MacDonald was one of my favorite writers.  My collection of his work includes all of the “Travis McGee” series.  He used a color in each of the “Travis McGee” titles … “Pale Grey for Guilt”; “Nightmare in Pink”, “Bright Orange for the Shroud”; etc. My project of reducing the load on the bookshelves in the study hits a pause each time I consider getting rid of the MacDonald paperbacks.  A favorite quote from MacDonald, “A bore is someone who takes away your solitude without providing you with company.”

Here are the next two entries for the 2018 puzzle (see January 1, 2018)

2 = 2+0*18

3= 2*0+Sqrt(1+8)

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Monday, January 1, 2018  We can start the 2018 puzzle. (The 2017 puzzle is in the example section where exact answers from 1 to 40 are listed with a very few exceptions).  I believe all solutions from one to a hundred are possible, at least to the nearest hundredth, for the 2018 puzzle.

1=2*0+1^8

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Sunday, December 31, 2017  Happy New Year Eastern Time Zone!  It is 2018 there.  2018 is barely composite.  2018=2*1009 and 1009 is prime.  Best wishes and good luck to all.  And to all a good night.

Saturday, December 30, 2017 I made my New Year’s resolution to read 100 books in the year 2017.  I have finished 103 books.  This may be my best exercise year ever.  I have only missed 9 days this year.  This is 97.5 percent.  One more book read will put me at two a week.

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Thursday, December 28, 2017  These frigid mornings are keeping our outdoor cat on her heating pad!  But she seems to be handling this winter weather ok.  I am glad she has her warm box to retreat from the bitter cold.  I finished reading “The Lost World” by Arthur Conan Doyle.  This may motivate me to read more of Doyle’s non-Sherlockian work.

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Tuesday, December 26, 2017  Santa Claus was good to me!  My wish for the New Year is from the written record in the gospel of LUKE of the angel’s song “Peace on Earth, Good Will to all”  This world needs peace and good will!

I finally got a formula for my “cycle three” function.  (see the discussion of December 8 and following) The function described below has the property that if a is any real number then f(a)=b where b is a real number not equal to the number a.  f(f(a))=c where c is a real number not equal to a or b.  and to complete the “3 cycle” f(f(fa)))=a.  INT stands for the greatest integer function.

f(x) = x + -.5 +1.5*(-1)^INT(x-3INT(x/3))

for example f(73.221)=71.221; f(71.221=72.221; and f(72.221)=73.221 back to the beginning.

I was proud of my efforts of following this somewhat silly problem down the rabbit hole.

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Sunday, December 24, 2017  Christmas Eve!  I think that I heard Santa Claus outside.  But I see nary a snowflake.  When I stepped out after dark a Hoot Owl wished me Merry Christmas.  That is my wish for all my former students and friends.

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Friday, December 22, 2017  Today is Christmas Eve’s Eve Eve!  I am starting to feel like a little boy once again and I searched through my Grandfather’s Butler’s Desk for my 60 year old Christmas stocking!

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Wednesday, December 20, 2017  On this day which is a tie for the shortest daylight day of 2017 I climbed the steps of “Saul’s Mound” which is one of North America’s tallest “Indian” mounds to see if I could feel the spirits.  I think that perhaps I did.  The view from the top was beautiful with the subtle look the landscape has when nearly all the leaves of the West Tennessee hardwoods have been shed.

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Tuesday, December 19, 2017  Here is the cycle three function as described without a formula.  If the integer part of x is zero modulo 3 or two modulo 3 then x maps to x+1.  But if the integer part of x is one modulo 3 then x maps to x-2.  For example if we start with 8.29355 then f(8.29355)=9.29355.  f(9.29355)=10.29355 and finally we get back to the original input since f(10.29355)=8.29355.  Note that 8 is 2mod3, 9 is zero mod 3 and 10 is one mod3.  The only remaining difficulty for this puzzle is to write a function using the greatest integer function INT so that one written formula does all three cases.  As for the value of this exploration … to quote Ebeneezer Scrooge, I would be a year older and not a dollar richer.  But God bless me just the same!

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Monday, December 18, 2017  I have made some progress trying to get a formula for a cycle three function. (See December 13 and preceding discussions.  I think I am close to a formula using INT.

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Saturday, December 16, 2017  I have developed a severe case of “writer’s block” over the last few weeks.  Maybe it is the time of year.  My work on my memoirs has hit a standstill.  An essay with philosophical, political and religious overtones has been cooking in my head but nothing is going down on “paper”.  I have put some math puzzles that I have been working with on the back burner.  On the bright side … I have kept my New Year’s resolution to read 100 books in 2017 (I am halfway through my 100th book)!  However, it was a bit much and next year my goal is 50.  But here is one math bit for the season. Theorem:  People who send Christmas cards receive fewer than they send (on average)  Proof:  There exist people who get cards that send none.  This decreases the total received by the senders enough that the total received by  the senders must be fewer than the total they send. Q.E.D.  This lackluster display of logic is somewhat like the letterbox method of proof.  In any case, send a card to someone who will appreciate it!

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Wednesday, December 13, 2017  I have a formula that will do the “three cycle” for any real number between 1 and 3.  Using INT for the greatest integer function the formula f(x) = 1+x – 3*INT(x/3)  For example f(2.4475)=3.4475 and f(3.4475)=1.4475 and f(1.4475)=2.4475 which is back to the original number.  I haven’t found a way to adjust this to work for real numbers outside of the one to three domain.

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Monday, December 11, 2017  Someone suggested to me that I could use a modulo three plan.  It would work for positive integers,  For example using modulus 3 arithmetic. and the formula f(x)=x+1we get f(0)=1; f(1)=2; and then f(2)=0 and the cycle three repeats.  But it doesn’t work for all real numbers, only for 0,1,2.  f(8)=8+1=0=0 and we will never get back to the original value of 8.

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Sunday, December 10, 2017  This “triple cycled” function that I am searching for can not be a polynomial … I think that can be easily proved.  Although the identity function f(x)=x is in a trivial way has cycle three.  But there is the non interesting way x=f(x)=f(f(x))=f(f(f(x))) … ad nauseum.  But even though there is no polynomial function that does what I am after, I am hoping to find a function with a formula.

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Friday, December 8, 2017  I am working on a different math investigation.  It’s called in my notes a “triple cycled” function.  It is quite east to do a “two cycled” function.  The requirements for a two cycle function call it f(x) are:

first f(x) is not equal to x for all real numbers (for most anyway); second f(f(x))=x for ALL real numbers x.  For example f(a)=b but f(f(a)=a

I am trying to find a function so that f(a)=b, f(f(a))=c and f(f(f(a))) is always a. Wish me luck.

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Wednesday, December 6, 2017  I tried my hand at making a birthday cake for my wife.  She had requested a buttermilk pound cake.  It turned out well.  There is something quite refreshing in a simple recipe that turns out well.  Eggs, flour, butter, sugar, shortening, buttermilk, and vanilla flavoring were the only ingredients.  No frosting used … just a prime number of candles poked in the top.

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Sunday, December 3, 2017  We were lazy and stayed home from church today.  When I was young (less than 40!) I attended church Sunday morning and night and Wednesday evening and usually sang in at least two church choirs.  But these days I feel secure that an occasional miss won’t extend my stay in Purgatory.

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Friday, December 1, 2017  It was only a few years ago when I pondered how incongruous it was to have our tenth month named “December”

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Thursday, November 30, 2017  I was a substitute teacher today at Madison Academic High School.  The students were heavily involved in the inane policy of “end of course” testing.  Never mind that this is two weeks before the end of the course! I had time to scribble around with some more “fixed point” explorations with the aid of a good calculator.

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Tuesday, November 28, 2017   Tonight I doodled around and explored some “fixed point” problems.  I took out my old TI-83 graphing calculator and entered 1.  Then I keyed in the formula .25ans^2 – 1 (ans is the variable name that stores the previous answer) Pressing enter the result was -.75.  Pressing enter again ran the -.75 through the equation and gave a result of -.859375.  Pressing enter 3 more times gave answers -.81536865, -.83379349, and -.8261971039.  Continuing to run answers back through the formula by pressing enter the results reached a limit of -.8284271247…  Does anyone recognize this number?

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Saturday, November 25, 2017  I probably need to leave the equation first mentioned on November 16 and let it rest.  Maybe I will be inspired to find a pattern later.  One more comment for now …  if you look at the ratios of B/A for the ten integer solutions discovered so far, the ratios are approaching a very familiar limit.  Listed below are some decimal approximations for B/A

.2500  .4000 .5454  .5625  .5750  .5763  .5772  .5772727  .577338  .5773447

If this appears to you to be suspiciously close to tangent(pi/6) you have got it.  And it turned out so easy to prove that I gave myself a “dope slap”.  However it did not seem to help me find a pattern to the integer solutions!

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Thursday, November 23, 2017  Happy Thanksgiving!  I wrote a short BASIC program for my TI-84 graphing calculator and used it to help me find four more integer solutions to A^2 – 3B^2 = 13.  As for finding any sort of pattern to these solution, it must be confessed that effort has been unsuccessful.  Listed below are the ten solutions:

(A,B)=(4,1); (A,B)=(5,2); (A,B)=(11,6); (A,B)=(16,9); (A,B)=(40,23); (A,B)=(59,34)

(A,B)=(149,86); (A,B)=220,127); (A,B)=(556,321); (A,B)=(821,474)

These seem to be the ten smallest solutions.  I don’t believe I have missed any.

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Tuesday, November 21, 2017  It took a good bit of persistence on my part to find two more integer solutions to the equation A^2 – 3B^2 =13, but here is the revised list including all six solutions that I have found:

(A,B)=(4,1); (A,B)=(5,2); (A,B)=(11,6); (A,B)=(16,9); (A,B)=(40,23); (A,B)=(59,34)

As far as I know this equation has no importance … I pulled it out of thin air while I was doodling some mathematics.  There is no pattern to the solutions … at least no pattern obvious to me.  I would welcome solutions or a discovered pattern by any readers of this journal.  My email address is on the home page.

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Sunday, November 19, 2017  Since the comments in Thursday’s Journal entry, I have found two more integer solutions to A^2 – 3B^2 = 13  Listed below are the 4 solutions I have found so far:

(A,B)=(4,1); (A,B)=(5,2); (A,B)=(11,6); (A,B)=(16,9)

You can see some of my notes on this problem in the EXAMPLES section on this site.

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Thursday, November 16, 2017  There is only one week until Thanksgiving.  It comes early this year.  I found myself scribbling some math on a napkin today as I (for some random reason) tried to find integer solutions A,B to the equation A^2 – 3B^2 = 13.  Two solutions popped out quickly since 4^2 – 3(1^2) =13 and 5^2 – 3(2^2) =13.  So (A,B) can be (4,1) and also (5,2).

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Tuesday, November 14, 2017  I sing with a small Barbershop Chorus.  We had to cancel our meeting tonight because so many of our singers had conflicts that caused them to miss.  I got somewhat depressed having to do without the wonderful fellowship and being able to sing and feel those wonderful four part harmony chords.

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Friday, November 10, 2017 Take all the real numbers between zero and one.  Throw out all those numbers that have digit 5 in the 1st digit after the decimal point.  That will leave 9/10 of the real numbers we started with.  Now from what’s left eliminate all numbers with digit 5 in the third position after the decimal point. That leaves 99/100 of what we had after the first removal … so we have (.9) times (.99) of our original set left.  Next throw out all the decimal numbers that have the digit 5 in the sixth position to the right of the decimal point.  That leaves 999/1000 of the numbers we have left after the second removal so that there are (.9)times(.99)times(.999) of the original set of numbers remaining… which is .890109 of the original set.  If this process is continued we will get a nowhere dense set of numbers, meaning that given any two numbers in the set there is a continuous gap somewhere between the two.  This gives us the “Cantor” set discussed in the November 6 discussion.

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Wednesday, November 6, 2017  Following up Sunday’s (11/5) discussion in a partial answer to the “what number do you get?” question … if you stop the product at (.99999) you get approximately .8900110889.  It can be shown that the product, no matter how far it is carried out, never gets lower than .89001.  This allows a description of a Cantor set that does not have measure zero.  Most Cantor sets that are described in the literature, as do most other “nowhere dense sets” have measure zero.  For example the set of all real number decimals between zero and one that do not contain the digit 5 has measure zero.  But using the infinite product as previously described a Cantor set can be defined that has measure at least .89.

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Sunday, November 5, 2017  If 9/10 is multiplied by itself repeatedly the product approaches closer and closer to zero.  For example (.9) raised to the 100th power is approximately .00002656 and (.9) to the 200th power is approximately .0000007055.  A question came to me , What number do you get if you multiply:

(.9) times (.99) times (.999) times (.9999) etc and on to infinity?  The answer to this actually has something to do with the DP function discussed in previous entries of this journal.

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Friday, November 3, 2017  It was a pleasant experience to be able to be a substitute teacher at Madison Academic H.S. yesterday and today for pre calculus and Algebra 2.  The students were polite and cooperative and we had some good discussions on vectors and exponents and radicals.  These students were better students han I was in my high school days!

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Monday, October 30, 2017  To follow up yesterday’s journal entry let us describe a unusual number set which will be denoted by S.  S will be the set to which a real number x belongs if and only if x is in some M(n) where n is a whole number.  Here is where I was going with this discussion and the question is …. can you name a number that is not in set S?  Or how about this question … How many numbers are not in set S?  Or … who cares … besides math geeks like Mr. Scott?  Actually, I think the set of real numbers not in set S is VERY interesting!

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Sunday, October 29, 2017  Using the DP function described 10/18/2017  we can construct infinitely many sets that are nowhere dense.  Denote each set M(n) where this set contains all real numbers that do not contain the pattern of digits designated by DP(n).  For example M(1) contains all real numbers without the digit 1 anywhere in its decimal notation.  M(35) contains all real numbers without the two digit block 25 anywhere in its decimal notation.  It can be shown that for any whole number n, M(n) is nowhere dense.

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Friday, October 27, 2017
I will be working to make this web site to replace my former site at teacherweb.com
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Thursday, October 26, 2017 We just returned from Mobile, AL. While there we got to see the movie “Victoria and Abdul” starring Judy Dench. Great movie!
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Wednesday, October 18, 2017  It looks like we can get a one to one function with just a few rules and a formula.  for a 2 digit number the output pattern can be determined by subtracting 10.  For example DP(35)=25 and DP(98)=88.  If you get a one digit answer then start the two digit pattern with 0.  For a 3 digit input subtract 110.  For example to find the out put of 523 you get 523-110 which is 423.

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Friday, October 13, 2017  In an attempt to come up with a formula for the DP function that is referred to in yesterday’s comments, perhaps it will be useful to notice the following:
DP(1)=1; DP(11)=01; DP(111)=001; DP(1111)=0001
Notice that the DP function does not map each whole number to a number, but rather each whole number is mapped to a finite digit pattern.

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Thursday, October 12, 2017  To accomplish my goal of building the set described in Tuesday’s comments, I need to define a function that maps the whole numbers onto all possible finite digit patterns.  I am naming this function DP.  To somewhat establish the function I am defining
DP(0)=0; DP(1)=1; DP(2)=2 … DP(9)=9

DP(10)=00; DP(11)=01; DP(12)=02 … DP(19)=09
DP(20)=10; DP(21)=11; DP(22)=12 ,,, DP(29)=19
etc  ………..
DP(90)=80; DP(91)=81; DP(92)=82 … DP(99)=89
DP(100)=90; DP(101)=91 ……. DP(109)=99
DP(110)=000; DP(111)=001  …

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Tuesday, October 10, 2017  I am working on a set of real numbers that is the union of countably many nowhere dense sets that contains all rational numbers and many irrational numbers.