The Nature of Mathematics Michael Randy Gabel, Ph.D. Associate Professor Mathematics and Integrative Studies George Mason University June 30, 2015

The Nature of Mathematics The Core Ideas The Creative Process

To Remember Core math concepts vs scientific method Neither is a recipe. New Polls Histo – and maybe some intro slides. Possible to have navigation buttons? Generalization (part of conjecture) – Pythagorean Theorem with polygon sides, decimals base 15 or 7 or 12, the 8x8 and Fibonacci series. Observe, Deduce, Conjecture (Use these words sometimes?) Need an ending. Hand Outs: 8x8 and decimals. Take notecards. Skip Bellhop? Mention thinking and then answering.

A Little QUIZ Name a song or the title of piece of music. (2) Name a theorem in mathematics. Card: one side: left song right side music. Then a bit later, say what the Pythagorean Theorem says.

? OR http://scienceworld.wolfram.com/biography/Pythagoras.html

PPT built in clips.

Statement of the theorem CORE IDEAS in Mathematics Statement of the theorem

Theorem: Consider a right triangle. 90 Attach a square to each side: ---->>> Then: AREA (I) + AREA (II) = AREA (III). = c2 a2 + b2

The sum of the areas of the squares on the sides of a right triangle equals the area of the square on the hypotenuse.

http://www.youtube.com/watch?v=2pWSwfVDiq8 http://1.bp.blogspot.com/_P1KuD6CE6G0/S20pVGCJ5TI/AAAAAAAAAXo/FPQApigkpfQ/s320/scarecrow_oz.gif http://www.youtube.com/watch?v=2pWSwfVDiq8 -- seconds 42 to 53

Statement of the Theorem CORE IDEAS in Mathematics Statement of the Theorem Proof of the Theorem

The PROOF: We need to see why AREA (I) + AREA (II) = AREA (III)

Statement of the Theorem Proof of the Theorem CORE IDEAS in Mathematics Statement of the Theorem Proof of the Theorem Conjecture [The Creative Process] [Observe - Think - Intuit - Puzzle]

http://www.youtube.com/watch?v=DAvr-m0ZWqE#t=0m28s

Pythagoras - Revisited Think

Observe

1 / 2 = . 5 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1/2=.500000000000000000000000000000000000 1/3=.333333333333333333333333333333333333 1/4=.250000000000000000000000000000000000 1/5=.200000000000000000000000000000000000 1/6=.166666666666666666666666666666666666 1/7=.142857142857142857142857142857142857 1/8=.125000000000000000000000000000000000 1/9=.111111111111111111111111111111111111 1/10=.100000000000000000000000000000000000 1/11=.090909090909090909090909090909090909 1/12=.083333333333333333333333333333333333 1/13=.076923076923076923076923076923076923 1/14=.071428571428571428571428571428571428 1/15=.066666666666666666666666666666666666 1/16=.062500000000000000000000000000000000

1/17=.058823529411764705882352941176470588 1/18=.055555555555555555555555555555555555 1/19=.052631578947368421052631578947368421 1/20=.050000000000000000000000000000000000 1/21=.047619047619047619047619047619047619 1/22=.045454545454545454545454545454545454 1/23=.043478260869565217391304347826086956 1/24=.041666666666666666666666666666666666 1/25=.040000000000000000000000000000000000 1/26=.038461538461538461538461538461538461 1/27=.037037037037037037037037037037037037 1/28=.035714285714285714285714285714285714 1/29=.034482758620689655172413793103448275 1/30=.033333333333333333333333333333333333 1/31=.032258064516129032258064516129032258

1/2 = .5 1/4 = .25 1/5 = .2 1/8 = .125 1/3 = .3333333333333333333333333 ...... 1/6 = .1666666666666666666666666........ 1/7 = .142857142857142857142857142857142857..... 1/7 = .142857142857142857142857142857142857.....

1/2 = .5 1/4 = .25 1/5 = .2 1/8 = .125 1/3 = .3333333333333333333333333 ...... 1/6 = .1666666666666666666666666........ 1/7 = .142857142857142857142857142857142857..... 1/9 = .01111111111111111 [whew] 1/10 = .1 1/13 = .076923076923076923076923076923076923076923.. 1/21 = .047619047619047619047619047619047619047619..

Conjecture: The decimal expansion of 1/n either stops, eventually repeats a single digit, or repeats in blocks of length 6. 1/11 = .090909090909090909090909 .....

1/11 = .090909090909090909090909 ..... Theorem: [MORE PARTS?] If n is neither even nor a multiple of 5, then (a) n goes into 9 or 99 or 999 or 9999 or 99999 (etc) (b) The smallest number of 9s that n goes into is the block size in the decimal expansion of 1/n. 1/11 = .090909090909090909090909 .....

Big Theorem on Decimal Expansions of 1/n Theorem: Let n be a natural number. Write n as n=2a·5b·m, where neither 2 nor 5 divides m. Then The decimal expansion of 1/n eventually repeats in blocks. m divides 9 or m divides 99 or m divides 999 or m divides 9999 , etc. The smallest number of 9s that m goes into is the block size in the decimal expansion of 1/n. The non-repeating part of this decimal expansion has length the larger of a and b.

Write n as n=2a·5b·m, where neither 2 nor 5 divides m. Developing Intuition [Generalization] Theorem: Let n be a natural number. Write n as n=2a·5b·m, where neither 2 nor 5 divides m. Then The decimal expansion of 1/n eventually repeats in blocks. The non-repeating part of this decimal expansion has length the larger of a and b.

[random event simulation] Developing Intuition [random event simulation]

How Likely Is It? Toss a fair coin 10 times and get exactly 5 heads and 5 tails? Toss a fair coin 1000 times and get exactly 500 heads and 500 tails? Not very likely Somewhat likely Quite likely Really likely

1 head out of 2 50% 5 heads out of 10 25% 500 heads out of 1000 3% From 490 to 510 heads out of 1000 49%

(Learning through crafted confusion) Paradoxes [Puzzles] (Learning through crafted confusion) The Strange Square

The Dishonest Bellhop:   Three men go into a hotel and the bellhop takes them upstairs to their room. He tells them that the room costs $30 and so each man gives the bellhop $10. The bellhop walks back down the stairs and gives the $30 to the manager at the front desk. But the manager tells the bellhop that the room is only $25 and gives the bellhop 5 single dollar bills to return the men upstairs. As he is walking up the stairs, the bellhop begins to think about the three men in the room and the five dollar bills in his hand and decides to keep two of the dollar bills for himself and just return three dollars to the three men. And that is what he does. He keeps $2 and gives each man back $1. Thus each man paid just $9 for the room, not the original $10. SO, of the original $30, each of the three men paid $9; that makes $27. The bellhop has $2. That makes $29. What happened to that other dollar?

The Strange Square Hide this one?

Think Back I get it. PPT built in shapes I don’t get it.

X3 = X • X • X X5 = X • X • X • X •X X3 • X5 = (X • X • X)(X • X • X • X •X) = X • X • X • X • X • X • X •X = X3+5 = X8 X3 • X5= X3+5 = X8

X-1 http://3.bp.blogspot.com/_JDH0sOofT-s/ScJhNHZWpII/AAAAAAAACgQ/tauU0Liq3AU/s1600/peanuts

I don’t get it.

Definition: X-1 = 1/X

X3 • X5= X3+5 = X8 X3 • X-1 = X3-1 = X2 (X • X • X) • (what?) = X • X (X • X • X) • ( 1/X ) = X • X X3 • (1/X) = X2 And so, X-1 is DEFINED to equal to 1/X.

Statement of the theorem CORE IDEAS in Mathematics Statement of the theorem Proof of the Theorem Conjecture Definition

Statement of the Theorem CORE IDEAS in Mathematics Statement of the Theorem Proof of the Theorem Conjecture Definition

Statement of the Theorem CORE IDEAS in Mathematics Statement of the Theorem Proof of the Theorem Conjecture Definition

Statement of the Theorem CORE IDEAS in Mathematics Statement of the Theorem Proof of the Theorem Conjecture Definition