The Nature of Mathematics Michael Randy Gabel Ph

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

A Little QUIZ (1) Name a song or the title of piece of music. (2) Name a theorem in mathematics.

OR ?

CORE IDEAS in Mathematics Statement of theorem

Theorem: Consider a right triangle. 90 Attach a square to each side: ---->>> Then: AREA (I) + AREA (II) = AREA (III). 2 2 2 = c a + b

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

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)

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

Pythagoras - Revisited Think

Observe

Big Theorem on Decimal Expansions of 1/n Theorem: Let n be a natural number. Write n as n=2 a· 5 b·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 9999 , etc. § The smallest number of 9 s 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.

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% 5000 heads out of 10000 1% From 490 to 510 heads out of 1000 49%

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

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

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

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

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