### Pythagorean Theorem

```Pythagorean
Theorem
• Chapter 12
• Fred Stenger
Larry L. Harman
Everyone knows the Theorem
but who know him
 Pythagoras (569-500 B.C.E.) was born in Greece, and did
much traveling through Egypt, learning, among other
things, mathematics.
 Not much more is known of his early years.
 Pythagoras became famous by founding a group, the
Brotherhood of Pythagoreans, which was devoted to the
study of mathematics.
 The group was almost cult-like in that it had symbols,
rituals and prayers.
 In addition, Pythagoras believed that "Numbers rule the
universe,"and the Pythagoreans gave numerical values to
many objects and ideas. These numerical values, in turn,
were endowed with mystical and spiritual qualities.
a2 + b2 = c 2
Because the Theorem appears so
natural, it most likely appeared in many
other cultures.
The Theorem has been found in many
parts of the ancient world including:
Mesopotamia, Egypt, India, China
Oldest reference is from India, in the
first millennium BC.
a2 + b2 = c 2
Pythagoreans recognized irrational
numbers, but refused to accept them –
(square root spiral).
One of the proofs uses the concept of
area by squaring the length of the lines
of the original right triangle.
Euclid’s Proof of a2 + b2 = c2
http://www.cut-theknot.org/pythagoras/morey.shtml
Algebraic Proof
The embedded square.
President Garfield is even credited with a
proof.
Can you find the geometric proof.
Geometric Proof
The simplest proof of all. Uses the concept
of similar triangles.
Pythagorean Triples
 Integer solutions to a2 + b2 = c2
32 + 4 2 = 5 2
 A formula where m>n,
a= m2-n2
b= 2mn
c= m2+n2
Pythagoras and Fermat
Pythagorean Theorem led to Fermat’s
conjecture that there were no solutions to
an + bn = cn
when n is greater than 2.
This Theorem was not actually proved until
1993 by Andrew Wiles.
Timeline
From early times to about 600 BC
various cultures appeared to have
discovered the concept of a2 + b2 = c2.
569 to 500 BC – Era of Pythagoras.
~300 BC -Euclid’s Elements
1600’s-Fermat
References
 http://www.geom.uiuc.edu/~demo5337/Group3/hist.h
tml
 http://www.cut-the-knot.org/pythagoras/morey.shtml
 Berlinghoff, william P and Fernando Q. Gouvea.
Math through the Ages – A Gentle History for
Teachers and Others. Farmington: OxtonHouse,
2002.
 http://mathforum.org/library/drmath/view/55811.html
```