• 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.
What do you know about
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.
What do you know about
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
Algebraic Proof
The embedded square.
President Garfield is even credited with a
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.
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
 Berlinghoff, william P and Fernando Q. Gouvea.
Math through the Ages – A Gentle History for
Teachers and Others. Farmington: OxtonHouse,

Pythagorean Theorem