The Man and the Theorem

A Greek coin showing Pythagoras

• Pythagoras was born on the Greek island of Samos in c. 475 BC

• He travelled to Egypt to learn mathematics and astronomy.

• Founded a school in Samos called the Semicircle.

• He founded a secret sect in Croton (Southern Italy)

A Greek stamp showing Pythagoras

• Women were allowed to join this sect

• The members were vegetarians but beans were excluded from their diet

• Clay tablets (1800 BC and 1650 BC) show that the Babylonians already knew about the Theorem

• The Egyptians could have used it to construct right angles when they build the pyramids

A Babylonian tablet

• Pythagoras was probably the first to prove the theorem.

• He is reputed to have proved the theorem while hiding in a cave from the tyrant Polycrates. The cave of

Pythagoras at the foot of Mount Kerki, in

Samos. • Legend has it that he sacrificed

an ox to thank the gods!

• The theorem states that in any right angled triangle ….

b c

a

The square of the hypotenuse is equal to the sum of the squares of the other sides.

b c

a

c2 = a2 + b2

• There are nearly 400 proofs of the theorem!

James A. Garfield 30th President of the

United States

• Among them is a proof by an American president.

• Area of red triangle: ½ a b

b

a

b

a

c

c• Area of blue triangle: ½ a b

• Area of green triangle: ½ c 2

• Area of trapezium:

½ (a + b)2

b

a

b

a

c

c

• Therefore

½ (a + b)2 = ½ a b + ½ a b + ½ c 2

(a + b)2 = 2a b + c 2

a 2 + b 2 + 2a b = 2a b + c 2

a 2 + b 2 = c 2

Area of square = c 2

Area of each triangle = ½ ab

Area of central square= ½ (a b) 2

Area of square = c 2

= 4x ½ ab + (a b)2

= 2ab + a 2 2ab + b 2

c 2 = a 2 + b 2

a

b c

Consider a cuboid of length a, width b and height c.

a

b c

We want to find the distance d from one corner to the other

d

x

x2 = a2 + b2

d 2 = x2 + c2 hence d 2 = a2 + b2 + c2

• There are cases when the lengths of the sides of a right-angled triangle have integral values

3

4

5• The 3, 4, 5 right-angled triangle

is such a case

The numbers 3, 4 and 5 are said to form a

Pythagorean Triple 52 = 32 + 42

• There are an infinite number of Pythagorean triples

• Here are two more examples …

5

12

13

257

24

finis