A Suggested Proof of the Pythagorean Theorem

Image result for Pythagoras. Size: 85 x 101. Source: classicalwisdom.com

Pythagoras, 570 BC to 495 BC

We shall state the Pythagorean Theorem as follows:

Given the equation

x2+y2=z2, where x and y are integers,    (1)

  1. x2+y2=z2 has integer solutions for z if y>x>0. 
  2. x2+y2=z2 has non-integer solutions for z if y>x>0.

We shall call Eq 1 “Pythagoras’ Equation”.

Following is a suggested proof of the Pythagorean Theorem.

We begin by reordering Pythagoras’ Equation:

z2=x2+y2.                                                (2)

Let us define the positive integer p to be such that

y=x+p.                                                    (3)

Then

z2=x2+(x+p)2                                               (4a)

    = x2+(x2+2xp+p2)                                   (4b)

    =2x2+2xp+p2.                (4c)

Thus

z=(2x2+2xp)+p2]1/2.                (5)

  1. Using x=1 and p=1, z=51/2 and z is non-integer.
  2. Using x=2 and p=1, z=131/2 and  z is non-integer.
  3. Using x=3 and p=1, z=251/2=5 and z is integer.

QED.