Proof that the square root of 2 is irrational

Let the number whose square is 22 == 2\sqrt{2}

Suppose that 2\sqrt{2} is the ratio of two whole numbers with no common fators aa and bb:

2=ab\sqrt{2} = \frac{a}{b}

Squaring both sides of the equation gives:

2=a2b22 = \frac{a^2}{b^2}

Multiplying both sides by b2b^2 gives:

2b2=a22b^2 = a^2

2b22b^2 must be even because 22 is a factor. Therefore a2a^2 must be even.

The square root of any even number is even. Therefore aa must be even.

If aa is even it must contain 22 as a factor. Therefore a=2da = 2d where dd is some whole number.

Substituting 2d2d for aa we get:

2b2=(2d)22b^2 = (2d)^2 2b2=4d22b^2 = 4d^2 b2=2d2b^2 = 2d^2

Applying the same logic we used to show that aa is even we can see that bb is also even.

If aa and bb are both even 22 must be a common factor. Yet we eliminated common factors at the start.

This contradiction prooves that 2\sqrt{2} cannot be the ratio of two whole numbers and must be irrational.


  • Mathematics for the Nonmathematician (pp 66-68) - Morris Kline
Tags: Mathematics