Proof that Root Two is Irrational: How to Prove that Root Two is Irrational Using Proof By Contradiction

Proving that the square root of two is irrational is an example of a proof by contradiction.

The basic method is to suppose that square root two is rational and then show that this leads to a contradiction (an equation that is obviously not true).

A rational number is one that can be written as a ratio of two integers (a fraction):

Rational number = p / q , where p and q are integers (whole numbers).

An irrational number can't be written in this way.

1. Suppose that square root two is rational

√2 = p / q

where p and q are integers. Assume that this fraction is expressed in its simplest form: i.e.,p and q have no common factors (this will be important later).

2. Show that this leads to a contradiction

First, square both sides:

2 = p² / q²

Rearrange:

q² = 2 p²

Now, we can draw some conclusions about p and q.

q² must be even, because it is equal to 2 times an integer.

Therefore, q must also be even. (Because: even x even = even; odd x odd = odd)

Because q is even, we can write q it as 2 times an integer. We call this integer k.

q = 2 k

Therefore,

q² = (2 k)² = 4 k²

We put this back into the equation q² = 2 p² to get:

4 k² = 2 p²

Rearrange to get an equation for p²

p² = 2 k²

This means that p² is even (because it is equal to two times an integer). Therefore, p is also even.

This is where the contradiction lies. If p is even, and q is even, then they both have q as a factor. Our original assumption stated that p and q had no common factors.

Our assumption that the square root of two is rational has led to a contradiction. Therefore, it must be untrue.

We have proved that the square root of two is irrational.