# Euclid's Proof that √2 is Irrational

Euclid proved that √2 (the square root of 2) is an irrational number.

He used a proof by contradiction.

First Euclid **assumed** √2 was a rational number.

A rational number is a number that can be in the form p/q where p and q are integers and q is not zero.

He then went on to show that in the form p/q it can **always be simplified**.

But we can't go on simplifying an integer ratio forever, so there is a **contradiction**.

So √2 must be an irrational number.

We will go into the details of his proof, but first let's take a look at some useful facts:

## Rational Numbers and Even Numbers

First, let's look at some interesting facts about even numbers and rational numbers:

Any integer multiplied by 2 gives an even number.

- 2×3 = 6, 6 is an even number.
- 2×16 = 32, 32 is an even number.
- etc

The square of an even number is always an even number (multiplying two even numbers gives an even number).

Likewise if a number is even and is a square of an integer, then its square root must be even.

In fact the square root of 256 is 16.

Rational numbers or fractions must have a **simplest** form.

We can go further and simplify it to 7/25.

But it cannot be simplified further since 7 and 25 have no common factors.

## The Proof

Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.

^{2}/q

^{2}

^{2}= p

^{2}

p^{2} must be even (since it is 2 multiplied by some number).

Since p^{2} is even, then p is also even (square root of a perfect square is even).

Since p is even, it can be written as 2m where m is some other whole number (because an even number can be written as 2 multiplied by a whole number).

Substituting p=2m in the above equation:

^{2}= p

^{2}

^{2}= (2m)

^{2}

^{2}= 4m

^{2}

^{2}= 2m

^{2}

q^{2} is an even number (since it is written as 2 multiplied by some number).

So q is an even number.

Since q is even, it can be written as 2n where n is some other whole number.

Now we have **p = 2m** and **q = 2n** and remember we assumed that √2 = p/q:

**simpler than**p/q.

But now we can **repeat the whole process again** using m/n and simplify it to something else (say g/h).

We can then do that again ... and again ... and again ... !

But a rational number cannot be simplified forever. There must eventually be a **simplest** rational number, but in our case there is not: we have a **contradiction**!

So something is definitely wrong here. √2 cannot be written as p/q or it can be simplified **forever**.

So √2 cannot be rational and so must be irrational.

**Infinite Descent**because it uses the fact that there is no infinite sequence of decreasing positive integers, and is a special case of Proof by Contradiction.