Euclid's Proof that √2 is Irrational

 

DRAFT

 

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

Proof by Contradiction

The proof was by contradiction. In a proof by contradiction, the contrary is assumed to be true at the start of the proof. After logical reasoning at each step, the assumption is shown not to be true.

For example, I can prove you can't always win at checkers.

Let us start with the contrary: you can always win at checkers.

I now set up a board, let you take the first move, then turn the board around and let you take the opponent's move. Both sides cannot win, and you are playing both sides!

So the idea that you can always win is absurd, so I have proven that you cannot always win.

In this case, Euclid assumed √2 was a rational number. (A rational number is a number that is expressed in the form p/q where p and q are integers and q is not equal to zero.) Euclid assumed √2 was a rational number equal to p/q.

Rational Numbers and Even Numbers

In order to understand the proof, one should bear in mind the following facts about rational numbers and even numbers:

* Any integer multiplied by 2 gives an even number. The definition of an even number is it is divisible by 2. Example, 2×3=6, 6 is an even number. 2×16=32, 32 is an even number.

* The square of an even number is always an even number. The converse is also true. That is, if the number is even and a perfect square, then the square root of the number is also even.

For example, the square of 14 is 196. 14 is an even number, and so is 196. Similarly, 256 is an even number and also a perfect square. The square root of 256 is 16, which is also an even number.

* Rational numbers or fractions can be simplified and be expressed in the simplest form.

For example, 16/64 is the same as ¼. In this case, we have cancelled the common factor 16 in the numerator and the denominator. Similarly, 17/85 can be written as 1/5. Here, we have first expressed the numerator and the denominator as a product of two numbers, one of which is 17. The numerator can be written as 17×1 and the denominator as 17×5. By canceling the common factor 17 in the numerator and the denominator, the rational number is expressed as 1/5.

* It is impossible to continue simplifying a rational number indefinitely. At some stage, the numerator and the denominator would be co-prime, that is, they would have no common factors other than 1. The rational number can no longer be simplified.

For example, 28/100 can be simplified and written as 14/50. It can further be simplified and written as 7/25. But thereafter, the rational number cannot be simplified since 7 and 25 are co-primes.

The Proof

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

√2=p/q

Squaring both sides,

2=p²/q²


The equation can be rewritten as


2q²=p²


From this equation, we know p² must be even (since it is 2 multiplied by some number). Since p² is an even number, it can be inferred that p is also an even number.


Since p is even, it can be written as 2m where m is some other whole number. This is because the definition of an even number is it can be written as 2 multiplied by a whole number. Substituting p=2m in the above equation:

2q²=p²=(2m)²=4m²
or
2q²=4m²


Dividing both sides of the equation by 2:


q²=2m²

By the same reasoning as before, q² is an even number (since it is written as 2 multiplied by some number). So q is an even number. Let q=2n where n is some whole number. We had assumed √2 to be equal to p/q. So:

√2=p/q=2m/2n


By canceling 2 in the numerator and the denominator of the Right hand side,

√2=m/n

We now have a fraction m/n simpler than p/q.

However, we now find ourselves in a position whereby we can repeat exactly the same process on m/n, and at the end of it, we can generate a simpler function, say g/h. This fraction can be put through the same process again, and the new fraction, say, e/f will be simpler again.

But we know that rational number cannot be simplified indefinitely. There must always be a simplest rational number and the original assumption that √2 is equal to p/q does not obey this rule.

So it can be stated that a contradiction has been reached.

If √2 could be written as a rational number, the consequence would be absurd. So it is true to say that √2 cannot be written in the form p/q. Hence √2 is not a rational number.


Thus, Euclid succeeded in proving that √2 is an Irrational number.