Why the Square Root of 2 is Irrational

The Square Root of 2

Is the square root of 2 a fraction?

Let us assume that it is, and see what happens.

If it is a fraction, then we must be able to write it down as a simplified fraction like this:

m/n

(m and n are both whole numbers)

And we are hoping that when we square it we get 2:

(m/n)2 = 2

which is the same as

m2/n2 = 2

or put another way, m2 is twice as big as n2:

m2 = 2 × n2

Have a Try Yourself

See if you can find a value for m and n that works!

Example: let us try m=17 and n=12:

m/n = 17/12

When we square that we get

172/122 = 289/144 = 2.0069444...

Which is close to 2, but not quite right

You can see we really want m2 to be twice n2 (289 is about twice 144). Can you do better?

 

Even and Odd

Now, let us take up this idea that m2 = 2 × n2

It actually means that m2 must be an even number.

Why? Because whenever we multiply by an even number (2 in this case) the result is an even number. Like this:

Operation Result Example
Even × Even Even 2 × 8 = 16
Even × Odd Even 2 × 7 = 14
Odd × Even Even 5 × 8 = 40
Odd × Odd Odd 5 × 7 = 35

And if m2 is even, then m must be even (if m was odd then m2 is also odd). So:

m is even

And all even numbers are a multiple of 2, so m is a multiple of 2, so m2 is a multiple of 4.

And if m2 is a multiple of 4, then n2 should be a multiple of 2 (remembering that m2/n2 = 2).

And so ...

n is also even

But hang on ... if both m and n are even, we should be able to simplify the fraction m/n.

Example: 2/12 can be simplified to 1/6

But we already said that it was simplified ...

around and around

... and if it isn't already simplified, then let us simplify it now and start again. But that still gets the same result: both n and m are even.

Well, this is silly - we can show that both n and m are always even, no matter that we have simplified the fraction already.

So something is terribly wrong ... it must be our first assumption that the square root of 2 is a fraction. It can't be.

And so the square root of 2 cannot be written as a fraction.

Irrational

We call such numbers "irrational", not because they are crazy but because they cannot be written as a ratio (or fraction). And we say:

"The square root of 2 is irrational"

It is thought to be the first irrational number ever discovered. But there are lots more.

Reductio ad absurdum

By the way, the method we used to prove this (by first making an assumption and then seeing if it works out nicely) is called "proof by contradiction" or "reductio ad absurdum".

Reduction ad absurdum: a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an absurd result. (from Wikipedia)

History

Many years ago (around 500 BC) Greek mathematicians like Pythagoras believed that all numbers could be shown as fractions.

And they thought the number line was made up entirely of fractions, because for any two fractions we can always find a fraction in between them (so we can look closer and closer at the number line and find more and more fractions).

Example: between 1/4 and 1/2 is 1/3. Between 1/3 and 1/2 is 2/5, between 1/3 and 2/5 is 3/8, and so on.

(Note: The easy way to find a fraction between two other fractions is to add the tops and add the bottoms, so between 3/8 and 2/5 is (3+2)/(8+5) = 5/13).

So because this process has no end, there are infinitely many such points. And that seems to fill up the number line, doesn't it?

And they were very happy with that ... until they discovered that the square root of 2 was not a fraction, and they had to re-think their ideas completely!

Conclusion

The square root of 2 is "irrational" (cannot be written as a fraction) ... because if it could be written as a fraction then we would have the absurd case that the fraction would have even numbers at both top and bottom and so could always be simplified.