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


(m and also n space both entirety numbers)

And we are hoping that when we square that we acquire 2:

(m/n)2 = 2

which is the exact same as

m2/n2 = 2

or put an additional way, m2 is twice as large as n2:

m2 = 2 × n2

Have a try Yourself

See if friend can find a worth for m and also n the works!

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

m/n = 17/12

When we square that we get

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

You are watching: Square root -2

Which is close to 2, yet not rather right

You can see we really desire m2 to be double n2 (289 is about twice 144). Have the right to you execute better?

Even and also Odd

Now, let us take increase this idea the m2 = 2 × n2

It actually way that m2 have to be an also number.

Why? since whenever us multiply through an even number (2 in this case) the result is an also number. Like this:

Operation an outcome 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 should be even (if m to be odd then m2 is additionally odd). So:

m is even

And all even numbers room a lot of of 2, therefore m is a lot of of 2, for this reason m2 is a multiple of 4.

And if m2 is a multiple of 4, climate n2 have to be a multiple of 2 (remembering that m2/n2 = 2).

And for this reason ...

n is likewise even

But cave on ... If both m and n are even, us should be able to simplify the fraction m/n.

But we already said the it was streamlined ...


... And if it isn"t currently simplified, climate let us simplify the now and also start again. However that still gets the exact same result: both n and m are even.

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

So miscellaneous is terribly wrong ... It must be our first assumption that the square source of 2 is a fraction. It can"t be.

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


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

"The square root of 2 is irrational"

It is assumed to be the an initial irrational number ever discovered. Yet there are lots more.

Reductio ad absurdum

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

Reduction advertisement absurdum: a type of logical discussion where one presume a insurance claim for the sake of argument and also derives one absurd or ridiculous outcome, and also then concludes the the original claim must have actually been wrong together it led to an absurd result. (from Wikipedia)


Many years back (around 500 BC) Greek mathematicians prefer Pythagoras believed that all numbers can be displayed as fractions.

And they thought the number heat was made up entirely of fractions, because for any two fractions us can constantly find a portion in in between them (so we can look closer and closer at the number line and find an ext and much more fractions).

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

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

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So due to the fact that this procedure has no end, there room infinitely countless such points. And also that appears to to fill up the number line, doesn"t it?

And castle were an extremely happy with that ... Till they uncovered that the square root of 2 was not a fraction, and also they had to re-think their principles completely!


The square source of 2 is "irrational" (cannot be written as a fraction) ... Due to the fact that if it could be written as a portion then us would have actually the absurd instance that the fraction would have also numbers in ~ both top and also bottom and also so could constantly be simplified.