Real number
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A real number is a rational or irrational number. Usually when people say "number" they mean "real number". There are also numbers which are not real. Those are called complex numbers.
One real number that is not rational is 
. This number is irrational. When you draw a square and it will have a 1 unit side, the length of the diagonal will be .
[edit] Proof that the square root of 2 is not rational
The number is not rational. Here is the proof. We will show that the number is not rational. We will see that if we believe that is rational, we get some kind of impossibility. We then conclude that the number is 'irrational' (not rational).
- Let's think that the number is rational. So there are some numbers a,b such that
. - We can say that a or b is odd. If a and b were even, then we could shorten the fraction (for example, instead of writing
we would write 
). - If we square both sides then we get a2 / b2 = 2 and a2 = 2 b2.
- The left side is 2b2. This number is even. So the left side must be even too. So a2 is even. If you square an odd number you will get an odd number. And if you square an even number you will get an even number. So a is even.
- Because a is even, we can write: a = 2k.
- We put the equation from the point 3. into 6. We get 2b2 = (2k)2
- We can use an exponentiation rule (see the article) - we get 2b2 = 4k2.
- We divide both sides by 2. So b2 = 2k2. This means that b is even.
- In 2 we said that a is odd or b is odd. But in 4 we said that a is even. In 7 we said that b is even. This is impossible.
We found that it is impossible that is a rational number. So is irrational.
