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 $\sqrt{2}$ . This number is irrational. When you draw a square and it will have a 1 unit side, the length of the diagonal will be $\sqrt{2}$ .

[edit] Proof that the square root of 2 is not rational

The number $\sqrt{2}$ is not rational. Here is the proof. We will show that the number is not rational. We will see that if we believe that $\sqrt{2}$ 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 $a/b=\sqrt{2}$ .
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 $\frac{2}{4}$ we would write $\frac{1}{2}$ ).
If we square both sides then we get a² / b² = 2 and a² = 2 b².
The left side is $2 b 2$ . This number is even. So the left side must be even too. So $a 2$ 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: $a = 2 k$ .
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 $2 b 2 = 4 k 2$ .
We divide both sides by 2. So $b 2 = 2 k 2$ . 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.