Irrational numbers and what they are.

The person who invented irrational number was drowned in the sea!. So, be careful to talk about the irrational numbers ;)

Source

It feels good to understand why √2 is an irrational number.

Proof of the irrationality of √2

The reductio ad absurdum proof of the irrationality of √2 is as follows:

1. Take a right triangle whose sides are 1 unit in length
2. By the Pythagorean theorem, the diagonal is √2
3. Suppose that √2 is a natural number, or that it is the ratio of two natural numbers. In both cases, it can be represented as the ratio of two natural numbers, √2=m/n
4. Suppose that m/n has been reduced to its lowest common form by division
5. It follows that either m and n are both odd, or that m is odd and n is even, or that m is even and n is odd (if not, we could reduce m/n even further by dividing both numbers by 2)
6. Square both sides of √2=m/n, so that 2=m2/n2
7. Then 2n2=m2, so that m2 is even, and therefore m is even
8. If m is even, then m=2x, where x is some other natural number
9. Squaring this, it follows that m2=4x2=2n2
10. It follows that n2=2x2, and therefore n2 is even, which means that n, being a natural number, must be even
11. So we've reached a contradiction: although we assumed that m and n cannot both be even, it now turns out they both are. It therefore follows that √2 cannot be expressed as the ratio of two natural numbers, and must therefore be in another class of numbers