# Why does sqrt2 sqrt2 sqrt2 2

The following evidence was already known in ancient times. It was passed down from Euclid (more precisely: Euclides of Alexandria). It is guided `` indirectly '', i.e. we initially assume on a trial basis that its opposite is true.

So suppose that 2 is a rational number. Then it can be called Fractional number of the form, where and are natural numbers. We assume that & # 223 this fraction has already been completely reduced, so that & # 223 and are coprime (i.e. have no common factor). In particular, then and not both even numbers be.

Since the square of 2 is 2, it follows from (1) i.e. 2 is twice as big as 2. We can also write that as. It follows that & # 223 2 divided by 2, so it is an even number. Hence there is also a just Number (because the square of an even number is even, the square of an odd number is odd).

The whole point now is that & # 223 2 is not only even, but even has 4 as a divisor: Since is even, it can be written as, where is a natural number. Consequently, in (3) is inserted, from which we conclude that & # 223 2 twice as big as 2 (in formulas: 2 = 2 2). That shows that & # 223 2 a just Number is, and therefore also.

Overall, we have concluded that & # 223 is both and even numbers are, and that contradicts the observation made above that this is not the case.

From the assumption that 2 is rational, we have constructed a (logical) contradiction, which proves:

2 cannot be written as a fraction of the form (1) and is therefore an irrational number.

comment:

2 is the length of the diagonal of the unit square (according to the Pythagorean theorem: 12 + 12 = Diagonal2 ). The irrationality of 2 shows that the ratio of diagonal to side length in the square is not rational, i.e. that even the simplest geometrical figures cannot be constructed by "juxtaposing" copies of a "smallest elementary length". This realization probably triggered one of the first fundamental crises in mathematics in the fifth century BC.