Square root of 2#Proof by infinite descent

Euclid handed a proof that the square root of 2 is irrational. This is one of the most important statements of mathematics. The number-theoretic proof of Euclid is performed indirectly by contradiction and is considered the first proof by contradiction in the history of mathematics.

The below mentioned proof was given by Euclid in Book X of the Elements. But irrational proportions were already the Pythagorean Archytas known of Euclid's theorem proven already proved in more general terms. Previously it was believed, the worldview of the Pythagoreans had been made by the discovery of incommensurability in question, because they would have meant the whole of reality must be expressed by integer ratios. According to the current state of research but that is not true.

A geometric proof that diagonal and side of the square or in the regular pentagon can have no common measure leg had been BC discovered in the late 6th or early 5th century by the Pythagoreans Hippasos of Metapontum.

Evidence

Assertion

The square root of 2 is an irrational number.

Evidence

The proof is carried out indirectly by the method of proof by contradiction, that is, it is shown that the assumption that the square root of 2 is a rational number, leads to a contradiction (Latin: reductio ad absurdum ).

It is thus assumed that the square root of 2 is rational and thus can be represented as a fraction. It is also believed that and relatively prime integers, so the fracture is in abbreviated form:

This means that the square of the fraction is equal to 2:

Or converted:

Since an even number, there's just. It follows that the number is even.

The number can thus be represented by:

This is obtained with the above equation:

And from this by dividing by 2

With the same reasoning as before, it follows that and therefore is an even number.

Since and are divisible by 2, we get a contradiction to coprimality.

This contradiction shows that the assumption that the square root of 2 is a rational number is incorrect, and therefore the opposite must be true. This is the claim that is irrational, proven.

Generalization

The idea of ​​the proof of Euclid can be applied to the general case of the - th root of any natural number which is not a -th power, extend:

If no -th power (not represented as a natural number ), then it is irrational.

Proof: Instead of simple even-odd reasoning used here generally to the existence of a unique prime factorization of natural numbers. The proof is indirect again: Suppose that with natural numbers. It is to indicate that then is a -th power, that is, even being a natural number. First, follows by simple transformation that applies. Be an arbitrary prime number. In the prime factorization of or kick or exactly on the multiplicity or or. Then follows immediately, with definitely so. Since this holds for every prime, divides must be in fact, that is a natural number and is the -th power.

Simple consequence of the Irrationalitätssatz:

Is irrational for all natural numbers > 1 ( because n is not n- th power of a natural number can be> 1).