In the previous post How is that irrational numbers? we reached the point of discovery of a type of numbers that, that then Century V BC, were totally unknown, it was therefore, of some inexpressible numbers, the irrational numbers.
We say tormenting because remember friends that for the Pythagoreans, geniuses of the moment we refer to mathematics, everything could be expressed, and represented, with known numbers, that is, with integers and fractions.
But this was not possible with the diagonal of the square of side 1 (the simplest of the squares), since *there was no whole number, nor fraction, whose square was equal to 2 * As we mentioned in the previous post, therefore, the number that could assume that role would represent something like an inexpressible number, would be irrational.
Demonstration.
The method used by the Pythagoreans was Reduction to Absurd (remember the creator of the method, Hipócrates de Chios, already referred to in a previous post ([Hopócrates ¿Medical or Mathematical?](Https: // steemit.com/spanish/@josmar2511/hipocrates-medico-o-matematico)).
They began by supposing, the Pythagoreans, that there existed a fraction a / b whose square was equal to two, in as much:
Assumed hypothesis, which should lead to a conclusion that would represent an absurdity
Take as a fraction the simplest, or smallest, that is, the irreducible whose terms, numerator and denominator are cousins, meanwhile, they do not admit more denominator in common than unity. Therefore, a and b can not be the two even numbers.
Then if:
then
so
is even, because it is equal to "a" double. But only the square of a pair is even, so "a" is even.
If "a" is even, then it is a double, for example, of a number "c", so it could be written:
Let's go back, now, to equality
and replace "a" by its equivalent in
We now have the following
so that:
then
dividing both members by 2, it results:
so
It is a double, therefore:
is even, and since the square of a pair is only an even number, necessarily b is even.
The Pythagoreans then obtained, in the demonstration, that as well as b turned out to be even numbers, but this was absurd because they began by considering, in the supposed hypothesis, that both, a and b , they can not be even at the same time. The whole contradiction comes from having supposed, the absurd, that there was a fraction whose square was 2.
This already shows that there is no fraction whose square is equal to 2, that is, the diagonal of a square can not be represented as the quotient between two integers and those numbers that can not be expressed as the quotient between two integers are named of ¡IRRATIONALS!