Definition of irrational numbers

Rational numbers have a correspondence with sequences with a recurring number. We might now wonder what happens with the decimal expressions that correspond to the sequences of digits without any regularity. The numbers associated with this kind of expressions are the irrational numbers.

Some irrational numbers are: $$$\sqrt{2}=1,4142135623730950488 \ldots$$$ $$$\pi=3,141592653589793238462\ldots$$$ $$$e=2,71828182845904523536\ldots$$$

We could give more digits, but we would see that there is no period, meaning that they are not rational in any case.

To verify if a number is rational or irrational the best option is not always to calculate its digits.

Let's study the number $$\sqrt{2}$$ to prove that it is not rational.

Let's see the expression $$\sqrt{2}=\dfrac{p}{q}$$ where $$p$$ and $$q$$ are integers without any factors in common. We multiply by $$q$$ and square the expression, obtaining as a result $$2q^2=p^2$$.

If we factorize it into prime numbers we see that on the left side there is an odd number of twos and that on the right side there is an even number.

We can observe that in the factorization of the square of an integer, all the prime factors appear an even number of times. Then, we can say that $$\sqrt{2}$$ is not rational.

To verify that the numbers $$\pi$$ and $$e$$ are not rational it is necessary to use other most complicated tools. The difference is based on the fact that $$\sqrt{2}$$ is a constructible irrational, while $$\pi$$ and $$e$$ are not.