What is a rational number?

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A rational number is every number that can be written as a fraction, in which the numerator is a whole number and the denominator is a whole number different from zero. The set of rational numbers is called `QQ`. Here are some examples:

  • All the whole rational numbers. For instance: `-2" ; "0" ; "3" ; "45`
  • All the finite tenths are rational. For instance: `-3,5" ; "4,9" ; "8,65`
  • All the infinite tenths, which are periodic, are rational. For instance: `-1,(9)" ; "52,12(43)`

It should also be noted that the set of rational numbers (`QQ`) includes all numbers in the set of whole numbers (` ZZ`), which in turn includes all numbers in the set of natural numbers (`NN`). This inclusion can be seen in the following scheme:

set of rational numbers

So, what are rational numbers? In other words, a rational number is a number that is represented by a finite decimal part or by repeating decimals. Repeating decimals are numbers whose decimal parts are composed of infinitely-repeating sequences of digits. That is, when you divide two whole numbers, you get an integer or a decimal number. In the case of a decimal number, it can have a finite set of digits, or an infinite set. In the case of infinite, if they were obtained by dividing two whole numbers then they are always periodic. This means that there is always a set of digits in the decimal part of the number that is repeated periodically. I draw your attention to the fact that sometimes this set of figures that is repeated is so large that it is not even visible on the most modern calculating machines. For example, try the following division: `12-: 51`. The repetition of digits, that is, the period, is only visible from the 17th digit!

What is an irrational number?

number pi

Every real number that is not rational is irrational. So, it is a number that cannot be written as a fraction with a whole numerator and denominator.

  • Some of the very well known mathematical constants are irrational. For instance: `pi = 3,14159" ; "e = 2,71828`
  • Some square roots are irrational. For instance: `sqrt (2)" ; "sqrt (3)`
  • All the nom-periodic infinite tenths are irrational. For instance: `-4,8542576..." ; "7,50234529...`

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