The **rational numbers** are all numbers that can be expressed as a **fraction**, that is, as the quotient of two whole numbers. The word ‘*rational*‘derives from the word’*reason*‘, which means proportion or quotient. For example:* 1, 50, 4.99, 142*.

In the **mathematical operations** that are done daily to solve everyday questions, almost all the numbers that are handled are rational, since the category includes all whole numbers and a large part of those that have decimals.

Both rational fractional numbers and **irrational** (its counterpart) are infinite categories. However, these behave differently: rational numbers are understandable and, insofar as they can be represented by fractions, their value can be approximated with a simply mathematical criterion, this is not the case with irrational numbers.

### Examples of rational numbers

Rational numbers are listed here as an example. In the cases of being these in turn **fractional numbers**, its expression is also indicated as a quotient:

- 142
- 3133
- 10
- 31
- 69.96 (1749/25)
- 625
- 7.2 (36/5)
- 3,333333 (10/3)
- 591
- 86.5 (173/2)
- eleven
- 000,000
- 41
- 55.7272727 (613/11)
- 9
- 8.5 (17/2)
- 818
- 4.52 (113/25)
- 000
- 11.1 (111/10)

Most of the operations that are carried out between rational numbers necessarily result in another rational number: this does not happen, as we have seen, in all cases, as in the case of the radication operation and neither of the potentiation.

Other typical properties of rational numbers are the **equivalence and order relations **(the possibility of making equalities and inequalities), as well as the existence of inverse and neutral numbers.

The three most important properties are:

- The associative
- The distributive
- The commutative

These are simply demonstrable from the inherent condition of all rational numbers to be able to be expressed as quotients of whole numbers.

### Recurring numbers

A very particular category of rational numbers, which is often misleading, is that of **periodic numbers**: these are made up of infinite numbers but can be expressed as a fraction.

There are many recurring issues. The simplest of them is the one born from **divide the unit into three equal parts**, equivalent to 1/3 or 0.33 plus infinite decimal places: not because of its infinity condition it becomes irrational.

### Irrational numbers

The** irrational numbers** They are those that fulfill the most recognized functions for the purposes of mathematics and geometry: undoubtedly the most important number in this science of ideal figures is the **number pi (π)**, which expresses the length of the perimeter of a circle whose diameter (that is, the distance between two opposite points) is equal to 1.

The number pi is approximately **3.14159265359**, and the prolongation can be extended to infinity to meet its definition of inability to express itself as a fraction.

The same happens with the length of the diagonal of a square taking each of the sides of that square as equal to unity: that number is the square root of 2, which is 1.41421356237. Both numbers, as the most important of the irrationals, have multiple functions derived from their primary role in geometry.