# Rational numbers

A rational number is a number that can be written in the form of a common fraction of two integers. In other words, it is a number that can be represented as one integer divided by another integer. The following are some examples.

## Properties of rational numbers

Rational numbers, as a subset of the set of real numbers, shares all the properties of real numbers. Below are some specific properties of rational numbers, some of which differentiate them from irrational numbers.

### Closure

One of the properties of rational numbers that separates them from their irrational counterpart is the property of closure. Rational numbers are closed under the operations of addition, subtraction, multiplication, and division. This means that performing any of these operations using two rational numbers will always result in another rational number:

2 + 2 = 4

2 - 2 = 0

2 × 2 = 4

2 ÷ 2 = 1

All of the results are rational numbers, and the result of these operations will always be rational given that the initial two values are rational numbers. This is not true of irrational numbers, which can either result in rational or irrational numbers depending on the original values.

### Additive inverses

All rational numbers have an additive inverse. Given a rational number a/b, its additive inverse is:

Also, given a non-zero rational number, a/b, its multiplicative inverse is:

The multiplicative inverse is also known as the reciprocal.

Below are some other general things to note about rational numbers.

- Rational numbers can be written in the form of a terminating decimal (the decimal ends) or a repeating decimal (the decimal does not end but has repeating digits).
- Non-terminating decimals are not rational numbers because they cannot be expressed in the form of a common fraction.
- The denominator of the common fraction used to express a rational number cannot be 0.
- All integers are rational numbers since the denominator of the common fraction can be 1.

Examples

1. The examples used above can all be converted into either terminating decimals or repeating decimals:

2. The square root of 2 is not a rational number because its decimal never ends so we have no way to express it in the form of a common fraction:

## Rational numbers and other number sets

There are many different sets of numbers that are commonly used throughout mathematics. Many of them overlap, and it can be helpful to know the various differences between number sets and how they relate to each other.

The set of rational numbers is typically denoted as Q. It is a subset of the set of real numbers (R), which is made up of the sets of rational and irrational numbers.

The set of rational numbers also includes two other commonly used subsets: the sets of integers (Z) and natural numbers (N). Rational numbers include all of the integers as well as all the values between each integer, while integers include all of the natural numbers in addition to their negative values.

The following image depicts the relationships described above (excluding irrational numbers):