Addition and Subtraction of rational numbers are carried out in the same way as the arithmetic operations like addition and subtraction on integers and fractions. Remember that the set of integers is defined as

From this it is evident that the set of natural and whole numbers are sub sets of integers, that is, ℕ ⊆ ℤ and W⊆ ℤ.

Using the set of integers, we define the set of rational numbers as follows:

Definition

A rational number is a number that can be expressed in the form

where: p and q are integers, q ≠ 0. The set of rational numbers, denoted by ℚ is the set described by

Do not confuse rational numbers with that of fractions. Fractions include only positive numbers, while rational numbers include both positive and negative numbers. Fractions are a part of rational numbers, while rational numbers are a broad category that includes other types of numbers. So we say that the set of fractions is a sub set of the set of rational numbers. The following are some examples of rational

Note that integers are included in the set of rational numbers. This is because all integers such as 5, 0 and –9 can be expressed as