# What was the need of extending the number system to Rational Numbers?

The need to extend the number system to rational numbers arose  due to the mathematical operations we perform on the integers or natural numbers.
We know  that  the operation of addition, subtraction and multiplication are closed for integers and natural numbers. It means that addition of two or more integers  is an integer, difference of two more integers is an integer and multiplication of two or more integers is an integer.Thus there was no need for any other number system for these operation to be defined.
While when we consider the operation of division, we find that  dividing an integer by another integer may or may not be an integer.
For example, $12÷3=4$
is an integer
while
is not an integer.
Therefore, we defined rational numbers as numbers which can be expressed in the form of $\frac{p}{q}$ where p,q are integers and q # 0. Such that all the above mentioned operations are closed in the group of rational numbers, with the exception when 0 is taken in the denominator.

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