# Why do we need rational and irrational numbers?

Apr 3, 2016

See explanation.

#### Explanation:

All subsets of real numbers were created to extend the mathematical operations we can perform on them.

First set was natural numbers ($\mathbb{N}$) .

In this set only addition and multiplication could be done.

To make substraction possible people had to invent negative numbers and expand natural numbers to integer numbers ($\mathbb{Z}$)

In this set multiplication, addition and substraction were possible but some division operatins could not be done.

To extend the range to all 4 basic operations (addition, substraction, multiplication and division) this set had to be extended to set of rational numbers ($\mathbb{Q}$)

But even in this set of numbers not all operations were possible.

If we try to calculate the hypothenuse of an isosceles right triangle, whose catheti have length of $1$ we get a number $\sqrt{2}$ which is an example of irrational number.

If we add rational and irrational numbers we get the whole set of real numbers ($\mathbb{R}$)