Why do we need rational and irrational numbers?
All subsets of real numbers were created to extend the mathematical operations we can perform on them.
First set was natural numbers (
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 (
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 (
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
If we add rational and irrational numbers we get the whole set of real numbers (