Irrational Numbers

The decimal expansion of a rational number is either terminating or is a non-terminating and repeating decimal. A decimal expansion which is neither terminating nor is repeating represents an irrational number.

The rational numbers are inadequate to measure all lengths in terms of a given unit. This inadequacy necessitates the extension of rational numbers to irrationals (which are not rational). For example, consider a square ABCD, each of whose sides is 1 unit. The diagonal BD is of length √2 units. Now, √2 is not a rational number, as there is no rational, whose square is 2.

Real Numbers

The system of numbers consisting of all rational and irrational numbers is called the system of real numbers.