Complex Number System

The number system is the gradual development from natural numbers to integers, from integers to rational numbers and from rational numbers to the real numbers.

This real number system is not sufficient to solve equations of the form x² + 9 = 0. There does not exist any real number which satisfies x² = −9. 

The mathematical need to have solutions for equations of the above form led to extend the real number system to a new kind of number system that allows the square root of negative numbers.

Square root of −1 is denoted by the symbol i, called the imaginary unit.

A complex number is of the form a+ib where a and b are real numbers and i is called the imaginary unit.

i² = −1.

If z = a + ib, then a is called the real part of z, denoted by Re(z) and b is called the imaginary part of z and is denoted by Im(z).