# Definition

A number of the form x + iy, where x, y R, and i = is called a complex number and “i” is called iota.

A complex number is usually denoted by z and the set of complex number is denoted by C, i.e.,

C = {x + iy; x R, y R, i = }

For example, 5 + 3i, –1 + i, 0 + 4i, 4 + 0i, etc. are complex numbers.

Notes:
• For any positive real number a, we have .
• The property  is valid only if at least one of a and b is non-negative. If aand b are both negative then .
• i4n = 1, i4n + 1 ii4n + 2 = –1, i4n + 3 = –i, where n is any integer.
• If x and y are two real numbers, then anumber of the form z = x iy is called acomplex number. Here, “x” is called the real part of z and “y” is known as the imaginary part of z. The real part of z is denoted by Re(z) and the imaginary part by Im(z).
• A complex number z is purely real if its imaginary part is zero, i.e., Im(z) = 0 and purely imaginary if its real part iszero, i.e., Re(z) = 0.