Complex numbers

A complex number is a number that can be expressed in the form a + bi where a and b are real numbers and i is the imaginary unit, satisfying the equation i²=−1. The number a is called the real part and the number b is called the imaginary part of the complex number.

Any real number r can be expressed as a complex number r + 0i.

The complex number a + bi can be represented visually as follows:

Complex number

The phase and norm (modulus) are alternative ways to define a complex number.

Create a complex number

You create a complex number using the COMPLEX function or by entering the number directly in the form a+bi. COMPLEX(a,b) returns the number a + bi. For example, COMPLEX(2,3) returns 2+3i.

Basic mathematical operations

Multiplication or division of a complex number with a real number
A * r	both real and imaginary part of the complex number A are multiplied by the real number r.
A / r	both real and imaginary part of the complex number A divided by the real number r.
Addition or subtraction of a complex number with a real number
A + r	the real number r is added to the real part of the complex number A.
A − r	the real number r is subtracted from the real part of the complex number A.
Addition or subtraction of complex numbers
A + B	the corresponding parts of the two complex numbers are added.
A − B	the corresponding parts of the two complex numbers are subtracted.
Multiplication of complex numbers
A * B	The multiplication of two complex numbers is defined as follows: $$ (a+bi) (c+di) = (ac-bd) + (bc+ad)i $$
Division of complex numbers
A / B	The division of two complex numbers is defined as follows: $$ \frac{a + bi}{c + di} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i $$