2. Basic Operations with Complex Numbers

by M. Bourne

Addition and subtraction of complex numbers works in a similar way to that of adding and subtracting surds. This is not surprising, since the imaginary number j is defined as `j=sqrt(-1)`.

Addition of Complex Numbers

Add real parts, add imaginary parts.

Subtraction of Complex Numbers

Subtract real parts, subtract imaginary parts.

Example 1- Addition & Subtraction

a. `(6 + 7j) + (3 − 5j) =`

`(6 + 3) + (7 − 5)j = 9 + 2j`

b. `(12 + 6j) − (4 + 5j) =`

`(12 − 4) + (6 − 5)j = 8 + j`

Continues below

Multiplication of Complex Numbers

Expand brackets as usual, but care with `j^2`!

Example 2 - Multiplication

Multiply the following.

a. `5(2 + 7j)`

b. `(6 − j)(5j)`

c. `(2 − j)(3 + j)`

d. `(5 + 3j)^2`

e. `(2sqrt(-9)-3)(3sqrt(-16)-1)`

f. `(3 + 2j)(3 − 2j)`

Multiplying by the conjugate

Example 2(f) is a special case.

`3 + 2j` is the conjugate of `3 − 2j`.

In general:

`x + yj` is the conjugate of `x − yj`


`x − yj` is the conjugate of `x + yj`.

Notice that when we multiply conjugates, our final answer is real only (it does not contain any imaginary terms.

We use the idea of conjugate when dividing complex numbers.

Division of Complex Numbers

Earlier, we learned how to rationalise the denominator of an expression like:


We multiplied numerator and denominator by the conjugate of the denominator, `3 + sqrt2`:




We did this so that we would be left with no radical (square root) in the denominator.

Dividing with complex numbers is similar.

Example 3 - Division

a. Express


in the form x + yj.

b. Simplify:



1. Express in the form a + bj:


2. Express in the form a + bj.



Search IntMath, blog and Forum

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Algebra Lessons on DVD


Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.