# 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).

## 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

and

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:

5/(3-sqrt2)

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

5/(3-sqrt2)xx(3+sqrt2)/(3+sqrt2)

=(15+5sqrt2)/(9-2)

=(15+5sqrt2)/7

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

(3-j)/(4-2j)

in the form x + yj.

b. Simplify:

(1-sqrt(-4))/(2+9j)

### Exercises

1. Express in the form a + bj:

(4+sqrt(-16))+(3-sqrt(-81))

2. Express in the form a + bj.

sqrt(-4)/(2+sqrt(-9))

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