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# 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. See also Simplest Radical Form. 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

## Multiplication of Complex Numbers

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

### Example 2 - Multiplication

Multiply the following.

a. 5(2 + 7j)

5(2 + 7j) = 10 + 35j

b. (6 − j)(5j)

(6 − j)(5j)

= 30j − 5j^2

= 30j − 5(−1)

= 5 + 30j

c. (2 − j)(3 + j)

(2 − j)(3 + j)

= 6 − 3j + 2j − j^2

= 6 − j − (−1)

= 6 − j + 1

= 7 − j

d. (5 + 3j)^2

We apply the algebraic expansion (a+b)^2 = a^2 + 2ab + b^2 as follows:

(5 + 3j)^2 = 25 + 2(5)(3)j + 9(j^2)

= 25 + 30j + 9(-1)

= 25 + 30j - 9

= 16 + 30j

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

(2sqrt(-9)-3)(3sqrt(-16)-1)

=(2j(3)-3)(3j(4)-1)

=(6j-3)(12j-1)

=72(j^2)-36j-6j+3

=-69-42j

f. (3 + 2j)(3 − 2j)

(3 + 2j)(3 − 2j)

= (3)^2 − (2j)^2

= 9 − 4j^2

= 9 + 4

= 13

### 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)

To simplify the expression, we multiplied numerator and denominator by the conjugate of the denominator, 3 + sqrt2 as follows:

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 by a complex number is a similar process to the above - we multiply top and bottom of the fraction by the conjugate of the bottom.

### Example 3 - Division

a. Express

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

in the form x + yj.

The conjugate of 4 − 2j is 4 + 2j. We multiply the top and bottom of the fraction by this conjugate.

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

=(12+6j-4j-2j^2)/(16-4j^2)

=(12+2+6j-4j)/(16+4)

=(14+2j)/20

=(7+j)/10

b. Simplify:

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

We multiply the top and bottom of the fraction by the conjugate of the bottom (denominator).

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

 =(1-2j)/(2+9j) xx (2-9j)/(2-9j)

=(2-9j-4j+18j^2)/(4-81j^2)

=(-16-13j)/(4+81)

=(-16-13j)/85

### Exercises

1. Express in the form a + bj:

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

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

=(4+4j)+(3-9j)

=7-5j

2. Express in the form a + bj.

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

(2j)/(2+3j) xx (2-3j)/(2-3j)
 = (4j-6j^2)/(4+9)
=(6+4j)/13