 # Complex Numbers - Basic Operations

## Definition of Complex Numbers

A complex number z is a number of the form
z = a + b i
where a and b are real numbers and i is the imaginary unit defined by
$i = \sqrt{-1}$
a is called the real part of z and b is the imaginary part of z.
Note that the set
R of all real numbers is a subset of the complex number C since any real number may be considered as having the imaginary part equal to zero..

## Complex Conjugate

The conjugate of a complex number a + b i is a complex number equal to
a - b i

Examples: Find the conjugate of the following complex numbers.
a) 2 - i , b) -3 + 4i , c) 5 , d) -5i
Solution to above example
a) 2 + i
b) -3 - 4i
c) 5
d) 5i

Addition of two complex numbers a + b i and c + d i is defined as follows.
(a + b i) + (c + d i) = (a + c) + (b + d) i
This is similar to grouping like terms: real parts are added to real parts and imaginary parts are added to imaginary parts.
Example: Express in the form of a complex number a + b i.
• (2 + 3i) + (-4 + 5i)
• (3i) + (-5 + 6i)
• (2) + (-2 + 9i)

Solution to above example.
• (2 + 3i) + (-4 + 5i) = (2 - 4) + (3 + 5) i = - 2 + 8 i
• (3i) + (-5 + 6i) = (0 - 5) + (3 + 6) i = -5 + 9 i
• (2) + (-2 + 9i) = (2 - 2) + (9) i = 9i
Addition can be done by grouping like terms.
(2 + 3i) + (-4 + 5i) = 2 + 3 i - 4 + 5 i = -2 + 8 i
Calculator
to add complex numbers for practice is available.

## Subtraction of Complex Numbers

The subtraction of two complex numbers a + b i and c + d i is defined as follows.
(a + b i) - (c + d i) = (a - b) + (b - d) i
Example: Express in the form of a complex number a + b i.
• (2 - 5i) - (-4 - 5i)
• (-7i) - (-5 - 6i)
• (2) - (2 + 6i)

Solution to above example
• (2 - 5i) - (-4 - 5i) = (2 - (-4)) + (-5 - (-5)) i = 6
• (-7i) - (-5 - 6i) = (0 - (-5)) + (-7 - (-6)) i = 5 - i
• (2) - (2 + 6i) = (2 - 2) - 6 i = -6 i
Note: subtraction can be done as follows:
(a + b i) - (c + d i) = (a + bi) + (- c - d i) and then group like terms
Example:
(2 - 5i) - (-4 - 5i) = 2 - 5 i + 4 + 5 i = 6

## Multiply Complex Numbers

The multiplication of two complex numbers a + b i and c + d i is defined as follows.
(a + b i)(c + d i) = (a c - b d) + (a d + b c) i
However you do not need to memorize the above definition as the multiplication can be carried out using properties similar to those of the real numbers and the added property i
2 = -1. (see the example below)
Example: Express in the form of a complex number a + b i.
(3 + 2 i)(3 - 3i)
Solution to above example
(3 + 2 i)(3 - 3i)
Using the distributive law, (3 + 2 i)(3 - 3 i) can be written as
(3 + 2 i)(3 - 3 i) = (3 + 2 i)(3) + (3 + 2 i)(-3 i) = 9 + 6 i - 9 i -6 i
2
Group like terms and use i
2 = -1 to simplify (3 + 2 i)(3 - 3 i)
(3 + 2 i)(3 - 3 i) = 15 - 3 i
Calculator
to multiply complex numbers for practice is available.

## Divide two Complex Numbers

We use the multiplication property of complex number and its conjugate to divide two complex numbers.
Example: Express in the form of a complex number a + b i.
• $\dfrac{8 + 4 i}{1-i}$
We first multiply the numerator and denominator by the complex conjugate of the denominator
$\dfrac{(8 + 4 i)\color{red}{(1+i)}}{(1-i)\color{red}{(1+i)}}$
Multiply and group like terms
$= \dfrac{8 + 4 i + 8 i + 4 i^2}{1 - i + i - i^2} = \dfrac{4 + 12i}{2}$
$= 2 + 6 i$
Calculator
to divide complex numbers for practice is available.

## Equality of two Complex Numbers

The complex numbers a + i b and x + i y are equal if their real parts are equal and their imaginary parts are equal.
a + i b = x + i y    if and only if    a = x and b = y
Example: Find the real numbers x and y such that 2x + y + i(x - y) = 4 - i.
For the two complex numbers to be equal their real parts and their imaginary parts has to be equal. Hence
2x + y = 4 and x - y = - 1
Solve the above system of equations in x and y to find
x = 1 and y = 2.

## Exercises

• Find the complex conjugate of the following complex numbers
a) 2 + 6 i
b) -8 i
c) 12
• Write the following expressions in the form a + b i
a) (2 - 8 i) + (-6 i)
b) -8 i + (3 - 9 i)
c) 6 - (3 - i)
d) (2 - 3 i)(7 - i)
e) $\dfrac{2+2i}{2-2i}$

### Solutions to above exercises

• Find the complex conjugate.
a) 2 - 6 i
b) 8 i
c) 12
• Write the following expressions in the form a + b i
a) (2 - 8 i) + (-6 i) = 2 - 14 i
b) -8 i + (3 - 9 i) = 3 - 17 i
c) 6 - (3 - i) = 3 + i
d) (2 - 3 i)(7 - i) = 11 - 23 i
e) $\dfrac{2+2i}{2-2i} = i$