- Definition of Complex Numbers
- Complex Conjugate
- Addition of Complex Numbers
- Subtraction of Complex Numbers
- Multiply Complex Numbers
- Divide two Complex Numbers
- Equality of two Complex Numbers

A complex number z is a number of the form

Note that the set \( \mathbb{R} \) of all real numbers is a subset of the complex number \( \mathbb{C} \) since any real number may be considered as a complex number having an imaginary part equal to zero..

The conjugate of a complex number \( a + b i \) is a complex number equal to

a) \( 2 - i \) , b) \( -3 + 4i \) , c) \( 5 \) , d) \( -5i \)

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 the following 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.

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:__

\( (2 - 5i) - (-4 - 5i) = 2 - 5 i + 4 + 5 i = 6 \)

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.

We use the multiplication property of complex number and its conjugate to divide two complex numbers.

__Example:__ Express \( \dfrac{8 + 4 i}{1-i} \) in the form of a complex number \( a + b i \).

Solution

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.

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 \( y = 2 \).

1) Find the complex conjugate of the following complex numbers

a) \( 2 + 6 i \)

b) \( -8 i \)

c) \( 12 \)

2) 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) \( \dfrac{2+2i}{2-2i} \)

1) Find the complex conjugate.

a) \( 2 - 6 i \)

b) \( 8 i \)

c) \( 12 \)

2) 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) \( \dfrac{2+2i}{2-2i} = i \)