# 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 $$\mathbb{R}$$ 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..

## 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 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.

## 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 $$\dfrac{8 + 4 i}{1-i}$$ 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.

## 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

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

### Solutions to above exercises

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