# Complex Numbers

## INTRODUCTION

A complex number is a number that can be represented by an expression of the form:  a + bi

where a and b are real numbers, and i is a symbol with the property

i2  = −1

We call a the real part of the complex number and we call b the imaginary part.

## ADDING AND SUBTRACTING COMPLEX NUMBERS

To add or subtract complex numbers, simply add or subtract their real parts and their imaginary parts separately.

$$(a+bi)\pm (c+di) = (a\pm c)+(b\pm d)i$$

Example:  If $$x=3+2i$$ and $$y=-2-5i$$, find $$x+y$$.

Solution:

\begin{align} x+y &= (3+2i)+(-2-5i) \\ &= (3-2)+(2-5)i \\ &= (1)+(-3)i \\ &= 1-3i \end{align}

## MULTIPLYING COMPLEX NUMBERS

Multiplication is defined so that the usual laws hold: i.e., do FOIL as usual, but simplify at the end using the fact that $$i^2 = -1$$.

$$\displaylines{(a+bi)x(c+di) &=& ac+adi+bci+bdi^2 \\ &=& (ac-bd)+(ad+bc)i}$$

Example: If $$x=3 +2i$$ and $$y=-2-5i$$, find $$xy$$

Solution:
\begin{align} xy &=(3+2i)(-2-5i) \\ &= (3)(-2)+(3)(-5i)+(2i)(-2)+(-2i)(-5i) \\ &= (-6)+(-15i)+(-4i)+(10i^2) \\ &= -6-15i-4i-10(-1) \\ &= -6-15i-4i+10 \\ &= 4-19i \end{align}