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# 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) ± (c + di) = (a ± c) + (b ± d )i

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

Solution:

x + y = (3 + 2i) + (−2 − 5i)

= (3 − 2) + (2 − 5)i

= (1) + (−3)i

= 1 − 3i

## 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 i2  = −1 .

(a + bi) × (c + di) = ac + adi + bci + bdi2

= (ac − bd ) + (ad+ bc)i

Example:  If x = 3 + 2i and y = −2 − 5i , find xy.

Solution:
xy = (3 + 2i)(−2 − 5i)
= (3)(−2) + (3)(−5i) + (2i)(−2) + (2i)(−5i)
= (−6) + (−15i) + (−4i) + (−10i2 )
= −6 − 15i − 4i − 10(−1)
= −6 − 15i − 4i + 10
= 4 − 19i