7.8 1

Section 7.8 Complex Numbers The imaginary number i Simplifying square roots of negative numbers Complex Numbers, and their Form The Arithmetic of Complex Numbers Complex Conjugates Division of Complex Numbers Powers of i

7.8 2

Quadratic Equations with Non-Real Solutions

Try to solve the equation:

No real solutions – but perhaps we can extend our number system beyond real numbers …

The Complex Number System also containsall Real Numbers:

0532 xx

CR

7.8 3

Imaginary Numbersand the Complex Number System The Number i

i is the unique number for which

An Imaginary Number can only be written inform a + bi where a and b are real numbers, b≠0 3 + i 8i 0.3 – 2i 3 – πi etc

A Complex Number can be a Real Number or an Imaginary Number(both a and b can be 0)

11 2 iandi

7.8 4

Recall the Properties of Radicals

These properties are used to write square rootswith negative radicands in terms of i

i3199 i7228

i30256

7.8 5

Addition and Subtraction of Complex Numbers

7.8 6

Multiplying Complex Numbers

-6 + 4i + 9i – 6i2 =

-6 + 4i + 9i + 6 = 13i

7.8 7

Warning … When you have the square root of a negative

number involved in any multiplication, always convert to i Form then multiply.

Without converting:

Correct way:

?52

wrongiswhich10)5)(2(52

correctiii 10105252 2

7.8 8

Complex Conjugates

?32 of conjugate theisWhat i

together?

conjugates hemultiply tyou when happensWhat

-2 + 3i

the product is always a Real Number

7.8 9

Dividing Complex Numbers

To divide complex numbers, we often have to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The answer should be written in the form a +bi.

iii

i

i

ii

ibyDivide

261

265

26

5

125

5

5

5

5

1

5

1

51

7.8 10

ExamplesPerform the operations on the given complex numbers.

Write answers in the form a +bi.

7.8 11

Powers of i

7.8 12

Powers of i

7.8 13

Solving Quadratic EquationsWe have solved quadratic equations for rational number

solutions using factoring and the principle of zero products.

In future sections, we’ll learn some ways to find solutions that are irrational numbers or complex numbers. If a negative radicand results, we can put it in terms of i

For example:

7.8 14

What Next? Quadratic Equations! Present Section 8.1