Lesson 1-5 The Complex Numbers pg .25

object. T o add, subtract, multiply and divide complex numbers

Throughout history of mathematics, new kinds of numbers have been invented to to fill deficiencies in the number system.

• In earliest times the counting numbers existed1,2,3,4,…. The Egyptians and the Greeks invented

the rational numbers, so named because they are ratios of integers, to represent fractional parts of quantities. The Greeks also discovered that some numbers were not rational. For example, the ratio of the length of a diagonal of a square to the length of a side cannot be represented as the quotient of two integers. We know this ratio is

an irrational number.

2

s

s

2s

Complex Numbers

In the 16th century, a few mathematicians begin to work with number whose squares were negative numbers. The word imaginary gradually came to be used to describe such numbers as

Today the phrase imaginary number seems a little unfortunate since these numbers are firmly established in mathematics. They are routinely used in advanced mathematics, AC circuits, quantum mechanics to name just a few fields.

1 15and

We define the imaginary unit i with the following property.

Even though there is now a definition for the square root of negative numbers, we can not assume that all the square root properties of positive numbers will also be true for negative real numbers.

21 _ _ 1i and i

We then define the square root of any negative number as follows

_ 0, _if a a i a

Example 1: Simplify

9 2 25 9 2 25i i

.) 25 25 5

.) 7 7

a i i

b i

.) 25a .) 7b

3 2 5

3 10

7

i i

i i

i

Complex Numbers a + bi

Real Numbers imaginary numbers (b = 0) (b ≠ 0)

E.g. 0,-7,3∏,½, e.g. 7i,4-3i,

2 5i

Example 3: (2+3i) + (4 + 5i) = 6 + 8iExample 4: (2+3i)(4 + 5i) = 8 + 10i + 12i + 15i2

= 8+ 22i + (15)(-1) = -7 + 22i

The complex numbers a+ bi and a – bi are called complex conjugates. Their sum is a real number and their product is a nonnegative real number

Example 5Express 5-2i in the form a + bi. 4+ 3iSolution: Multiply the numerator and denominator by

the conjugate of the denominator 4 + 3i

2

5 2 5 2 4 3 20 15 8 6

4 3 4 3 4 3 16 9

i i i i i i

i i i i

20 23 6

16 9

14 23 14 23

25 25 25

i

ii

Homework 2-46 even pp. 28-29

11.)(6 )(6 )i i (6 6) (6 ) (6 ) ( )i i i i

236 6 6i i i

236 i 36 1 37

14.)( 3 4 2 )( 3 4 2 )i i 2

( 3 3) 4 2 3 4 2 3 (4 2 ) )i i i 2

( 9 16 4 )i 3 16 2 ( 1) 3 32 35

119.)

2 5i

2 2

1 2 5 2 5

2 5 2 5 (2 10 10 (5 ))

i i

i i i i i

2 5

(4 25( 1))

i

2 5 2 5

4 25 29

i i

2 5

29 29i

Video on complex numbers

http://www.youtube.com/watch?v=9Fm8aUyf1Yo&feature=related