Square Root of a Negative Number

You’re not allowed to have a negative

number under the radical sign.

Let me show you how it

works

The Imaginary Unit

The Imaginary Unit is denoted as:i

1i The Imaginary

Unit is used to find the square root of negative numbers.

4 14 2i

81 181 9i

4 25 14 25 )4(5 i 20i

These are all examples of Pure Imaginary Numbers.

A Pure Imaginary Number is the product of any real number and i.

Complex NumbersThe sum of a real number and a pure imaginary number is called a Complex Number.

The standard form of a Complex Number isa ibExamples: 2 3i

6 4i27 i

9 i 3 How is that a

complex number.

There’s no i.3 03 i

I never imagined that imaginary

numbers would be so easy.

Properties of Complex Numbers

Adding, subtracting, multiplying, and dividing complex numbers works the same way as with regular binomials with real numbers and variables.

Let’s take a look at some examples.

2 3 5x x x

6 9

73

x

x

9 2x

210 29 21

( )( )23 75 x x

x x

52 3( )( )x x x11

56x

xx

5 7 2i i i

3 4

32

i

i

5 i

2

( )(

6 8 8

34 22 )i i

i i

74 3( )( )i i i9

36i

ii

Operations with Complex Numbers

16 25

6 51 11 2

4 5i i 9i

20 45

0 51 12 4

5 51 14 9

2 35 5i i

5 5i5 98 3 50

98 01 15 3 5

15 2 3 24 19 25 5 2 3 2( ) )7 (5i i

35 2 15 2i i 50 2i

I’d like to push the easy button

now.

Powers of i

1i 1 i

2i 1 1 13i 2 1i i 1i i4i 2 2i i 1 1 1

i

1

i

1

5i 4 1i i 1i i

6i 4 2i i 1 1 17i 4 3i i 1 i i8i 4 4i i 1 1 1

i

1

i

1

9i i

10i 111i i12i 1

13i i

14i 1

15i i

16i 1

Hey, it’s Sam Ting

again.

Every 4th power repeats. So you

only need to know the first 4 powers

of i.

When the power of i is greater than 4, divide by 4 and use the remainder to find the simplified value.

27i i

Simplifying Powers of i

37i374

9.25 4 9 36 37 136 37i i

46i 464

11.5 4 11 44 46 244 46 2i i 2 1i

55i 154

55

3.7 4 13 52 55 352 55 3i i 3i i

64i 644

16.0 4 16 64 64 064 64 4i i 4 1i

Complex ConjugatesComplex Conjugates are two complex numbers that are the same with the exception of the sign in the middle.

The product of a pair of Complex Conjugates is always a positive real number.

This should be pretty easy. We did this conjugate stuff before with

radicals

Complex ConjugatesComplex Conjugates are two complex numbers that are the same with the exception of the sign in the middle.

The product of a pair of Complex Conjugates is always a positive real number.

( )(2 2 3 )3i i

Let’s use foil first.

26 64 9i i i 4 19( ) 4 913

( )(5 5 4 )4i i 220 2 1625 0i i i

16 125 ( ) 25 16

41

Hey, I can use a stupid human trick for this

too!

2 2( )( )b ba a ai bi

This should be pretty easy. We did this conjugate stuff before with

radicals.

Multiplying Complex Conjugates

Multiply each number by its complex conjugate.

3 7i

25 i

6 5i

1 i

( )(3 )7i 9 49 58

( )( 25 )i 25 4 29

( )(6 )5i 36 25 61

( )(1 )i 1 1 2

Dividing Complex Numbers

Complex Conjugates are used to divide by a complex number the same way that regular conjugates are used to rationalize a denominator with a radical in it.4 235ii

Dividing Complex Numbers

Complex Conjugates are used to divide by a complex number the same way that regular conjugates are used to rationalize a denominator with a radical in it.4 235ii

55

33ii

212 10 629

025i i i

220 123

)4

6(i

21434

2i

2220 634

i

14 2234 34

i 7 1117 17

i .41 .65i

321ii

11

22ii

231

6 24

i i i

7 2( 13

5)i

3 257i

157i

1 75 5

i .2 1.4i

Can I do this on my calculator?

Holy schnikies, it

works!

Asi De Facil

Complex Numbers Homework

Amsco Online TextbookChapter 5: Page 21627 – 37 Odd #’s Only47 – 53 Odd #’s Only

Multiplicative Inverse of a Complex Number

Write the multiplicative inverse, in standard a + bi form, of3 4i

The multiplicative inverse of a complex number is its reciprocal.

3 4i1

3 4i33

44ii

39

416i

325

4i

3 425 25

i .12 .16i

Write the multiplicative inverse, in standard a + bi form, of1 5i

1 5i1

1 5i11

55ii

11

525

i

126

5i

1 526 26

i .04 .19i

This complex number stuff is not that complex.

Complex Numbers Homework

Amsco Online Textbook

Chapter 5: Page 21627 – 53 Odd #’s Only

Graphing Complex NumbersWhen a complex number is graphed in the Complex Plane, the horizontal axis is the Real axis and the vertical axis is the Imaginary axis.

Real

Imaginary

Graph the following complex numbers.

a) 2 + 3i

b) 2 - 4i

c) -4 + 2i

d) -2 - 3i

4i

3i

2i

i

-i

-2i

-3i

-4i

2 + 3i

2 - 4i

-4 + 2i

-2 - 3i

So easy, even a caveman can do

it.

More Graphing Complex Numbers

When a complex number is graphed in the Complex Plane, the horizontal axis is the Real axis and the vertical axis is the Imaginary axis.

Let Z1 = 2 + 4i and let Z2 = -3 - 2i

a) Graph both on the same axes.

b) Determine Z1 + Z2.

2 + 4i

-3 - 2i

( ) (2 3 2 1 2)4i ii

-1 + 2i

Real

Imaginary

That was easy