Imaginary & Complex Numbers

Once upon a time…

1 no real solution

-In the set of real numbers, negative numbers do not have square roots.

-Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.

-These numbers were devised using an imaginary unit named i.

1i

-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.

-The first four powers of i establish an important pattern and should be memorized.

Powers of i1 2 3 41 1i i i i i i

Divide the exponent by 4No remainder: answer is 1.remainder of 1: answer is i.

remainder of 2: answer is –1.remainder of 3:answer is –i.

i4 1

i

i2 1

i i3

Powers of i1.) Find i23

2.) Find i2006

3.) Find i37

4.) Find i828

i1i1

Complex Number SystemReals

Rationals(fractions, decimals)

Integers(…, -1, -2, 0, 1, 2, …)

Whole(0, 1, 2, …)

Natural(1, 2, …)

Irrationals(no fractions)

pi, e

Imaginary

i, 2i, -3-7i, etc.

1.) 5 1*5 1 5 5i

1*7 1 7 7i

1*99 1 99

3 11i

Simplify.3.)

2.) 7 4.)

3.) 995.)

i 3 3 11

-Express these numbers in terms of i.

You try…6.7.

7

36

1608.

i 7

6i4 10i

To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

94i

22 5i 2 52 21i

( 1) 21 21

Multiplying47 2i

2 5i

3 7

2 1 5i 2 5i i

i i3 7

9.

10.

11.

a + biComplex Numbers

real imaginary

The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

7.) 7 9i i 16i

8.) ( 5 6 ) (2 11 )i i 3 5i

9.) (2 3 ) (4 2 )i i 2 3 4 2i i 2 i

Add or Subtract12.

13.

14.

Multiplying & Dividing Complex Numbers

Part of 7.9 in your book

REMEMBER: i² = -1

12 2i 12( 1) 12

2 27 i 49( 1) 49

Multiply3 4i i

27i

1)

2)

You try…3)

4)

7 12i i

211i

284i )1(84 84

2211 i )1(121 121

28 8i 21i 26i228 29 6i i

28 29 6( 1)i 28 29 6i 22 29i

ii 2734 5)

Multiply

You try… ii 1032 6)

2103206 iii

i1716

210176 ii 110176 i

10176 i

25 35i 35i 249i

25 49( 1)

25 4974

You try… ii 7575 7)

Conjugate-The conjugate of a + bi is a – bi

-The conjugate of a – bi is a + bi

Find the conjugate of each number…

3 4 i 3 4 i

4 7i 4 7i

5i 5i

6 6

8)

9)

10)

11)

iiBecause 06 as same theis 06

11ii

2

14 41

ii

14 4

2i

7 2i

Divide…

5 91

ii

12)

5 5 9 91

2

2

i i ii i i

3 53 5

ii

2

9 199 25

ii

9 19

34i

2 33 5ii

13)You try…

6 10 9 159 15 15 25

2

2

i i ii i i