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 i 1 2 3 41 1i i i i i i= = − = − =

Divide the exponent by 4 No 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 i

1.) Find i23

2.) Find i2006

3.) Find i37

4.) Find i828

i−=1−=i=1=

Complex Number System

Reals

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. 1.)

2.) 7− −2.)

3.) 99−3.)

= ⋅ ⋅i 3 3 11

-Express these numbers in terms of i.

You try…

4.

5.

−7

− −36

−1606.

= i 7

= −6i= 4 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 5= −2 21i= −

( 1) 21= − − 21=

Multiplying 47 2i ⋅

2 5i ⋅ −

− − ⋅ −3 7

= ⋅ − ⋅2 1 5i = ⋅2 5i i

= − ⋅i i3 7

1.

2.

3.

a + bi Complex 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 Subtract

1.

2.

3.

Multiplying & Dividing Complex Numbers

REMEMBER: i² = -1

12= 2i 12( 1)= − 12= −

2 27 i= 49( 1)= − 49= −

Multiply 3 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+ 26i+228 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 −5i6 6

8)

9)

10)

11)

iiBecause 06 as same theis 06 −+

11ii

++

2

14 41

ii

− +=−

14 42

i− += 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 1934

i− −=

2 33 5−+ii

13) You try…

= − − +− + −6 10 9 159 15 15 25

2

2

i i ii i i