Complex Numbers

� Any number in form a+bi, where a and b are real numbers and i is imaginary.

What is an imaginary number?

Definition of imaginary numbers:

−b2 = b2 ⋅ −1=bi

i = −1

i2 =−1

Simplify complex numbers

Ex: −4i ⋅7i=−28⋅ i2 =−28⋅−1=Remember i

2 =−1

28

Ex#2: −8⋅ −5 =

i 8⋅ i 5 = Remember that −1=i

i2 ⋅ 40= −1⋅2 10=

−2 10

Ex#3: i19

i19 =i18⋅i i18⋅i = i2( )

9⋅ i

i2( )9⋅ i= −1( )

9⋅ i

Answer: -i

Try these problems:

1. −5i ⋅−3i =

2. 3 −6⋅ −3=

3. i13=

-15

9i 2

i

When adding or subtracting complex numbers, combine like terms.

Ex: 8−3i( ) + 2+5i( )8+2( ) + −3i+5i( )

10+2i

Try these on your own

1. (16−4i)−(12−3i)

2. (3i+7)+(34−6i)

3. (−4+7i)+(8−9i)

4. (−1−i)−(−3−4i)

ANSWERS:

1. 4−i

2. −3i+41

3. 4−2i

4. 2+3i

Multiplying complex numbers.

To multiply complex numbers, you use the same procedure as multiplying polynomials.

Examples to do together:

3⋅ −12=

−36= −1⋅ 36

6i

Lets do another example.

3−2i( ) 5+3i( )

15+9i−10i−6i2F O I L

15+9i−10i+6 i2 =−1Next

Answer:

Now try these:

21-i

1. −5⋅ 20

2. 4+5i( ) 4−5i( )

3. (3−2i)2

Next

Answers:

1. 10i

2. 41

3. 5−12i

Conjugates

In order to simplify a fractional complex number, use a conjugate.

What is a conjugate?

a b−c d and a b+c d

are said to be conjugates of each other.

Ex: 3 2i −5 and 3 2i +5

Lets do an example:

Ex: 8i

1+3i

8i1+3i

⋅1−3i1−3i

Rationalize using the conjugate

Next

8i−24i2

1+9=8i+24

10

4i +125

Reduce the fraction

Lets do another example

Ex: 4+i2i

4+i2i

⋅ii=

4i +i2

2i2

Next

4i +i2

2i2=

4i−1−2

Try these problems.

1. 3

2 −5i

2. 3- i2- i

1. 2 +5i9

2. 7+i

5