Complex Numbers

2

1Definitions

Graphing

3Absolute Values

2

Imaginary Number (i)

Defined as:

Powers of i

1i

1i

12 i

ii 3

14 i

Complex Numbers

A complex number has a real part & an imaginary part.

Standard form is:

bia

Real part Imaginary part

Example: 5+4i

4

Definitions

Pure imaginary number Monomial containing i

Complex Number An imaginary number combined with a real

number Always separate real and imaginary parts

ii

5

3

5

2

5

32

The Complex plane

Imaginary Axis

Real Axis

Graphing in the complex plane

i34 .

i52 .i22 .

i34

.

Absolute Value of a Complex Number

The distance the complex number is from the origin on the complex plane.

If you have a complex number the absolute value can be found using:) ( bia

22 ba

Examples

1. i52

22 )5()2(

254 29

2. i622 )6()0(

360

366

9

Simplifying Monomials

Simplify a Power of i Steps

Separate i into a power of 2 or 4 taken to another power

Use power of i rules to simplify i into -1 or 1 Take -1 or 1 to the power indicated Recombine any leftover parts

10

Operations

Simplify a Power of iSimplify

11

Simplifying Monomials Example Square Roots of Negative NumbersSimplify

12

Addition & Subtraction

Add and Subtract Complex Numbers Treat i like a variableSimplify

ii 4523

ii 4523

i22

ii 3146

ii 3146

i7

Ex: )33()21( ii

ii 3231 i52

Ex: )73()32( ii )73()32( ii

i41

Ex: )32()3(2 iii iii 3223

i21

Addition & Subtraction Examples

)7332 ii

14

Multiplying Complex Numbers Multiply Pure Imaginary Numbers Steps

Multiply real parts Multiply imaginary parts Use rules of i to simplify imaginary parts

15

Monomial Multiplication Example

Multiply Pure Imaginary NumbersSimplify

16

Multiplication Example

Multiply Complex NumbersSimplify ji 5731

)57(3)57(1 iii 2152157 iii 2152157 iii

)1(152157 ii152157 ii

i1622

17

Solving ax2+b=0

Equation With Imaginary SolutionsSolve

Note: ± is placed in the answer because both 4 and -4 squared equal 16

Multiply the numerator and denominator by the complex conjugate of the complex number in the denominator.

7 + 2i3 – 5i The complex conjugate

of 3 – 5i is 3 + 5i.

Multiplying Complex Numbers

19

Dividing Complex Numbers

Divide Complex Numbers No imaginary numbers in the

denominator! i is a radical

Remember to use conjugates if the denominator is a binomial

Simplify

i

i

i

iEx

21

21*

21

113 :

)21)(21(

)21)(113(

ii

ii

2

2

4221

221163

iii

iii

)1(41

)1(2253

i

41

2253

i

5

525 i

5

5

5

25 i

i 5

21

Division Example

Simplify

7 + 2i3 – 5i

21 + 35i + 6i + 10i2

9 + 15i – 15i – 25i221 + 41i – 10

9 + 25

(3 + 5i)(3 + 5i)

11 + 41i 34

Try These.

1. (3 + 5i) – (11 – 9i)

2. (5 – 6i)(2 + 7i)

3. 2 – 3i 5 + 8i

4. (19 – i) + (4 + 15i)

Try These.

1. (3 + 5i) – (11 – 9i) -8 + 14i

2. (5 – 6i)(2 + 7i) 52 + 23i

3. 2 – 3i –14 – 31i 5 + 8i 89

4. (19 – i) + (4 + 15i) 23 + 14i