Complex Numbers. Definition of pure imaginary numbers: Any positive real number b, where i is the...

Complex Numbers

Definition of pure imaginary numbers:

Any positive real number b,

where i is the imaginary unit and bi is called the pure

imaginary number.

b2 b2 1 bi

Definition of pure imaginary numbers:

i 1

i2 1

i is not a variable it is a symbol for a specific

number

Simplify each expression.

1. 81 81 1 9i

2. 121x5 121x4 1 x 11x2i x

3. 200x 100 1 2x 10i 2x

4. 8i 3i 24i2 241

Remember i2 1

Simplify each expression.

24

5. 5 20 i 5i 20Remember that 1 i

i2 100 110Remember i

2 1

10

Definition of Complex Numbers

Any number in form a+bi, where a and b are

real numbers and i is imaginary unit.

Definition of Equal Complex Numbers

Two complex numbers are equal if their real parts are equal and their imaginary

parts are equal.If a + bi = c + di,

then a = c and b = d

When adding or subtracting complex numbers, combine like terms.

Ex: 8 3i 2 5i 8 2 3i 5i

10 2i

Solving quadratic functions with complex numbers.

1222 x

-2 -2 subtract 2

102 x Take the square root of both sides

10101 ix simplify

8 7i 1211i

8 12 7i 11i

418i

Simplify.

Simplify.

9 12 6i 2i

3 8i

9 6i 122i

Multiplying complex numbers.

To multiply complex numbers, you use the

same procedure as multiplying polynomials.

Simplify.

8 5i 2 3i

16 24i 10i 15i2F O I L

16 14i 15 31 14i

Simplify.

3018i 10i 6i2F O I L

3028i 6 2428i

62i 5 3i

CONJUGATESEach imaginary unit has a conjugate. Two imaginary units are conjugates if and only if their products are a real number.

i

i

i

2

42

3

i

i

i

2

42

3

4)1)(4(4)2)(2(

2016416884)42)(42(

9)1(99)3)(3(

2

2

2

iii

iiiii

iii

Dividing complex numbers.

To divide complex numbers you must multiply the numerator and denominator by the conjugate of the denominator.

i

i

2

4

What’s the conjugate of the denominator?

5

48

224

48

2

2

2

42

2

i

iii

ii

i

i

i

i

Your turn. Write in standard form by dividing.

i

i

6

32

Graphing complex numbers

y-axis is the imaginary axis. x-axis is the real numbers

Identify the numbers plotted.