5.4 – Complex Numbers

Our Imaginary Friends

Uses of Imaginary Numbers

Contrary to its name, the “imaginary number” exists in a similar fashion to the “real number,” often describing physical characteristics that we can detect, observe, and measure. Common applications are circuit analysis in electrical engineering and vibration analysis in mechanical engineering.

Want to learn more? http://www.picomonster.com/imaginary-numbers-lesson-1/

Fractals: What are they?

A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales.

Examples of Fractals

Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes, crystals, lightning, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels, and ocean waves. DNA and heartbeats can be analyzed as fractals.

5.4 – Complex NumbersObjectives:

1. Be able to find, identify, and name imaginary numbers

2. Be able to add and subtract imaginary numbers

3. Be able to multiply and divide imaginary numbers

Vocabulary:imaginary unit - i, complex numbers, standard form, imaginary numbers, pure imaginary numbers, conjugate

i 1

i2 1

Objective #1 – Be able to find, identify, and name imaginary numbers.

5.4 – Complex Numbers

complex number = a + bia is the real partb is the imaginary part

a + 0i are the real numbersa + bi are the imaginary numbers0 + bi are the pure imaginary numbers

Objective #1 – Be able to find, identify, and name imaginary numbers.

5.4 – Complex Numbers

Examples of Complex Numbers

4 + 0i = 4 (Real Number)

2 – 3i (Imaginary Number Real Number and Imaginary Number Combo)

0 + 6i = 6i (Pure imaginary Number)

The Square Root of a Negative Number

Properties

r i r 2i r r

Examples

5 12

5i 2 3i

8 16

Examples

2 2i 4i

Examples

26i 224i

6 24

Examples

212i 27i

12 7

Objective #2 – Be able to add and subtract imaginary numbers.

5.4 – Complex Numbers

( ) ( ) ( ) ( )a bi c di a c b d i ( ) ( ) ( ) ( )a bi c di a c b d i

Real part Imaginary part

Examples

( 1 2 ) (3 3 )i i

2 5i

Examples

(2 3 ) (3 7 )i i

1 4i

Examples

2 (3 ) (2 3 )i i i

1 2i

Solving Quadratic Equations with i

2 16x

4x i

25 18 3x

Solving Quadratic Equations with i

3x i

1. Simplify the radical expression.

2. Solve the equation.

3. Write the expression as a complex number in standard form.

Exit Slip

54

24 7 65x

(2 11 ) (6 )i i

3 6i

3 2x i

4 12i

Complete the following problems individually. You may use your notes. Remain quiet until all exit slips have been collected.

pg. 277 #18-28 even, 39-42

Homework

Homework…

5.4 – Complex Numbers

Our Imaginary Friends

i2 1REMEMBER

Objective #3 – Be able to multiply and divide imaginary numbers.

5.4 – Complex Numbers

(2 )(4 3 )i i (3 )i i

(2 3 )( 6 2 )i i

(1 2 )(1 2 )i i

Example

(2 )(4 3 )i i

11 2i

Example

(3 )i i

1 3i

Example

(2 3 )( 6 2 )i i

6 22i

Example

(1 2 )(1 2 )i i

5This answer is a REAL number!

2 7

1

i

i

3 11

1 2

i

i

Conjugates: a + bi and a - bimultiplying conjugates will always

give you a real number!

Objective #3 – Be able to multiply and divide imaginary numbers.

5.4 – Complex Numbers

2 7

1

i

i

Example

5 9

2 2

i

Example

3 11

1 2

i

i

5 i

Future electrical engineer? Check out pg. 279 #95, 96

Homework

pg. 278 #47-62 (every third problem)

That lesson was…