Download presentation

1
5.7 Complex Numbers 12/17/2012

2
**Quick Review Exponent Rule:**

If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1 Ex: 30 = = 1 Exponent Rule: When multiplying powers with the same base, you add the exponent. x2 • x3 = x5 y • y7 = y8

3
The square of any real number x is never negative, so the equation x2 = -1 has no real number solution. To solve this x2 = -1 , mathematicians created an expanded system of numbers using the IMAGINARY UNIT, i.

4
**Simplifying i given any powers**

The pattern repeats after every 4. So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Based on that remainder, you can determine it’s value. Step 1. 22÷ 4 has a remainder of 2 Step 2. i22 = i2 Step ÷ 4 has a remainder of 3 Step 2. i51 = i3 Do you see the pattern yet?

5
Checkpoint Find the value of 1. i 15 2. i 20 3. i 61 4. i 122

6
**Properties of Square Root of Negative Number**

7
**Example 1 Solve the equation. = 7x 2 49 – a. b. = 3x 2 5 – 29 SOLUTION**

Solve a Quadratic Equation Solve the equation. = 7x 2 49 – a. b. = 3x 2 5 – 29 SOLUTION Write original equation. = 7x 2 49 – a. Divide each side by 7. = x 2 7 – Take the square root of each side. = x + – 7 Write in terms of i. = x + – 7 i

8
**Example 1 b. = 3x 2 29 – 5 = 3x 2 24 – = x 2 8 – = x + – 8 = x + – 8 i**

Solve a Quadratic Equation b. = 3x 2 29 – 5 Write original equation. Add 5 to each side. = 3x 2 24 – Divide each side by 3. = x 2 8 – Take the square root of each side. = x + – 8 Write in terms of i. = x + – 8 i Simplify the radical. = x + – 2 i 8

9
**Checkpoint Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7**

Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7 – ANSWER 7, i 7 – 3. = x 2 20 – ANSWER 5, 2 5 – i 4. = x 2 3 2 + – ANSWER 5, i 5 – 5. = y 2 4 – 12 ANSWER 2, 2 – i

10
**Adding and Subtracting Complex Numbers**

Is a number written in the standard form a + bi where a is the real part and bi is the imaginary part. Add/Subtract the real parts, then add/subtract the imaginary parts Complex Number Adding and Subtracting Complex Numbers

11
**Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i (**

Example 2 Add Complex Numbers Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i ( SOLUTION Group real and imaginary terms. 2i 3 ( + i 1 – = 2 i Write in standard form. = 4 + i 11

12
**Write as a complex number in standard form. 2i 6 ( – 1**

Example 3 Subtract Complex Numbers Write as a complex number in standard form. 2i 6 ( – 1 SOLUTION Group real and imaginary terms. 2i 6 ( = 1 2 i + – -1 + 2i Simplify. = 5 + 0i Write in standard form. = 5 12

13
**Write the expression as a complex number in standard form.**

Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 6. ( 4 – ( 2i ( + 1 + 3i ( ANSWER i 5 + 7. i 3 ( – + 4i 2 ANSWER 3i 5 + 8. 6i 4 ( + 3i 2 – ANSWER 3i 2 + 9. 4i 2 ( + 7i – ANSWER 3i 4 –

14
**Write the expression as a complex number in standard form.**

Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 11. 2i 1 ( – + 5i 4 ANSWER 3i 5 + 12. i 2 ( – 4i 1 ANSWER 3i 3 +

15
**Write the expression as a complex number in standard form.**

Example 4 Multiply Complex Numbers Write the expression as a complex number in standard form. a. 1 ( 3i + – 2i b. 3i 6 ( + 3i 4 ( – SOLUTION Multiply using distributive property. 1 ( 3i + – 2i = 6i 2 a. 1 ( – 2i 6 = + Use i 6 2i – = Write in standard form.

16
**Example 4 b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = 24 6i – 9i 2 = 24 6i – 1 (**

Multiply Complex Numbers b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = Multiply using FOIL. 24 6i – 9i 2 = Simplify. 24 6i – 1 ( 9 = Use i 6i 33 – = Write in standard form. 16

17
Complex Conjugates Two complex numbers of the form a + bi and a - bi Their product is a real number because (3 + 2i)(3 – 2i) using FOIL 9 – 6i + 6i -4i2 9 – 4i2 i2 = -1 9 – 4(-1) = = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi)

18
**Write as a complex number in standard form. 2i 3 + 1 – a + bi SOLUTION**

Example 5 Divide Complex Numbers Write as a complex number in standard form. 2i 3 + 1 – a + bi SOLUTION 2i 3 + 1 – = • Multiply the numerator and the denominator by i, the complex conjugate of i. Multiply using FOIL. 1 2i 3 6i + – 4i 2 = 3 8i + 1 ( – 4 = Simplify and use i 8i + – 1 5 = Simplify. 5 1 – 8 i + = Write in standard form. 18

19
**Write the expression as a complex number in standard form.**

Checkpoint Multiply and Divide Complex Numbers Write the expression as a complex number in standard form. 13. i 2 ( – 3i ANSWER 6i 3 + 14. ( 2i 1 + i 2 – ANSWER 3i 4 + 15. i 2 + 1 – ANSWER 2 1 + 3 i

20
**Graphing Complex Number**

Imaginary axis Real axis

21
Ex: Graph 3 – 2i To plot, start at the origin, move 3 units to the right and 2 units down 3 2 3 – 2i

22
**Ex: Name the complex number represented by the points.**

Answers: A is 1 + i B is 0 + 2i = 2i C is -2 – i D is i D B A C

23
Homework 5.7 p.264 #17-20, 27/29, 33-35, 40, 43, 45, 46, 52-54, 64-71 5.6

24
**Checkpoint Solve the equation. 1. x 2 = – 3 2. = x 2 7 – 3. = x 2 20 –**

Solve a Quadratic Equation Solve the equation. 1. x 2 = – 3 2. = x 2 7 – 3. = x 2 20 – 4. = x 2 3 2 + – 5. = y 2 4 – 12

25
**Write the expression as a complex number in standard form.**

Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 6. ( 4 – ( 2i ( + 1 + 3i ( 7. i 3 ( – + 4i 2 8. 6i 4 ( + 3i 2 – 9. 4i 2 ( + 7i –

26
**Write the expression as a complex number in standard form.**

Checkpoint Add and Subtract Complex Numbers Write the expression as a complex number in standard form. 10. 2i 1 ( – + 5i 4 11. i 2 ( – 4i 1 Write the expression as a complex number in standard form. 12. i 2 ( – 3i 14. i 2 + 1 – 13. ( 2i 1 + i 2 –

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google