Download presentation

1
**Section 3.2 Beginning on page 104**

Complex Numbers Section 3.2 Beginning on page 104

2
**Imaginary Numbers and Complex Numbers**

𝑖= − 𝑖 2 =−1 Imaginary numbers allow us to simplify radicals with no real solutions. −4 = −1 4 =2𝑖 −3 = −1 3 =𝑖 3 −20 = − =2𝑖 5 A complex number written in standard form is a number 𝑎+𝑏𝑖 where 𝑎 and 𝑏 are real numbers. The number 𝑎 is the real part, and the number 𝑏𝑖 is the imaginary part. Real Numbers: −1 𝜋 Complex Numbers: 2+3𝑖 −5𝑖 Pure Imaginary Numbers: 3𝑖 −5𝑖

3
**Examples 1 and 2 Example 1: Find the square root of each number**

a) −25 b) −72 c) −5 −9 = −1 25 = − =−5 −1 9 =5𝑖 =6𝑖 2 =−5∙3𝑖 =−15𝑖 Example 2: Find the values of x and y that satisfy the equation 2𝑥−7𝑖=10+𝑦𝑖 ** Set the real parts equal to each other, set the imaginary parts equal to each other. Solve the resulting equations. ** 2𝑥=10 −7𝑖=𝑦𝑖 𝑥=5 −7=𝑦 y=−7

4
**The Sums and Differences of Complex Numbers**

To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Example 3: 8−𝑖 +(5+4𝑖) 7−6𝑖 −(3−6𝑖) 13− 2+7𝑖 +5𝑖 = 8+5 +(−𝑖+4𝑖) =13+3𝑖 = 7−3 +(−6𝑖−(−6𝑖)) =4+0𝑖 =4 =13−2−7𝑖+5𝑖 =11−2𝑖

5
**Multiplying Complex Numbers**

Example 5: Multiply. Write the answer in standard form. a) 4𝑖(−6+𝑖) b) (9−2𝑖)(−4+7𝑖) ** 𝑖∙𝑖= 𝑖 2 =−1 ** =−36+63𝑖+8𝑖−14 𝑖 2 =−24𝑖+4 𝑖 2 =−36+71𝑖−14(−1) =−24𝑖+4(−1) =−36+71𝑖+14 =−24𝑖−4 =−22+71𝑖 =−4−24𝑖

6
**Solving Quadratic Equations**

Example 6: Solve a) 𝑥 2 +4=0 b) 2 𝑥 2 −11=−47 𝑥 2 =−4 2 𝑥 2 =−36 𝑥= −4 𝑥 2 =−18 𝑥= −1 4 𝑥= −18 𝑥= − 𝑥=±2𝑖 𝑥=±3𝑖 2

7
**Finding Zeros of Quadratic Functions**

Example 7: Find the zeros of 𝑓 𝑥 =4 𝑥 2 +20 0=4 𝑥 2 +20 −20=4 𝑥 2 −5= 𝑥 2 𝑥= −5 𝑥= −1 5 𝑥=±𝑖 5 The zeros of the function are 𝑖 5 and −𝑖 5

8
**Practice Find the square root of the number.**

−4 2) −12 3) − −36 4) 2 −54 Find the values of x and y that satisfy the equation. 5) 𝑥+3𝑖=9−𝑖𝑦 6) 9+4𝑦𝑖=−2𝑥+3𝑖 Perform the operation. Write the answer in standard form. 8) 9−𝑖 +(−6+7𝑖) 9) 3+7𝑖 −(8−2𝑖) 10) −4− 1+𝑖 −(5+9𝑖) 11) (−3𝑖)(10𝑖) 12) 𝑖(8−𝑖) 13) (3+𝑖)(5−𝑖) Answers: ) 2𝑖 2) 2𝑖 ) −6𝑖 ) 6𝑖 ) 𝑥=9, 𝑦=−3 6) 𝑥=−4.5, 𝑦= ) 3+6𝑖 9) −5+9𝑖 ) −10−10𝑖 11) ) 1+8𝑖 ) 16+2𝑖

Similar presentations

© 2021 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