Warm Up November 18, 2013

•Complete the square•1. x2 + 2x – 80 = 0

Imaginary Numbers

Squaring Numbers

•What is ? 16•What is -?

16*When you multiply 2 negatives, you still get a positive number.

Square Rooting Numbers•What is +6 or -6•What is ?

hmmmmm…*Here arrives the need for a new number, any ideas?

Introducing Imaginary Numbers

•An imaginary number, i, is defined to be and can be used to simplify negative radicals. Remember radical is

an expression that uses a root. The sign

is

Examples

5i 10i 7i

Steps to Simplify Radicals

•STEP #1: Remove the √(−1)= i.•Step #2: Look for “squares”. Take the square root out of the radical.•Step #3: Leave anything else under the radical.

Remember “Squares”25

9

16 4

36

49

64

81100

121 144

Examples

ii3i

ii5i

ii4i

Property of i

= = -1= =1

Examples

- =

-15

Feeling like you are in an imaginary world???

•Be patient•Keep trying•The more you see it, the easier it gets

Introducing Complex Numbers•A complex number is a number

with 2 parts: real part and the imaginary part

written as: a + bi

Examples 5 + 3i 3 + 5i – 7 +

8i10i – 6

-4 + 13i

Examples 4 +

(2i)(4i) 10 +

28-8

10 + i2

5 + 3i

4 + i

4 + 2i

Introducing Conjugates

•The conjugate of a complex number has the same real part, but an opposite imaginary part.

Examples

6 – 4i

𝟐𝟓 −𝟑−𝟔 𝒊𝟓−𝟐𝐢 √𝟑

𝟔+𝟒 𝒊

Vocabulary Review•Real Number•Radical• Imaginary Number•Complex Number•Conjugate

𝑖√121•Remove the “i”• Look for a “square”

√−12111 𝑖•Real Part?• Imaginary Part?•Conjugate -

𝑖√72•Remove the “i”• Look for a “square”

√−72

6 𝑖√2•Real Part?• Imaginary Part?•Conjugate

𝑖√2∙36−6 𝑖√2

5+𝑖√50•Remove the “i”• Look for a “square”

5+√−50

5+5 𝑖√2•Real Part?• Imaginary Part?•Conjugate

5+𝑖√2 ∙505−5 𝑖√2

−6−9 𝑖• Combine like terms

3−14 𝑖−9+5 𝑖•Real Part?• Imaginary Part?•Conjugate−6+9 𝑖

25+10 𝑖•Distribute

(5 + 2i) (5 + 2i)

•Real Part?• Imaginary Part?•Conjugate21−20 𝑖

+10 𝑖+4 𝑖2• Simplify25+20 𝑖−421+20 𝑖

2

•Remove the “i”• Look for a “square”

•Real Part?• Imaginary Part?•Conjugate

2 23− 32 𝑖√3

3+ 32 𝑖√3

• Simplify

WOW!!!• Nice Job working hard today