1. Warm Up November 18, 2013
Complete the square
1. x2 + 2x – 80 = 0

2. Imaginary Numbers

3. Squaring Numbers What is 4 2 ? 16 What is - 4 2 ?
*When you multiply 2 negatives, you still get a positive number.

4. Square Rooting Numbers
What is 36
+6 or -6
What is −36 ?
hmmmmm…
*Here arrives the need for a new number, any ideas?

5. Introducing Imaginary Numbers
An imaginary number, i, is defined to be −1 and can be used to simplify negative radicals.
Remember radical is an expression that uses a root. The sign is

7. Examples
−25
−100
−49 5i
10i 7i

8. 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.

9. 36 64
Remember “Squares”
121
144 25 9 49 4 16 81
100

10. Examples
−90
−75
−48
i 𝟗𝟎
i 𝟗∙𝟏𝟎
3i 𝟏𝟎
i 𝟕𝟓
i 𝟑∙𝟐𝟓
5i 𝟑
i 𝟒𝟖
i 𝟑∙𝟏𝟔
4i 𝟑

11. Property of i
𝑖 = −1 𝑖 2 = -1 𝑖 3 = 1 𝑖 4 = 1

12. 𝒊 𝟒 𝒊 𝟒 𝒊 𝟏𝟓𝒊 𝟐 −𝒊 −𝟏 -15 −𝟏 Examples 𝒊 𝟔 =𝒊 𝟒 𝒊 𝟐 𝑖 9 𝑖 𝟏 · 𝑖 𝟓 =
𝑖 𝟏 · 𝑖 𝟓 =
3i · 5i =
- −1 =
𝒊 𝟒 𝒊 𝟒 𝒊 −𝟏
𝒊 𝟔 =𝒊 𝟒 𝒊 𝟐 −𝟏
𝟏𝟓𝒊 𝟐
-15 −𝒊

13. Feeling like you are in an imaginary world???
Be patient
Keep trying
The more you see it, the easier it gets

15. Introducing Complex Numbers
A complex number is a number with 2 parts: real part and the imaginary part written as: a + bi

16. Examples
5 + 3i
3 + 5i – 7 + 8i
10i – 6 i

17. 8 𝒊 𝟐 -8 Examples 10 + i 𝟑𝟔 4 + i 𝟒∙𝟕 2 4 + 2i 𝟕 5 + 3i 4 + −28
4 + −28
(2i)(4i)
−36 2
10 + i 𝟑𝟔 2
5 + 3i
8 𝒊 𝟐 -8
4 + i 𝟒∙𝟕
4 + 2i 𝟕

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

19. Examples 𝟐𝟓
−3 + 6i
5 + 2i 3
6 – 4i 𝟐𝟓
−𝟑 −𝟔 𝒊
𝟔+𝟒𝒊
𝟓 −𝟐𝐢 𝟑

20. Vocabulary Review Real Number Radical Imaginary Number Complex Number
Conjugate

21. −121 𝑖 121 11𝑖 -11𝑖 Remove the “i” Real Part? Imaginary Part?
𝑖 121
Remove the “i”
Look for a “square”
11𝑖
Real Part?
Imaginary Part?
-11𝑖
Conjugate

22. −72 𝑖 72 𝑖 2∙36 6𝑖 2 −6𝑖 2 Remove the “i” Real Part? Imaginary Part?
𝑖 72
Remove the “i”
𝑖 2∙36
Look for a “square”
Real Part?
Imaginary Part?
6𝑖 2
Conjugate
−6𝑖 2

23. 5+ −50 5+𝑖 50 5+𝑖 2∙50 5+5𝑖 2 5−5𝑖 2 Remove the “i” Real Part?
5+ −50
5+𝑖 50
Remove the “i”
5+𝑖 2∙50
Look for a “square”
Real Part?
Imaginary Part?
5+5𝑖 2
Conjugate
5−5𝑖 2

24. 3−14𝑖−9+5𝑖 −6−9𝑖 − 6+9𝑖 Real Part? Imaginary Part? Conjugate
Combine like terms
Real Part?
Imaginary Part?
− 6+9𝑖
Conjugate

25. 25+10𝑖 +10𝑖+4 𝑖 2 25+20𝑖 −4 21+20𝑖 21 −20𝑖 (5 + 2i) (5 + 2i)
Distribute
25+20𝑖 −4
Simplify
Real Part?
Imaginary Part?
21+20𝑖
21 −20𝑖
Conjugate

26. 6 − −27 2
6−𝑖
6−𝑖 3∙9 2
6−3𝑖 3 2
Remove the “i”
Look for a “square”
3− 3 2 𝑖 3
Simplify
Real Part?
Imaginary Part?
𝑖 3
Conjugate

27. WOW!!!
Nice Job working hard today