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

Imaginary Numbers

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

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

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

Examples −25 −100 −49 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.

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

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

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

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

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 + 8i 10i – 6 -4 + 13i

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

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

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

Vocabulary Review Real Number Radical Imaginary Number Complex Number Conjugate

−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

−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

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

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

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

6 − −27 2 6−𝑖 27 2 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? 3+ 3 2 𝑖 3 Conjugate

WOW!!! Nice Job working hard today