Complex Numbers

CCSS objective: Use complex numbers in polynomial identities

N-CN.1 Know there is a complex number i such that i2 = −1, and every complex number has the form a + bi with a and b real.

N-CN.2 Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

i0 1

Cycle of "i"

i1 i i

2 1

i3 i

i4 1

i5 i

Imaginary Numbers:

where i is the imaginary unit

b2 b2 1 bi

i is not a variable it is a symbol for a specific number

Simplify each expression.

9i

2. √-100

3. √-121

1. √-81

= √100 √-1

= √81 √-1

=

4. 8i 3i 24i2 241Remember i

2 1

Simplify each expression. 24

6. 4i ∙ 3i

7. 19i ∙ 17i =

= 12i2

Leave space in your notes for #5

= 12 ∙ -1

5. 5 20 i2 100 10Do separately ? Must combine first ?

8. √-100 ∙ √-81 =

Application: Imaginary Numbers

Monday: (all students) textbook p. 278, even only, # 2-10 & even only, 42-46; copy problem & circle your answer to be graded.

Do at the beginning of next class

So i12 i0

i12

Simplify: Enrichment

To figure out where we are in the “i”cycle divide

the exponent by 4 and look at the remainder.

124 =3 with remainder 0

1

So i17 i1

Simplify:Enrichment

Divide the exponent by 4 and look at the

remainder.

174 =4 with remainder 1

i

i17

Definition of Equal Complex Numbers

Two complex numbers are equal if their real parts are equal and their

imaginary parts are equal.

If a + bi = c + di, then a = c and b = d

Simplify:

When adding or subtracting complex numbers, combine like terms.

Ex: 8 3i 2 5i 8 2 3i 5i

10 2i

8 7i 1211i

8 12 7i 11i

418i

Simplify.

Simplify.

3 8i

9 6i 122i

9 – 6i -12 – 2i

Application

Textbook p. 278 #29-34– Non-Enrichment: p.278 #50-52

– Enrichment: p/ 278 #14-16

Must show work up to point of calculator entry Must write question Circle answer you want graded

Due next class

Multiplying Complex Numbers.

To multiply complex numbers, you use the same procedure as multiplying

polynomials.

F

O I

L

Multiplying: (a + b)(c + d)

Multiplying Polynomials (a + b)(c + d)

(a – b)(c – d)

(a + b)(c – d)

Multiplying Complex Numbers (a + bi)(c + di)

(a – bi)(c – di)

(a + bi)(c – di)

Simplify.

8 5i 2 3i

16 24i 10i 15i2F O I L

16 14i 15 31 14i

16 – 24i + 10i -15(-1)

Simplify.

3018i 10i 6i2F O I L

3028i 6 2428i

62i 5 3i

Group work Teams of no more than 2; both names on one paper if

fine. Everyone in team must understand how to do if asked.

look up and define the vocabulary term Conjugate

Textbook page 279 # 57-66 Due next class hint: Order of Operations Circle answer Write question