Operations with Complex Numbers

Unit 1 Lesson 2

Make Copies of:

• Comparing Polynomials and Complex Numbers Graphic Organizer

• Kuta-Operations with Complex Numbers WS

GPS Standard

• MM2N1b- Write complex numbers in the form a + bi

• MM2N1c- Add, subtract, multiply, and divide complex numbers

• MM2N1d- Simplify expressions involving complex numbers

Essential Questions

• How do I add and subtract complex numbers?• How do I multiply complex numbers?

Real Numbers Imaginary Numbers

RationalNumbers

IrrationalNumbers

COMPLEX NUMBERS

Standard Form of a Complex Number

a + bi

REAL PARTIMAGINARY

PART

Adding/Subtracting Complex Numbers

• Adding and subtracting complex numbers is just like any adding/subtracting you have ever done with variables.

• Simply combine like terms.• (6 + 8i) + (2 – 12i) = 8 – 4i• (7 + 4i) – (10 + 9i) = 7 + 4i – 10 – 9i = -3 – 5i

To Add Complex Numbers

• (a + bi) + (a + bi)• Drop the parentheses• Combine like terms• Remember: the real number comes first, then

the imaginary number

Examples

• (3 + 5i) + (2 – 7i)• (12 – 3i) + (2 + 4i)• (13 +24i) + (17+ 5i)• (3 – 6i) + (5 – 2i)• (8 – 3i) + (4 – 11i)

Test Prep Example

• What is (5 – 2i) + (6 + 4i)?

A) -3iB) 3iC) 11 + 2iD) 11 + 6i

Test Prep Example

• Perform the indicated operation. (2 + 3i) +(13 – 2i) =

• A) 15 + 5iB) 15 + iC) 11 – 5iD) -11 – i

To Subtract Complex Numbers:

• (a + bi) – (a + bi)• Change the minus sign to plus• Change the sign of each term in the second

set of parentheses• Drop parentheses• Combine like terms• Remember: real number comes first, then

imaginary number

Examples

• (6 + 7i) – (4 + 3i)• (8 + 2i) – (3 – 7i)• (12 – 7i) – (2 + 6i)• (3 – 8i) – (7 – 11i)

Test Prep Example

• Perform the indicated operation.(-9 + 2i) – (-12 + 4i) =

A) -21 – 6iB) -3 + 6iC) 3 – 2iD) 21 + 2i

Multiplying Complex Numbers

• This will be FOIL method with a slight twist at the end.

• An i2 will ALWAYS show up. You will have to adjust for this.

• (4 + 9i)(2 + 3i) = 8 + 12i + 18i + 27i2 = 8 + 30i – 27 = -19 + 30i

• (7 – 3i)(6 + 8i) = 42 + 56i – 18i – 24i2 = 42 + 38i + 24 = 66 + 38i

Examples

Test Prep Examples

• 1. (5 – 3i)(6 + 2i)Multiply and simplify.

• A) 24 – 8i• B) 36 – 8i• C) 36 + 8i• D) 24 + 8i

Binomial Squares and Complex Numbers

• You can still do the five-step shortcut, or you can continue to do FOIL.

• You will still have to adjust for the i2 that will show up.

• (7 + 3i)2 = 49 + 42i + 9i2 = 49 + 42i – 9 = 40 + 21i

• (8 – 9i)2 = 64 – 144i + 81i2 = 64 – 144i – 81= -17 – 144i

Example

Test Prep Example

• Which has the same value as (4 + 3i)2 ?

A) 7B) 7 + 24iC) 25D) 25 + 24i

D2S and Complex Numbers

• Situations that in the real numbers would have been differences of two squares (D2S) demonstrate in the complex numbers what are known as conjugates.

• (3 + 4i)(3 – 4i) = (3)2 – (4i)2 = 9 – 16i2 = 9 + 16 = 25

• When conjugates are used, there will be no i in the answer.

Examples

Test Prep Example

• 2.) Perform the indicated operation.(4 – 7i)(4 + 7i) =

A) -33B) 16 – 49iC) 16 – 105iD) 65

Test Prep Example

• What is the square of 4 – 7i?• A) 33 – 56i• B) -33 – 56i• C) -33 + 56i• D) 33 + 56i

Test Prep Example

• Which is equivalent to (3 + 2i)(2 + 5i)?• A) -4 + 19i• B) 16 + 19i• C) 6 + 29i

Test Prep Example

• What is a if a + bi = (2 – i)2 • A) a = 3• B) a = 5• C) a = 2• D) a = 1

Test Prep Example

• Simplify: -10 + √-16 2

• A) -5 + 2i• B) -5 – 4i • C) 20 + 4i• D) 30 + 2i

Test Prep Example

• Perform the indicated operation.• (3 – 8i)(4 + i) = • A) 4• B) 20 – 29i• C) 12 – 8i• D) 5 + 35i

Test Prep Example

• Multiply 2i(i – 2) over the set of complex numbers.

• A) 0• B) 2 – 4i• C) -2 – 4i• D) 2 + 4i

Graphic Organizer

• Comparing Polynomials and Complex Numbers.doc

Assignment

• Kuta-Operations with Complex Numbers.pdf

Support Assignment

• Pg 8: 1-27• Pg. 13: 1 - 26