10 7 Complex Numbers 1 Write imaginary numbers

10. 7 Complex Numbers 1. Write imaginary numbers using i. 2. Perform arithmetic operations with complex numbers. 3. Raise i to powers.

Imaginary unit: Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.

Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Examples: al e r ry a in im ag

Complex Numbers: a + bi real imaginary b = 0: Real numbers a = 0: Imaginary numbers

Add Complex Numbers Add the real parts – add the imaginary parts

Subtract Complex Numbers

Simplify. ( 4 + 7 i) – (2 + i) a) 2 + 7 i 2 b) 2 + 8 i c) 6 + 6 i d) 6 + 8 i Copyright © 2011 Pearson Education, Inc. Slide 10 - 9

Simplify. ( 4 + 7 i) – (2 + i) a) 2 + 7 i 2 b) 2 + 8 i c) 6 + 6 i d) 6 + 8 i Copyright © 2011 Pearson Education, Inc. Slide 10 - 10

Multiply Complex Numbers

Multiply Complex Numbers standard a + bi form

Multiply Complex Numbers

Multiply Complex Numbers Rewrite & Foil

Multiply. ( 4 + 7 i)(2 + i) a) 15 + 10 i b) 1 + 10 i c) 15 + 18 i d) 15 + 18 i Copyright © 2011 Pearson Education, Inc. Slide 10 - 15

Multiply. ( 4 + 7 i)(2 + i) a) 15 + 10 i b) 1 + 10 i c) 15 + 18 i d) 15 + 18 i Copyright © 2011 Pearson Education, Inc. Slide 10 - 16

Divide Complex Numbers

Divide Complex Numbers Binomial denominator standard a + bi form conjugate

Write in standard form. a) b) c) d) Copyright © 2011 Pearson Education, Inc. Slide 10 - 19

Write in standard form. a) b) c) d) Copyright © 2011 Pearson Education, Inc. Slide 10 - 20

Powers of i:

Powers of i:

Simplify: