Chapter 7 Roots Radicals and Complex Numbers Copyright

Chapter 7 Roots, Radicals, and Complex Numbers Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -

Chapter Sections 7. 1 – Roots and Radicals 7. 2 – Rational Exponents 7. 3 – Simplifying Radicals 7. 4 – Adding, Subtracting, and Multiplying Radicals 7. 5 – Dividing Radicals 7. 6 – Solving Radical Equations 7. 7 – Complex Numbers Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -2 2

§ 7. 7 Complex Numbers Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -

Recognize a Complex Number An imaginary number is a number such as It is called imaginary because when imaginary numbers were first introduced, many mathematicians refused to believe they existed! Every imaginary number has as a factor. The , called the imaginary unit, is denoted by the letter i. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -4 4

Recognize a Complex Number Imaginary Unit Square Root of a Negative Number For any positive real number n, Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -5 5

Recognize a Complex Number Every number of the form Where a and b are real numbers and I is the imaginary unit, is a complex number. Every real number and every imaginary number are also complex numbers. A complex number has two parts: a real part, a, and an imaginary part, b. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -6 6

Recognize a Complex Number Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -7 7

Recognize a Complex Number Complex numbers can be added, subtracted, multiplied, and divided. To perform these operations, we use the definitions that and. Definition of i 2 If , then Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -8 8

Add and Subtract Complex Numbers To Add or Subtract Complex Numbers 1. Change all imaginary numbers to bi form. 2. Add (or subtract) the real parts of the complex numbers. 3. Add (or subtract) the imaginary parts of the complex numbers. 4. Write the answer in the form a + bi. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -9 9

Add and Subtract Complex Numbers Example Add Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -10 10

Multiply Complex Numbers To Multiply Complex Numbers 1. Change all imaginary numbers to bi form. 2. Multiply the complex numbers as you would multiply polynomials. 3. Substitute – 1 for each i 2. 4. Combine the real parts and the imaginary parts. Write the answer in a + bi form. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -11 11

Multiply Complex Numbers Example Multiply Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -12 12

CAUTION! Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -13 13

Divide Complex Numbers The conjugate of a complex number a + bi is a – bi. For example, Complex Number Conjugate Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -14 14

Divide Complex Numbers To Divide Complex Numbers 1. Change all imaginary numbers to bi form. 2. Multiply both the numerator and denominator by the conjugate of the denominator. 3. Write the answer is a + bi form. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -15 15

Divide Complex Numbers Example Divide Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -16 16

Find Powers of i The successive powers of i rotate through the four values of i, -1, -i, and 1. in = i if n = 1, 5, 9, … in = 1 if n = 4, 8, 12, … in = -1 if n = 2, 6, 10, … in = -i if n = 3, 7, 11, … Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 7 -17 17