1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section A.8 Introduction to Complex Numbers

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Square Roots of Negative Numbers i and i 2

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Square Roots of Negative Numbers If a is a positive real number, then Note: The number i is not under the radical sign. To avoid confusion, we sometimes write

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Square Roots of Negative Numbers Simplify the following radicals.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Real and Imaginary Parts of Complex Numbers Complex Numbers The standard form of a complex number is a  bi, where a and b are real numbers. a is called the real part and b is called the imaginary part. If b  0, then a  bi  a  0i  a is a real number. If a  0, then a  bi  0  bi  bi is called a pure imaginary number (or an imaginary number).

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Real and Imaginary Parts of Complex Numbers Complex Numbers (cont.) Complex Number: a + bi real part imaginary part

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Real and Imaginary Parts of Complex Numbers Notes The term “imaginary” is somewhat misleading. Complex numbers and imaginary numbers are no more “imaginary” than any other type of number. In fact, all the types of numbers that we have studied (whole numbers, integers, rational numbers, irrational numbers, and real numbers) are products of human imagination.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Real and Imaginary Parts Identify the real and imaginary parts of each complex number. a.4 −2i

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Real and Imaginary Parts (cont.) c.7

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations with Complex Numbers Equality of Complex Numbers For complex numbers a  bi and c  di, if a  bi = c  di, then a  c and b  d.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations Solve each equation for x and y. a.(x  3)  2yi  7  6i b. 2y  3  8i  9  4xi

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition and Subtraction with Complex Numbers For complex numbers a  bi and c  di,

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition and Subtraction with Complex Numbers Find each sum or difference as indicated. a.(6  2i)  (1  2i)

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition and Subtraction with Complex Numbers (cont.)