1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.6 The Complex Number System

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Topics o The imaginary unit i and its properties o The algebra of complex numbers o Roots and complex numbers

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Imaginary Unit i and its Properties The Imaginary Unit i The imaginary unit i is defined as In other words, i has the property that its square is −1:

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Imaginary Unit i and its Properties Square Roots of Negative Numbers If a is a positive real number, Note that by this definition, and by a logical extension of exponentiation,

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1 As is customary, we write a constant such as 4 before letters in algebraic expressions, even if, as in this case, the letter is not a variable. Remember that i has a fixed meaning: i is the square root of −1. As is customary, again, we write the radical factor last. You should verify that is indeed −8.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1 (cont.) The simple fact that i 2 = −1 allows us, by our extension of exponentiation, to determine i n for any natural number n. This observation shows that −i also has the property that its square is −1.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Imaginary Unit i and its Properties Complex Numbers For any two real numbers a and b, the sum a + bi is a complex number. The collection  = {a + bi|a and b are both real} is called the set of complex numbers and is another example of a field. The number a is called the real part of a + bi, and the number b is called the imaginary part. If the imaginary part of a given complex number is 0, the number is simply a real number. If the real part of a given complex number is 0, the number is a pure imaginary number.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Algebra of Complex Numbers Simplifying Complex Expressions Step 1:Add, subtract, or multiply the complex numbers, as required, by treating every complex number a + bi as a polynomial expression. Remember, though, that i is not actually a variable. Treating a + bi as a binomial in i is just a handy device. Step 2:Complete the simplification by using the fact that i 2 = −1.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2 Simplify the following complex number expressions. Solutions: Treating the two complex numbers as polynomials in i, we combine the real parts and then the imaginary parts.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2 (cont.) We begin by distributing the minus sign over the two terms of the second complex number, and then combine as in part a.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2 (cont.) The product of two complex numbers leads to four products via the distributive property, as illustrated here. After multiplying, we combine the two terms containing i, and rewrite i 2 as −1.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2 (cont.) Squaring this complex number also leads to four products, which we simplify as in part c. Remember that a complex number is not simplified until it has the form a + bi.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3 Simplify the quotients.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3(cont.) Solutions: We multiply the top and bottom of the fraction by 3 + i, which is the complex conjugate of the denominator. The rest of the simplification involves multiplying complex numbers as in the last example. We would often leave the answer in the form unless it is necessary to identify the real and imaginary parts.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3(cont.) In this example, we simplify the reciprocal of the complex number 4 − 3i. After writing the original expression as a fraction, we multiply the top and bottom by the complex conjugate of the denominator and proceed as in part a.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3(cont.) This problem illustrates the process of writing the reciprocal of the imaginary unit as a complex number. Note that with this as a starting point, we could now calculate i –2, i –3, …

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4 Simplify the following expressions.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4 (cont.) Solutions: Each is converted to before carrying out the associated multiplications. Note that incorrect use of one of the properties of radicals would have led to adding 3 instead of subtracting 3.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4 (cont.) We have already simplified in Example 3c, so we quickly obtain the correct answer of −i. If we had incorrectly rewritten the original fraction as we would have obtained or i as the final answer.