1. Section 7.7
Complex Numbers

2. Objectives Basic Concepts Addition, Subtraction, and Multiplication
Powers of i
Complex Conjugates and Division

3. PROPERTIES OF THE IMAGINARY UNIT i
THE EXPRESSION

5. Example
Write each square root using the imaginary i. a. b. c. Solution a. b. c.

6. SUM OR DIFFERENCE OF COMPLEX NUMBERS
Let a + bi and c + di be two complex numbers. Then
(a + bi) + (c+ di) = (a + c) + (b + d)i Sum
and
(a + bi) − (c+ di) = (a − c) + (b − d)i. Difference

7. Example
Write each sum or difference in standard form. a. (−8 + 2i) + (5 + 6i) b. 9i – (3 – 2i) Solution a. (−8 + 2i) + (5 + 6i) = (−8 + 5) + (2 + 6)I = −3 + 8i b. 9i – (3 – 2i) = 9i – 3 + 2i = – 3 + (9 + 2)I = – i

8. Example
Write each product in standard form. a. (6 − 3i)(2 + 2i) b. (6 + 7i)(6 – 7i) Solution a. (6 − 3i)(2 + 2i) = (6)(2) + (6)(2i) – (2)(3i) – (3i)(2i) = i – 6i – 6i2 = i – 6i – 6(−1) = i

9. Example (cont)
Write each product in standard form. a. (6 − 3i)(2 + 2i) b. (6 + 7i)(6 – 7i) Solution b. (6 + 7i)(6 – 7i) = (6)(6) − (6)(7i) + (6)(7i) − (7i)(7i) = 36 − 42i + 42i − 49i2 = 36 − 49i2 = 36 − 49(−1) = 85

10. POWERS OF i
The value of in can be found by dividing n (a positive integer) by 4. If the remainder is r, then
in = ir.
Note that i0 = 1, i1 = i, i2 = −1, and i3 = −i.

11. Example
Evaluate each expression. a. i25 b. i7 c. i44 Solution a. When 25 is divided by 4, the result is 6 with the remainder of 1. Thus i25 = i1 = i. b. When 7 is divided by 4, the result is 1 with the remainder of 3. Thus i7 = i3 = −i. c. When 44 is divided by 4, the result is 11 with the remainder of 0. Thus i44 = i0 = 1.

12. Example
Write each quotient in standard form. a. b. Solution a.

13. Example (cont) b.