Complex operations
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This document provides examples and explanations for performing operations with complex numbers. It begins with graphing complex numbers on a complex plane. It then covers determining the absolute value of complex numbers, adding and subtracting complex numbers by combining real and imaginary parts, and multiplying complex numbers using distribution and the property that i^2 = -1. It also discusses evaluating powers of i by expressing them as one of four values: i, -1, -i, or 1. The document aims to teach students how to perform basic operations like addition, subtraction, multiplication, and evaluating powers with complex numbers.
Complex operations
- 1. Operations with Complex Numbers Warm Up Lesson Presentation Lesson Quiz
- 2. Warm Up Express each number in terms of i. 1. 9i 2. Find each complex conjugate. 3. 4. Find each product. 5. 6.
- 3. Objective Perform operations with complex numbers.
- 4. Vocabulary complex plane absolute value of a complex number
- 5. Just as you can represent real numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane. The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
- 6. Helpful Hint The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi.
- 7. Example 1: Graphing Complex Numbers Graph each complex number. A. 2 – 3i –1+ 4i • B. –1 + 4i 4+i • C. 4 + i • –i D. –i • 2 – 3i
- 8. Check It Out! Example 1 Graph each complex number. a. 3 + 0i b. 2i 2i • 3 + 2i • c. –2 – i • • 3 + 0i –2 – i d. 3 + 2i
- 9. Recall that absolute value of a real number is its distance from 0 on the real axis, which is also a number line. Similarly, the absolute value of an imaginary number is its distance from 0 along the imaginary axis.
- 10. Example 2: Determining the Absolute Value of Complex Numbers Find each absolute value. A. |3 + 5i| B. |–13| C. |–7i| |–13 + 0i| |0 +(–7)i| 13 7
- 11. Check It Out! Example 2 Find each absolute value. a. |1 – 2i| b. c. |23i| |0 + 23i| 23
- 12. Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts. The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions.
- 13. Helpful Hint Complex numbers also have additive inverses. The additive inverse of a + bi is –(a + bi), or –a – bi.
- 14. Example 3A: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (4 + 2i) + (–6 – 7i) (4 – 6) + (2i – 7i) Add real parts and imaginary parts. –2 – 5i
- 15. Example 3B: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (5 –2i) – (–2 –3i) (5 – 2i) + 2 + 3i Distribute. (5 + 2) + (–2i + 3i) Add real parts and imaginary parts. 7+i
- 16. Example 3C: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (1 – 3i) + (–1 + 3i) (1 – 1) + (–3i + 3i) Add real parts and imaginary parts. 0
- 17. Check It Out! Example 3a Add or subtract. Write the result in the form a + bi. (–3 + 5i) + (–6i) (–3) + (5i – 6i) Add real parts and imaginary parts. –3 – i
- 18. Check It Out! Example 3b Add or subtract. Write the result in the form a + bi. 2i – (3 + 5i) (2i) – 3 – 5i Distribute. (–3) + (2i – 5i) Add real parts and imaginary parts. –3 – 3i
- 19. Check It Out! Example 3c Add or subtract. Write the result in the form a + bi. (4 + 3i) + (4 – 3i) (4 + 4) + (3i – 3i) Add real parts and imaginary parts. 8
- 20. You can also add complex numbers by using coordinate geometry.
- 21. Example 4: Adding Complex Numbers on the Complex Plane Find (3 – i) + (2 + 3i) by graphing. Step 1 Graph 3 – i and 2 + 3i 2 + 3i on the complex • plane. Connect each of these numbers to the origin with a line segment. • 3 –i
- 22. Example 4 Continued Find (3 – i) + (2 + 3i) by graphing. Step 2 Draw a parallelogram that has 2 + 3i these two line segments • as sides. The vertex that • 5 +2i is opposite the origin represents the sum of the two complex numbers, 5 + 2i. Therefore, (3 – i) + • 3 –i (2 + 3i) = 5 + 2i.
- 23. Example 4 Continued Find (3 – i) + (2 + 3i) by graphing. Check Add by combining the real parts and combining the imaginary parts. (3 – i) + (2 + 3i) = (3 + 2) + (–i + 3i) = 5 + 2i
- 24. Check It Out! Example 4a Find (3 + 4i) + (1 – 3i) by graphing. 3 + 4i Step 1 Graph 3 + 4i and • 1 – 3i on the complex plane. Connect each of these numbers to the origin with a line segment. 1 – 3i •
- 25. Check It Out! Example 4a Continued Find (3 + 4i) + (1 – 3i) by graphing. Step 2 Draw a 3 + 4i parallelogram that has • these two line segments as sides. The vertex that is opposite the origin • 4+i represents the sum of the two complex numbers, 4 + i. Therefore,(3 + 4i) + (1 – 3i) = 4 + i. • 1 – 3i
- 26. Check It Out! Example 4a Continued Find (3 + 4i) + (1 – 3i) by graphing. Check Add by combining the real parts and combining the imaginary parts. (3 + 4i) + (1 – 3i) = (3 + 1) + (4i – 3i) = 4 + i
- 27. Check It Out! Example 4b Find (–4 – i) + (2 – 2i) by graphing. Step 1 Graph –4 – i and 2 – 2i on the complex plane. Connect each of these numbers to the origin with a line –4 – i • 2 – 2i segment. ● 2 – 2i •
- 28. Check It Out! Example 4b Find (–4 – i) + (2 – 2i) by graphing. Step 2 Draw a parallelogram that has these two line segments as sides. The vertex that is opposite represents the sum of the two –4 – i • complex numbers, –2 – • 2 – 2i 3i. Therefore,(–4 – i) + • –2 – 3i (2 – 2i) = –2 – 3i.
- 29. Check It Out! Example 4b Find (–4 – i) + (2 – 2i) by graphing. Check Add by combining the real parts and combining the imaginary parts. (–4 – i) + (2 – 2i) = (–4 + 2) + (–i – 2i) = –2 – 3i
- 30. You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i2 = –1.
- 31. Example 5A: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. –2i(2 – 4i) –4i + 8i2 Distribute. –4i + 8(–1) Use i2 = –1. –8 – 4i Write in a + bi form.
- 32. Example 5B: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (3 + 6i)(4 – i) 12 + 24i – 3i – 6i2 Multiply. 12 + 21i – 6(–1) Use i2 = –1. 18 + 21i Write in a + bi form.
- 33. Example 5C: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (2 + 9i)(2 – 9i) 4 – 18i + 18i – 81i2 Multiply. 4 – 81(–1) Use i2 = –1. 85 Write in a + bi form.
- 34. Example 5D: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (–5i)(6i) –30i2 Multiply. –30(–1) Use i2 = –1 30 Write in a + bi form.
- 35. Check It Out! Example 5a Multiply. Write the result in the form a + bi. 2i(3 – 5i) 6i – 10i2 Distribute. 6i – 10(–1) Use i2 = –1. 10 + 6i Write in a + bi form.
- 36. Check It Out! Example 5b Multiply. Write the result in the form a + bi. (4 – 4i)(6 – i) 24 – 4i – 24i + 4i2 Distribute. 24 – 28i + 4(–1) Use i2 = –1. 20 – 28i Write in a + bi form.
- 37. Check It Out! Example 5c Multiply. Write the result in the form a + bi. (3 + 2i)(3 – 2i) 9 + 6i – 6i – 4i2 Distribute. 9 – 4(–1) Use i2 = –1. 13 Write in a + bi form.
- 38. The imaginary unit i can be raised to higher powers as shown below. Helpful Hint Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, –1, –i, or 1.
- 39. Example 6A: Evaluating Powers of i Simplify –6i14. –6i14 = –6(i2)7 Rewrite i14 as a power of i2. = –6(–1)7 = –6(–1) = 6 Simplify.
- 40. Example 6B: Evaluating Powers of i Simplify i63. i63 = i i62 Rewrite as a product of i and an even power of i. = i (i2)31 Rewrite i62 as a power of i2. = i (–1)31 = i –1 = –i Simplify.
- 41. Check It Out! Example 6a Simplify . Rewrite as a product of i and an even power of i. Rewrite i6 as a power of i2. Simplify.
- 42. Check It Out! Example 6b Simplify i42. i42 = ( i2)21 Rewrite i42 as a power of i2. = (–1)21 = –1 Simplify.
- 43. Recall that expressions in simplest form cannot have square roots in the denominator (Lesson 1-3). Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator. Helpful Hint The complex conjugate of a complex number a + bi is a – bi. (Lesson 5-5)
- 44. Example 7A: Dividing Complex Numbers Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.
- 45. Example 7B: Dividing Complex Numbers Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.
- 46. Check It Out! Example 7a Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.
- 47. Check It Out! Example 7b Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify.
- 48. Lesson Quiz: Part I Graph each complex number. 1. –3 + 2i 2. 4 – 2i –3 + 2i • 4 – 2i •
- 49. Lesson Quiz: Part II 3. Find |7 + 3i|. Perform the indicated operation. Write the result in the form a + bi. 4. (2 + 4i) + (–6 – 4i) –4 5. (5 – i) – (8 – 2i) –3 + i 6. (2 + 5i)(3 – 2i) 7. 3+i 16 + 11i 8. Simplify i31. –i