Adding Complex Numbers Graphing Worksheet
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S. I. M. Mr. Plassmann Section 1.5: Complex Numbers Part I Objectives: 1. Identify and graph complex numbers. 2. Perform operations with complex numbers. 3. Calculate the Norm or Absolute Value of a complex number. Vocabulary: Complex Numbers Imaginary Unit Real Part Imaginary Part Complex Plane Real Axis Imaginary Axis Complex Conjugate Gaussian Integers P. O. T. D.: 1. See attached worksheet. 2. Make an addition and multiplication table for 7 . Use them to find the following numbers in 1 7: 5, 2 5 3, 43 , , 2 3 Solve the equation x 9 . 2 3. Lesson: I. Complex Numbers The complex numbers arose out of the study of algebra and the attempt to be able to solve any polynomial equations with integral coefficients. Though they were discovered as early as the 1 st centrury AD, they were not universally accepted until the extensive investigation of them by the German Mathematician Carl Friedrich Gauss (1777 – 1855). The term “imaginary” was actually meant to be derogatory because of the skepticism surrounding the acceptance of these ideas. Recall that complex numbers are numbers in the form a bi , where a and b are real numbers and i , the imaginary unit, is defined by i 1 . We refer to a as the real part and bi as the imaginary part of the complex number. Gauss was the first to interpret the complex numbers geometrically. That is he was the first to graph them. To do this he used the complex plane. A pair of axis used to graph the two parts of a complex number where the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. Example #1: Graphing Complex Numbers Graph the complex numbers 2 3i and 4i . II. Operations with Complex Numbers Operations with complex numbers may be performed according to the rules in the following table. Operations with Complex Numbers For any complex numbers a bi and c di , the following is true. Addition (a bi) (c di) (a c) (b d )i Subtraction (a bi) (c di) (a c) (b d )i Multiplication (a bi)(c di) (ac bd ) (ad bc)i a bi (ac bd ) (bc ad )i Division c di c2 d 2 Complex numbers can be added and subtracted by combining their real parts and their imaginary parts. Example #2: Adding and Subtracting Complex Numbers A. (2 5i) (8 i) B. (9 6i) (5 i) The product of two or more complex numbers can be found by using the same procedures you when multiplying binomials. Example #3: Multiplying Complex Numbers Simplify (1 3i)(2 4i) Pairs of complex numbers of the form a bi and a bi are called complex conjugates. The product of two complex conjugates is always a real number. To divide two complex numbers you always multiply the numerator and denominator by the complex conjugate of the denominator. Example #4: Dividing Complex Numbers 4 3i Simplify 2i III. The Absolute Value or Norm of Complex Numbers Remember that the absolute value of a number is simply defined as the distance the number is from the origin. Because we graph complex numbers on a rectangular coordinate system, the real and imaginary coordinates along with the distance from the point to the origin make a right triangle, we may use the Pythagorean theorem to calculate the distance (the Pythagorean theorem has the sum of two squares in it, like the P. O. T. D.!). Another way to find this value is to multiply the complex number by its conjugate. Algebraically this results in the same calculation as the Pythagorean theorem. Example #5: Finding the Norm of a Complex Number Use the Pythagorean theorem or the conjugate product to find the absolute value of the complex numbers in example #1. IV. Homework Assignment 1.5 PRACTICE: 1. Perform the following operations with complex numbers: A. 3 5i 10 12i B. 18 3i 2 6i 4 10i C. 4 18i 2 3i D. 5 3i 2. Find the reciprocal of the complex number 8 2 2i . 1 3 3. A. Graph the complex numbers 2, , 2, and 6i . 2 B. Calculate their norms (absolute value). S. I. M. Mr. Plassmann Section 1.5 Homework: Complex Numbers 1. A. Graph the complex numbers (7 2i ) and (1 5i) . B. Find their absolute value. 2. Perform the following operations with complex numbers: A. 8 9i 24 13i B. 29 7i 17 6i 8 7i C. 2 5i 3 4i D. 4 9i S. I. M. Mr. Plassmann Investigating Square Numbers 1. Below, make a list of all the perfect squares less than or equal to 100. That is all the square numbers. 2. Now, use your list to determine which numbers can be written as the sum of two squares. That is the numbers which may be written as the sum of two perfect squares. Circle the numbers in the list below which may be written as the sum of two squares. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 3. Any conjectures? What structure do these numbers exhibit?