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Adding Complex Numbers Graphing Worksheet document sample
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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?
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