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Complex Numbers 427
Lesson
Complex Numbers
Lesson 6– 9
BIG IDEABIG IDEA Complex numbers are numbers of the form a + bi, where i = √
__
- 1 , and are operated with as if they are polynomials in i.
Many aspects of an electrical charge, such as voltage (electric potential) and current (movement of an electric charge), affect the performance and safety of the charge. When working with these two quantities, electricians find it easier to combine them into one number Z, called impedance. Impedance in an alternating–current (AC) circuit is the amount, usually measured in ohms, by which the circuit resists the flow of electricity. The two–part number Z is the sum of a real number and an imaginary number, and is called a complex number.
What Are Complex Numbers?
Recall from the previous lesson that the set of numbers of the form bi , where b is a real number, are called pure imaginary numbers. When a real number and a pure imaginary number are added, the sum is called a complex number.
Definition of Complex Number
A complex number is a number of the form a + bi, where a and b are real numbers and i = √
__
In the complex number a + bi , a is the real part and b is the imaginary part. For example, – 8.5 - 4 i is a complex number in which the real part is – 8.5 and the imaginary part is – 4 (not 4 i or – 4 i ).
We say that a + bi and c + di are equal complex numbers if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di if and only if a = c and b = d. For example, if x + yi = 2 i - 3, then x = – 3 and y = 2.
QY
Mental Math
Write an inequality to represent the sentence. a. The weight w of my carry–on luggage must be less than 30 pounds. b. A medium–size sock can be worn by anyone with a shoe size s from 7 to 10. c. To ride this roller coaster, your height h must be at least 54 inches.
Mental Math
Write an inequality to represent the sentence. a. The weight w of my carry–on luggage must be less than 30 pounds. b. A medium–size sock can be worn by anyone with a shoe size s from 7 to 10. c. To ride this roller coaster, your height h must be at least 54 inches.
QY If a + bi = – 4 + √–3, what is a and what is b?
QY If a + bi = – 4 + √–3, what is a and what is b?
Vocabulary complex number real part, imaginary part equal complex numbers complex conjugate
428 Quadratic Functions
Chapter 6
Operations with Complex Numbers
All of the assumed properties of addition, subtraction, multiplication, and division of real numbers hold for complex numbers.
Properties of Complex Numbers Postulate
In the set of complex numbers: Addition and multiplication are commutative and associative. Multiplication distributes over addition and subtraction. 0 = 0 i = 0 + 0 i is the additive identity; 1 = 1 + 0 i is the multiplicative identity. Every complex number a + bi has an additive inverse – a + – bi and a multiplicative inverse _____ a +^1 bi provided a + bi ≠ 0. The addition and multiplication properties of equality hold.
You can use the properties to operate with complex numbers in a manner consistent with the way you operate with real numbers. You can also operate with complex numbers on a CAS.
Step 1 Add the complex numbers. (2 + 3 i) + (6 + 9 i) =? (4 - 3 i) + (7 + 5 i) =? (– 16 + 5 i) + (4 - 8 i) =? Step 2 Subtract the complex numbers. (4 - 3 i) - (6 + 5 i) =? (– 2 + i) - (7 + 9 i) =? (8 - 4 i) - (1 - i) =? Step 3 Describe, in words and using algebra, how to add and subtract two complex numbers. Step 4 Check your answer to Step 3 by calculating (a + bi) + (c + di) and (a + bi) - (c + di) on a CAS.
In the Activity, you should have seen that the sum or difference of two complex numbers is a complex number whose real part is the sum or difference of the real parts and whose imaginary part is the sum or difference of the imaginary parts.
The Distributive Property can also be used to multiply a complex number by a real number or by a pure imaginary number.
ActivityActivity
430 Quadratic Functions
Chapter 6
Complex conjugates are useful when dividing complex numbers. To divide two complex numbers, multiply both numerator and denominator by the conjugate of the denominator. This gives a real number in the denominator that you can then divide into each part of the numerator.
Example 3
Simplify 3 _____ 3 +-^^62 ii. Solution Multiply the numerator and denominator by 3 + 2 i, the conjugate of 3 - 2 i.
3 ______ + 6i
3 - 2i =^
3 ______ + 6i
3 - 2i ·^
3 ______ + 2i
3 + 2i Identity Property of Multiplication
= ______________ (3(3 - + 2i)(36i)(3 ++ 2i)2i) Multiplication of fractions
= _____?? Distributive Property (Expand.)
= ________ 9 -? 4i 2 Distributive Property (Combine like terms.)
= ________ 9 -? 4(?) Definition of i
= __ 13? + __ 13? i Distributive Property
(adding fractions)
Check Divide on a CAS. It checks.
The Various Kinds of
Complex Numbers
Because a + 0 i = a , every real number a is a complex number. Thus, the set of real numbers is a subset of the set of complex numbers. Likewise, every pure imaginary number bi equals 0 + bi , so the set of pure imaginary numbers is also a subset of the set of complex numbers.
The diagram at the right is a hierarchy of number sets. It shows how the set of complex numbers includes some other number sets.
GUIDEDGUIDED
rational numbers irrational numbers
complex numbers a + bi
nonreal numbers a + bi , b ≠ 0
pure imaginary numbers 0 + bi , b ≠ 0
real numbers a + 0 i
integers
natural numbers
rational numbers irrational numbers
complex numbers a + bi
nonreal numbers a + bi , b ≠ 0
pure imaginary numbers 0 + bi , b ≠ 0
real numbers a + 0 i
integers
natural numbers
Complex Numbers 431
Lesson 6– 9
Applications of Complex Numbers
The first use of the term complex number is generally credited to Carl Friedrich Gauss. Gauss applied complex numbers to the study of electricity. Later in the 19th century, applications using complex numbers were found in geometry and acoustics. In the 1970s, complex numbers were used in a new field called dynamical systems.
Recall that electrical impedance Z is defined as a complex number involving voltage V and current I. A complex number representing impedance is of the form Z = V + Ii.
The total impedance ZT of a circuit made from two connected circuits is a function of the impedances Z 1 and Z 2 of the individual circuits. Two electrical circuits may be connected in series or in parallel.
In a series circuit, ZT = Z 1 + Z 2. In a parallel circuit, ZT = ______ Z^1 Z^2 Z 1 + Z 2. Thus, to find the total impedance in a parallel circuit, you need to multiply and divide complex numbers.
Example 4
Find the total impedance in a parallel circuit if Z 1 = 3 + 2 i ohms and Z 2 = 5 - 4 i ohms.
Solution Substitute the values of Z 1 and Z 2 into the impedance formula for parallel circuits.
Z (^) T =
______Z^1 Z^2
Z 1 + Z (^2)
Z T = ____________(3^ +^ 2i)(5? +?^ -^ 4i) Substitution
= ?_____?^ +-^ ?i?i
= ?_____?^ +-^ ?i?i · ?_____?^ ++^ ?i?i Multiply numerator and denominator by
the conjugate of the denominator.
Z T = ___ 68? Definition of i and arithmetic
The total impedance is ___ 68? ohms.
The basic properties of inequality that hold for real numbers do not hold for nonreal complex numbers. For instance, if you were to assume i > 0, then multiplying both sides of the inequality by i , you would get i · i > 0 · i , or – 1 > 0, which is not true. If you assume i < 0, then multiply both sides by i , you get (changing the direction) i · i > 0 · i , or again – 1 > 0. Except for those complex numbers that are also real numbers, there are no positive or negative complex numbers.
GUIDEDGUIDED
parallel circuit
series circuit
parallel circuit
series circuit
Complex Numbers 433
Lesson 6– 9
- A complex number a + bi is graphed as the point ( a , b ) with the x – axis as the real axis and the y – axis as the imaginary axis. a. Refer to the graph at the right. S and T are the graphs of which complex numbers? b. Graph S + T. c. Connect (0, 0), S , S + T , and T to form a quadrilateral. What type of quadrilateral is formed?
- a. Solve the equation x^2 - 6 x + 13 = 0 using the Quadratic Formula. Write the solutions in a + bi form. b. How are the solutions to the equation x^2 - 6 x + 13 = 0 related to each other?
In 22 and 23, consider the complex numbers a + bi and a - bi.
- Find their sum and explain why it is a real number.
- Find their product and explain why it is a real number.
- Prove that, if two circuits connected in parallel have impedances Z 1 and Z 2 and the total impedance is ZT , then ___ Z^1 T
= ___ Z^1
1
+ __ Z^1
2
REVIEW
In 25 and 26, solve. Write the solutions as real numbers or multiples of i. (Lesson 6–8)
- a^2 - 3 = – 8 26. – r^2 = 196
- Explain where the ± comes from in Step 8 of the derivation of the Quadratic Formula in Lesson 6–7. (Lesson 6–2)
- Write a piecewise definition for the function g with equation g ( x ) = x + 2 . (Lessons 6–2, 3–4)
- The two graphs at the right show the relationships between a dependent variable R and independent variables m and p. Find an equation for R in terms of m and p. (You may leave the constant of variation as k .) (Lesson 2–8)
EXPLORATION
- Refer to Question 20. a. Graph z = 1 + i as the point (1, 1). b. Compute and graph z^2 , z^3 , and z^4. c. What pattern emerges? Predict where z^5 , z^6 , z^7 , and z^8 will be.
y
x 0 2 4
4 S 2
T
4 2 2 4
y
x 0 2 4
4 S 2
T
4 2 2 4
20 10 0
30
40
60
70
50
0 1 2 3 4 5 6
R
m
p is held constant.
20 10 0
30
40
60
70
50
0 1 2 3 4 5 6
R
p
m is held constant.
20 10 0
30
40
60
70
50
0 1 2 3 4 5 6
R
m
p is held constant.
20 10 0
30
40
60
70
50
0 1 2 3 4 5 6
R
p
m is held constant.
QY ANSWERS
- a = – 4, b = √ 3
- 25
