SAT SUBJECT TEST MATH LEVEL 1
MISCELLANEOUS TOPICS
CHAPTER 17
Imaginary and Complex Numbers
• Imaginary Numbers
• Complex Numbers
• Exercises
• Answers Explained
Approximately 47 of the 50 questions on the Math 1 test are on topics covered in Chapters 2–16. However, as indicated on the chart on WHAT TOPICS ARE COVERED?, there are generally three questions on miscellaneous topics not covered in those chapters. In Chapters 17–19, you will read about the three topics that occur most often among the miscellaneous questions: imaginary numbers, sequences, and logic.
If your study time is limited, you may want to skip this material and concentrate on the chapters about algebra and geometry, which are the sources of the overwhelming majority of the questions on the Math 1 test. If, however, you have the time, you should study this chapter carefully. If you review the material in Chapters 17, 18, and 19, you should be able to answer any question that comes up on these topics.
IMAGINARY NUMBERS
By KEY FACT A1, if x is a real number, then x is positive, negative, or zero. If x = 0, then x ^{2} = 0; and if x is either positive or negative, then x ^{2} is positive. So if x is a real number, x ^{2} CANNOT be negative.
In the set of real numbers, the equation x ^{2} = –1 has no solution. In order to solve such an equation, mathematicians defined a new number, i, called the imaginary unit, with the property that i ^{2} = –1. This number is often referred to as the square root of –1: i= . Note that i is not a real number and does not correspond to any point on the number line.
All of the normal operations of mathematics—addition, subtraction, multiplication, and division—can be applied to the number i.
Addition: |
i + i = 2i |
2i + 5i = 7i |
Subtraction: |
i – i =0 |
2i – 5i = –3i |
Multiplication: |
(i)(i) = i^{2} = –1 |
(2i)(5i) = 10i^{2} = – 10 |
Division: |
Key Fact P1
If x is a positive number, then .
EXAMPLE 1:
Don”t Get Fooled
= (2i)(2i) = 4i^{2} = –4
is not equal to
EXAMPLE 2: What is
For the Math 1 test, you must be able to raise i to any positive integer power. In particular, note:
i^{1} = i |
(a^{1} = a for any number) |
i ^{2} = –1 |
(by definition) |
i ^{3} = –i |
i.i i = (i.i )(i ) = i^{2}.i = –1(i ) = –i |
i ^{4} = 1 |
i ^{4} = i.i.i.i = (i.i)(i.i ) = (–1)(–1) = 1 |
i ^{5} =i |
i^{5} = i^{4}.i = 1.i = i |
i ^{6} = –1 |
i ^{6} = i^{5}.i = i.i = i^{2} = –1 |
i ^{7} = –i |
i^{7} = i^{6}.i = (–1)i = –i |
i ^{8} = 1 |
i ^{8} = i^{7}.i = (–i)(i ) = –i^{2} = –(–1) = 1 |
Note that the powers of i form a repeating sequence in which the four terms, i, –1, –i, 1 repeat in that order indefinitely.
As you will see in KEY FACT P2, this means that to find the value of i^{n} for any positive integer n, you should divide n by 4 and calculate the remainder.
Key Fact P2
For any positive integer n:
• If n is a multiple of 4, i ^{n}= 1
• If n is not a multiple of 4, i ^{n}= i ^{r}, where r is the remainder when n is divided by 4.
EXAMPLE 3: To evaluate i ^{375}, use your calculator to divide 375 by 4:
375 4 = 93.75 the quotient is 93
Then multiply 93 by 4:
93 × 4 = 372 the remainder is 375 – 372 = 3
So i ^{375} = i ^{3} = –i.
The concepts of positive and negative apply only to real numbers. If a is positive, a is to the right of 0 on the number line. Since imaginary numbers do not lie on the number line, you cannot compare them. It is meaningless even to ask whether i is positive or negative or whether 12i is greater than or less than 7i.