SAT SUBJECT TEST MATH LEVEL 2
PART 2
REVIEW OF MAJOR TOPICS
CHAPTER 3
Numbers and Operations
3.2 Complex Numbers
IMAGINARY NUMBERS
The square of a real number is never negative. This means that the square root of a negative number cannot be a real number. The symbol is called the imaginary unit, i2 = –1. Powers of i follow a pattern:
Power of i |
Intermediate Steps |
Value |
i1 |
i |
i |
i2 |
i · i = –1 |
–1 |
i3 |
i2 · i = (–1) · i = –i |
–i |
i4 |
i3 · i = (–i) · i = –i2 = –(–1) = 1 |
1 |
i5 |
i4 · i = 1 · i = i |
i |
In other words, powers of i follow a cycle of four. This means that in = in mod 4, where n mod 4 is the remainder when n is divided by 4. For example, i 58 = i2 = –1.
The imaginary numbers are numbers of the form bi , where b is a real number. The square root of any negative number is i times the square root of the positive of that number. Thus for example, , and .
EXERCISE
1. i29 =
(A) 1
(B) i
(C) –i
(D) –1
(E) none of these