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, i^{2} = –1. Powers of i follow a pattern:
Power of i |
Intermediate Steps |
Value |
i^{1} |
i |
i |
i^{2} |
i · i = –1 |
–1 |
i^{3} |
i^{2} · i = (–1) · i = –i |
–i |
i^{4} |
i^{3} · i = (–i) · i = –i^{2} = –(–1) = 1 |
1 |
i^{5} |
i^{4} · i = 1 · i = i |
i |
In other words, powers of i follow a cycle of four. This means that i^{n} = i^{n}^{ mod 4}, where n mod 4 is the remainder when n is divided by 4. For example, i ^{58} = i^{2} = –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. i^{29} =
(A) 1
(B) i
(C) –i
(D) –1
(E) none of these