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, 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