﻿ IMAGINARY NUMBERS - Complex Numbers - Numbers and Operations - REVIEW OF MAJOR TOPICS - SAT SUBJECT TEST MATH LEVEL 2 ﻿

## PART 2 ## REVIEW OF MAJOR TOPICS ## 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

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