Imaginary Numbers

The Imaginary Number I The number i The basic element of the set of imaginary numbers. It is defined as follows: i = \^~T, and fl — - L The number j is used to simplify square roots of negative numbers. For instance, if r is a positive real number, then V - r = z'Vrl Examples: V — 3 = iV? V — 36 — i"V36 = 6i Pure imaginary number Any number of the form bi, b *£ 0. Examples: 3i and i V? CAUTION

o&.

When a and b are negative,

V36 For example: V^ 1 6(- 1) = -6 Correct: xT !2i • 3z = 6z2 To avoid making mistakes, always express the square root of a negative number as a pure imaginary number before performing any other operation. Example 1

Simplify:

a. \/-98

Solution

a. V^W = iV9l

b. ^-9 --^25 b.

r

^9 • V^IS = i-s/9 • /V25 = 3i • 5i = ISi2 = - 15

Simplify. 2. V"31!?: 6. 7i • 5i 10. (-31

S. 3^-~ 9. (6/)2

12. (2/V3)2

Example 2

Simplify:

Solution

To rationalize the denominator of a fraction, you must eliminate the imaginary number i from the denominator. Use the fact mat i2 = - — 1.

a.

5i

a.

10

J_ 5i

10 .

J0_ _ hj5_ i\/5 ' i-s/5

4i

w 4i

lOiVB

4J 5

13.

14. '*

, V24 3/V8

18.

U.

16. 20.