2

2

2

i1i

• You can't take the square root of a negative number, right?

• When we were young and still in Algebra I, no numbers that, when multiplied by themselves, gave us a negative answer. 

• Squaring a negative number always gives you a positive.   (-1)² = 1. (-2)² = 4 (-3)² = 9

So here’s what the math people did: They used the letter “i” to represent the square root of (-1). “i” stands for “imaginary.”

1i

So, does

1really exist?

Examples of how we use

1i

16 16 1

4 i

4i

81 81 1

9 i

9i

Examples of how we use

1i

45 45 1

3 3 5 1

3 5 1

3 5 i

3 5i

2 2 2 5 5 1

200

2 5 2 1

10 2 i

10 2i

200 1

The first four powers of i establish an important pattern and should

be memorized.

Powers of i1 2 1i i i

3 4 1i i i

Divide the exponent by 4No remainder: answer is 1.Remainder of 1: answer is i.Remainder of 2: answer is –1.Remainder of 3: answer is –i.

i4 1

i i1

i2 1

i i3

Powers of i

Find i23

Find i2006

Find i37

Find i828

i1i1

Complex Number System

Reals

Rationals(fractions, decimals)

Integers(…, -1, -2, 0, 1, 2, …)

Whole(0, 1, 2, …)

Natural(1, 2, …)

Irrationals(no fractions)

pi, e

Imaginary

i, 2i, -3-7i, etc.

1.) 5 1 5 1 5 5i

1 7 1 7

7i1 99 1 99

3 11i

Express these numbers in terms of i.

2.) 7

3.) 99

i 3 3 11

You try…

4.

5.

7

36

1606.

i 7

6i

4 10i

94i

22 5i

2 5

2 21i

( 1) 21 21

Multiplying

47 2i

2 5i

3 7

2 1 5i 2 5i i

i i3 7

7.

8.

9.

To mult. imaginary numbers or an

imaginary number by a real number,

it’s important to 1st express the

imaginary numbers in terms of i.

a + biComplex Numbers

real imaginary

The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

7.) 7 9i i 16i

8.) ( 5 6 ) (2 11 )i i 3 5i

9.) (2 3 ) (4 2 )i i 2 3 4 2i i 2 i

Add or Subtract

10.

11.

12.

Examples2)3( 1. i

22 )3(i1( 3 3)

)3(13

26103 Solve 2. 2 x

363 2 x122 x

122 x

12ix 32ix

MultiplyingTreat the i’s like variables, then change any that are not to the

first power

Ex: )3( ii 23 ii )1(3 i

i31

Ex: )26)(32( ii 2618412 iii

)1(62212 i

62212 ii226

3 11:

1 2

iEx

i

)21)(21(

)21)(113(

ii

ii

2

2

4221

221163

iii

iii

)1(41

)1(2253

i

41

2253

i

5

525 i

5

5

5

25 i

i 5

WorkWorkp. 277

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