7-7 Imaginary and Complex Numbers

Why Imaginary Numbers? What is the square root of 9?What is the square root of 9?

9 ?

What is the square root of -9?What is the square root of -9?

9 3 because 33 9

no real numberno real number

New type of number was defined for this purpose. New type of number was defined for this purpose. It is called an It is called an Imaginary NumberImaginary Number

Imaginary numbers are NOT in the Real Set.Imaginary numbers are NOT in the Real Set.

The constant, The constant, ii, is defined as the , is defined as the square root of negative 1: square root of negative 1:

i 1

Multiples of Multiples of ii are called Imaginary are called Imaginary NumbersNumbers

The square root of -9 is an imaginary The square root of -9 is an imaginary number...number...

9 9 1 3 i 3i

To simplify a square root with negative To simplify a square root with negative coefficient coefficient insideinside radical, write it as an radical, write it as an imaginary number.imaginary number.

Powers of Powers of ii::i

i2 1

i3 i2 i 1i i

i4 i2 i2 1 1 1

This pattern repeats:This pattern repeats:

i5 i4 i1i i

i6 i4 i2 1 1 1

i7 i4 i3 1 i i

i8 i4 i4 11 1

Multiples of i We can find higher powers of We can find higher powers of ii using this repeating pattern: using this repeating pattern: ii, -1, -, -1, -ii, 1, 1

i85 ?What is the highest number less than or equal to 85 that is What is the highest number less than or equal to 85 that is divisible by 4?divisible by 4? 84

So the answer is: So the answer is: i85 1i1 i

ii 214

Powers of i - Practice ii2828

ii7575

ii113113

ii8686

ii10891089

11--ii ii-1-1 ii

Negative Exponents

1i 1

1

1

1

ii

1

1Ex:

Ex:6i 16 i 1

1

11)(

22124

iiii

Odd negative powers are opposite

Even negative powers are the same!

Simplify:Simplify:

36 i6Ex 1:

Ex 2: 20 52i

Multiply

Ex 3Ex 3

Ex 4Ex 4

Ex 5Ex 5

)2(5 i i10

36 23182 i

)3(2 62i 6

Complex Numbers Complex NumberComplex Number : : a + bi ,a + bi ,

Where a and b are real #s and i is Where a and b are real #s and i is imaginary part imaginary part

real and imaginary numbers are real and imaginary numbers are not not like termslike terms, ,

Examples: 3 - 7Examples: 3 - 7ii, -2 + 8, -2 + 8ii, -4, -4ii, 5 + 2, 5 + 2ii

Complex #s

Real #s Imaginary #s

Irrational #sRational #s

Add and Subtract

Combine Like Terms Combine Like Terms

(the real & imaginary parts).(the real & imaginary parts). Example: Example:

((33 + + 44ii) + () + (-5-5 - 2- 2ii) =) = -2 -2 + + 22ii

PracticeAdd these Complex Numbers:Add these Complex Numbers:

(4 + 7(4 + 7ii) - (2 - 3) - (2 - 3ii)) (3 - (3 - ii) + (7) + (7ii)) (-3 + 2(-3 + 2ii) - (-3 + ) - (-3 + ii))

= 2 +10= 2 +10ii

= 3 + 6= 3 + 6ii

= = ii

Assignment

7-7/323/1-41, 43-48, 56-647-7/323/1-41, 43-48, 56-64