5.4 Imaginary Numbers Day 36

8/2/2019 5.4 Imaginary Numbers Day 36

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2

2

2


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i2i =


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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) = 1. (-2) = 4 (-3) = 9


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So heres what the mathpeople did: They used theletter i to represent thesquare root of (-1). istands for imaginary.

2i =

So, does 2really exist?


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Examples of how weuse

2i =

22 22 2 =

2 i=

2i=

22 22 2 =

2 i=

2i=


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Examples of how weuse

2i =

22 22 2 =

2 2 2 2=

2 2 2=

22 i=

22i=


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2 2 2 2 2 2=

222

2 2 2 2=

22 2 i=

22 2i=

222 2=


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The first four powers ofi establishan important pattern and should

be memorized.

Powers ofi1 2 1i i i= =

3 4 1i i i= =

4


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Divide the exponent by 4No remainder: answer is1.Remainder of 1: answer

is i.

i4 1=

i i1

=

i2 1=

i i3

=


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Powers of i

Find i23

Find i2006

Find i37

Find i828

i=

1=

i=

1=


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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.

xpress ese num ers n erms


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1.) 5 1 5= 1 5= 5i=

1 7= 1 7=

7i=

1 99=

1 99=

3 11i=

xpress ese num ers n ermsof i.

2.) 7

3.) 99

= i 3 3 11


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94i=

2

2 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.


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To mult. imaginarynumbers or an

imaginary numberby a real number,

its important to 1stexpress the

imaginary numbers

in terms of i.


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a + bi

Complex Numbers

real imaginary

The complex numbers consist of allsums a + bi, where a and b are realnumbers and i is the imaginary unit.

The real part is a, and the imaginarypart is bi.


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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.


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Examples

2)3(1. i22

)3(i=

1( 3 3)=

)3(1=

3=

26103Solve2. 2 =+x

363 2 =x122 =x

122

=x

12ix =

32ix =

M l i l i


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MultiplyingTreat the is like variables, thenchange 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 += i

i226 =


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5.4 Imaginary Numbers Day 36