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9-69-6ComplexComplexNumbersNumbers

Solve: Solve: 04/20/2304/20/239-69-6

012 x

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Imaginary NumbersImaginary NumbersImaginary Number:Imaginary Number:

Simplifying Imaginary Numbers:Simplifying Imaginary Numbers:

1i

3 3i

4 4i i2

18 18i 29i 23i

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Complex NumbersComplex Numbers

Complex numbers: Complex numbers: When a real number is combined with an imaginary number

- a number in the form of a + b- a number in the form of a + bii where: where:• a is the a is the real partreal part• bbii is the is the imaginary partimaginary part

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Operations on Complex NumbersOperations on Complex Numbers

Let x = 3 + 2Let x = 3 + 2ii and y = 1 – 3 and y = 1 – 3ii. Find:. Find:

a)a)x + y:x + y:

b)b)x – y:x – y:

c)c)xy:xy: So: Treat “i ” like

a variable and follow all algebra rules.

Factoring: The difference ofFactoring: The difference of perfect squares… perfect squares…

Factor the following:Factor the following:

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42 x )2)(2( xx

32 x )3)(3( xx

52 x)5)(5( xx

)5(2 x

)5)(5( ixix These are an example of Complex Conjugates.

Where do we find Where do we find complex conjugates?complex conjugates?

Solve the following equation:Solve the following equation:

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0452 2 xx

Other stuff…Other stuff…Factor Theorem still holds true:Factor Theorem still holds true:

If “If “i i ” is a solution, then (x – ” is a solution, then (x – ii) is a factor. ) is a factor. (and vise-versa)(and vise-versa)

All real numbers can be written as All real numbers can be written as complex numbers, and so can imaginary complex numbers, and so can imaginary numbers:numbers:

ex: 4ex: 4ii = 0 + 4 = 0 + 4ii

ex: 2 = 2 + 0ex: 2 = 2 + 0ii

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