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5.4 Complex Numbers By: L. Keali’i Alicea

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Goals 1)Solve quadratic equations with complex solutions and perform operations with complex numbers. 2)Apply complex numbers to fractal geometry.

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Imaginary Unit Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can! The imaginary unit is ¡. ¡= It is used to write the square root of a negative number.

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Property of the square root of negative numbers If r is a positive real number, then Examples: ±

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*For larger exponents, divide the exponent by 4, then use the remainder as your exponent instead. Example:

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Examples

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Complex Numbers A complex number has a real part & an imaginary part. Standard form is: Real part Imaginary part Example: 5+4i

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The Complex plane Imaginary Axis Real Axis

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Graphing in the complex plane i- axis “Real” number axis

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Adding and Subtracting (add or subtract the real parts, then add or subtract the imaginary parts) Ex:

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Multiplying Treat the i’s like variables, then change any that are not to the first power Ex:

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Absolute Value of a Complex Number The distance the complex number is from the origin on the complex plane. If you have a complex number the absolute value can be found using:

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Examples 1. 2. Which of these 2 complex numbers is closest to the origin? -2+5i

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Assignment 5.4 A (All)

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