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Complex numbers summary

Complex numbers summary - e i = cos i sin and by looking at...

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Complex numbers: a summary by Dr Colton Physics 123 Section 2, Fall 2010 We will soon be using complex numbers as a tool for describing waves. Chapter 1 of Physics phor Phynatics deals with this, but here is my own quick summary. Colton’s short complex number summary : A complex number x + iy can be written in rectangular or polar form, just like coordinates in the x - y plane. o The rectangular form is most useful for adding/subtracting complex numbers. o The polar form is most useful for multiplying/dividing complex numbers. The polar form ( A , θ ) can be expressed as a complex exponential Ae i . For example, consider the complex number 3 + 4 i : = (3, 4) in rectangular form, = (5, 53.13º) in polar form, and = 5 e 0.9273 i in complex exponential form, since 53.13º = 0.9273 rad. The complex exponential form follows directly from Euler’s equation:
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Unformatted text preview: e i = cos + i sin , and by looking at the x- and y-components of the polar coordinates. • By the rules of exponents, when you multiply/divide two complex numbers in polar form, ( A 1 , 1 ) and ( A 2 , 2 ), you get: o multiply: A 1 e i 1 × A 2 e i 2 = A 1 A 2 e i ( 1+ 2) = ( A 1 A 2 , 1 + 2 ) o divide: A 1 e i 1 ÷ A 2 e i 2 = ( A 1 /A 2 )e i ( 1-2) = ( A 1 / A 2 , 1 – 2 ) • The polar form is closely related to what Serway & Jewitt refer to as “phasors” (used to describe sinusoidal voltages in chapter 33), and is often written as: A ∠ . The “ ∠ ” symbol is read as, “at an angle of”. Thus you can write: (3 + 4i) × (5 + 12i) = 5 ∠ 53.13 ° × 13 ∠ 67.38 ° = 65 ∠ 120.51 ° (since 65 = 5 × 13 and 120.51 = 53.13 + 67.38)...
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