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# ho_complex - Taylors Theorem If you look up Taylors theorem...

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Taylor’s Theorem If you look up Taylor’s theorem in any decent calculus book, it will tell you that you can take any function and express it as a sum of polynomial terms. You can use Taylor’s theorem to find alternative ways of calculating transcendental functions such as sine, cosine and exponentiation. . . cos θ = 1 - θ 2 2! + θ 4 4! - θ 6 6! . . . sin θ = θ - θ 3 3! + θ 5 5! - θ 7 7! . . . e x = 1 + x + x 2 2! + x 3 3! + x 4 4! . . . For fun, take the derivative of each of these polynomial expressions. If everything was done correctly, the derivative of the function for sine should give you the function for cosine, the derivative of cosine should give you negative sine, and the derivative of e x should give you e x . Pretty cool, huh? You should also try plotting these polynomials on your calculator (do the trig functions in radians ). Imaginary Numbers In short, the numbers you’re used to playing with are members of the set of real numbers. There’s another set that most people don’t have occasion to use very often - the set of imaginary numbers. Take the square root of a positive real number - you’ll get a real number for an answer. Take the square root of a negative number, and you’ll get an imaginary number. - 1 i Imaginary numbers are multiples of i . Complex Numbers A complex number is simply a number with real and imaginary parts. For instance, 3 is a real number, 2 i is an imaginary number, 3 + 2 i is a complex number.

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ho_complex - Taylors Theorem If you look up Taylors theorem...

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