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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 Spring '10
 Corbin
 Complex number, Aeiθ

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