18.03 Class 6, Feb 21, 2006
Roots of Unity, Euler's formula, Sinusoidal functions
[1] Roots of unity
Let a > 0 . Since i^2 = 1 ,
(+ i sqrt(a))^2 =  a :
Negative real numbers have square roots in C.
Any quadratic polynomial with real coefficients has a root in C ,
by the quadratic formula
x^2 + bx + c
=
0 has roots (b + sqrt(b^2  4c))/2
In fact:
"Fundamental Theorem of Algebra":
Any polynomial has a root in C (unless it is a constant function).
Special case: z^n = 1 : "nth roots of unity"
n
=
2 :
z
=
+ 1
In general, if z^n = 1 , then z^n = 1 , but Magnitudes Multiply,
so z = 1 : roots of unity lie on the unit circle.
n = 3 : Angles Add, so if z^3 = 1 then the argument of z is 0
.....
no, not quite: it could be 2 pi / 3 , since three times that is 2 pi.
It's better to think of the argument of 1 as a choice:
0, or 2 pi, or 2 pi, or 4 pi, or
....
This gives
( 1 + sqrt 3 i ) / 2 .
Or it could be 4 pi / 3 , which gives
( 1  sqrt 3 i ) / 2
That's it, there's no other way for it to happen. The cube roots of
unity
start with 1 and divide the unit circle evenly into 3 parts.
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 Winter '08
 Staff
 Real Numbers, Square Roots, Cos, Complex number, Euler, sinusoidal function

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