NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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A rational number is a real
number that can be expressed as a ratio of two
integers:
Applying property (2) of exponents, we have
Thus, the only thing new is 
.
Since
we see that is the 
th root of 
.
This is sometimes written
The th root
of a real (or complex) number is not unique. As we all know, square
roots give two values (e.g., 
). In the general case of th roots,
there are
distinct values, in general.
How do we come up with
different numbers which when raised to the th power
will yield ? The
answer is to consider complexnumbers in polar
form. By Euler's
Identity, the real number 
can
be expressed, for any integer 
, as
. Using this form for
, the
number
can be written as
We can now see that we get a different
complex number for each 
. When 
, we get the same thing as when 
. When
, we
get the same thing as when 
, and
so on, so there are only cases using this construct. The th root
for can be
called the ''primitive th root of '', since integer powers of it give all of the others.
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