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The definition and domain of exponentiation has changed several times over the
years. The original operation xy was only defined when y
was a positive integer. The domain of the operation of exponentiation has been
extended for the same reason George Leigh Mallory climbed Mount Everest:
"Because it's there", and also for the useful results provided by
these extensions.
The original definition of exponentiation is this: x real, y
positive integer, xy = 1 * x * ... * x,
where 1 is multiplied by x, y times. This operation has a number of properties, including
- x1 = x
- xnxm = xn+m,
x real, n, m integer.
- If x is positive, then xn is positive.
Now, we can try to see how far we can extend the domain of
exponentiation so that the above properties (and others) still hold. This
naturally leads to defining the operation xy on the domain x positive
real; y rational, by setting xp/q = the qth
root of xp. This
operation agrees with the original definition of exponentiation on their
common domain, and also satisfies (1), (2) and (3). In fact, it is the
unique operation on this domain that does so. This operation also has
some other properties:
- If x>1, then xy is an increasing function of y.
- If 0<x<1, then xy is a decreasing function of y.
Again, we can again see how far we can extend the domain of exponentiation
while still preserving properties (1)-(5). This leads naturally to the
following definition of xy on the domain x positive real; y real:
If x>1, xy is defined to be
supq{xq} , where q runs over a
ll rationals less than or equal to y.
If x<1, xy is defined to be
infq{xq} , where q runs over a
ll rationals bigger than or equal to y.
If x=1, xy is defined to be 1.
Again, this operation satisfies (1)-(5), and is in fact the only operation
on this domain to do so.
The next extension is somewhat more complicated. As can be proved using
the methods of calculus or combinatorics, if we define e to be the number
e = 1 + 1/1! + 1/2! + 1/3! + ... = 2.71828...
it turns out that for every real number x,
ex = 1 + x/1! + x2/2! + x3/3! + ...
This series always converges regardless of
the value of x.
-
ex is also denoted exp(x).
One can also define an operation ln(x) on the positive reals, which is the
inverse of the operation of exponentiation by e. In other words, exp(ln(x))
= x for all positive x. Moreover,
- If x is positive, then x is e(y
ln(x)). (To show this, exponent rule 2 says xnxm = xn+m,
which means (xn)m = xnm.
Using this rule, e(y
ln(x)) = (e ln(x))y = xy.)
Because of this, a natural extension of exponentiation to complex exponents is
exp(z) = 1 + z/1! + z2/2! + z3/3! + ...
for all complex z (not just the reals, as before), and to define
xz = exp(z ln(x))
when x is a positive real and z is complex.
This is the only operation xy on the domain x positive real, y complex
which satisfies all of (1)-(7). Because of this and other reasons, it
is accepted as the modern definition of exponentiation.
From the identities
sin(x) = x - x3/3! + z5/5! - z7/7! + ...
cos(x) = 1 - x2/2! + z4/4! - z6/6! + ...
which are the Taylor series expansion of the
trigonometric sine and cosine functions respectively.
From this, one sees that, for any real x,
exp(ix) = cos(x) + i sin(x)
-
Thus, we get Euler's famous formula
ep i = -1
and
e2p i = e0 = 1
One can also obtain the classical addition formulae for sine and cosine
from (8) and (1).
All of the above extensions have been restricted to a positive real for
the base. When the base x is not a positive real, it is not as
clear-cut how to extend the definition of exponentiation. For example,
(-1)1/2 could well be i or -i, (-1)1/3
could be -1, 1/2+(sqrt(3)/2)i, or 1/2-(sqrt(3)/2)i, and so on. Some values of x and y
give
infinitely many candidates for xy, all equally plausible.
And x=0 has its own special problems. These problems can all be
traced to the fact that the exp function is not injective on the complex
plane, so that ln is not well defined outside the real line. There are
ways around these difficulties (defining branches of the logarithm, for
example), but we shall not go into this here.
Some people say 00 is not defined, while others say it is 1. Defining 00 as 1 is useful because it gives correct answers to formulas like the number of ordered n-tuples that can be made with m symbols. The formula is mn. If m and n are both zero the formula gives the answer: one. This is right. The number of 0-tuples that can be made with 0 symbols is one. It is
( ). Interestingly, the number of 1-tuples, 2-tuples or any other tuples that can be made with 0 symbols is zero -- there is no way to represent a 3-tuple, for example, without using symbols. Thus the formula gives correct answers for all m and n that are nonnegative.
The operation of exponentiation has also been extended to other systems
like matrices and operators. The key is to define an exponential
function by (6) and work from there.
Internet References
http://db.uwaterloo.ca/~alopez-o/math-faq/math-faq.html
-- Frequently Asked Questions in Mathematics
Related Pages in this website
Hyperbolic Functions -- sinh(x)
and cosh(x), which are defined in terms of exp(x)
Understanding Logarithms and
Exponents
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