Oh, the ambiguity!
Introduction
Most of us learned in high
school that 00 is somehow undefined or ambiguous. In college or
university, your calculus professor will confirm this, citing the ambiguity
resulting from different limits. We have Lim (x à 0): x0 = 1. But we also have Lim
(y à 0): 0y = 0. There is an obvious
discontinuity in the function z = xy at
(0, 0). (See 00
with Continuous Exponents)
Your algebra professor, on the
other hand, may tell you that you can assume that 00 = 1 on the
natural numbers--for convenience mostly. They may justify it with analogies to
various conventions, e.g. usually the convention of so-called empty
products--the product of no numbers?
Many simply define it to be 1. It’s apparently not something you can actually
prove. As for 00 being undefined on the real numbers, and
exponentiation being entirely consistent in both domains, that is a mere coincidence. We are talking about entirely different functions here, they
will say. If that strikes you as being just a bit too, well, “hand wavy” for
your liking, read on!
Here, I will develop the
exponentiation function on the natural numbers with 00 undefined,
given only the operations of addition and multiplication on N. I
use what I believe to be a novel approach that looks at all possible
functions that satisfy the usual requirements for an exponentiation function on
N. In so doing, we can justify
leaving 00 undefined, as it is on the set of real numbers R. I will also look at some implications
for the usual laws of exponents on N
for undefined 00.
Exponentiation Defined as
Repeated Multiplication on N
When you were first introduced to
exponents in elementary or high school, you probably started with the exponents
greater than or equal to two. After all, you need at least 2 numbers to
multiply. For all a ε N, we have:
a2 = a.a
a3 = a.a.a = a2. a
a4 = a.a.a.a = a3. a
a5 = a.a.a.a.a = a4. a
and so on.
This infinite sequence of
equations can be recursively summarized in just two equations for all a, b ε N as follows:
1. a2 = a. a
2. ab+1 = ab. a
These two equations, by themselves, do not, however, tell us
anything about exponents 0 or 1. It turns out that there are infinitely many
such exponent-like functions on N
that satisfy these equations. Proof (21 lines)
Fortunately, these infinitely
many functions differ only in the value
assigned to 00. Proof (194 lines)
This suggests that, in our
definition of exponentiation on N, we should simply leave 00
undefined. To this end, we can construct (i.e. prove the existence of) a unique
partial function
for exponentiation on N. Proof
(618 lines)
Thus we can define
exponentiation on N as follows:
1.
ab ε N (for a or b ≠ 0)
2.
01 = 0
3.
a0 = 1 (for a ≠ 0)
4.
ab+1 = ab. a (for a or b ≠ 0)
The Laws of Exponents on N
for undefined 00
Using the above definition, we
can derive the 3 Laws of Exponents on N:
1.
The Product of Powers Rule: ab. ac
= ab+c (for a ≠ 0 OR
both b, c ≠ 0) Proof
2.
The Power of a Power Rule: (ab)c
= ab.c (for a ≠ 0 OR
both b, c ≠ 0) Proof
3. The Power of a Product Rule: (a.b)c
= ac. bc
(for c ≠ 0 OR both a, b ≠ 0) Proof
Interestingly, these restrictions would not apply if 00
was defined to be either 0 or 1. So, adding these laws to the requirements for
exponentiation would narrow down the infinitely many possibilities to only two.
But we would still be left with some ambiguity—is 00
equal to 0 or 1? Oh, the ambiguity!
My Recommendation
If, as in most well used,
centuries-old textbook applications where an informal proof will usually
suffice, it is convenient and probably
safe to assume 00=1 on the natural numbers, if not on the reals. If,
however, you are pushing the limits of number theory and only a formal proof
will do, you should take the extra care and avoid assuming any particular value
for 00 in both the real
numbers and the natural numbers.
There are easy work-arounds in many cases. Applications of the binomial theorem to
obtain the usual expansions of (x+y)n, for example, will result
in terms including 00 when either x or y are zero. In such
cases, however, there is a simple work-around, e.g. (x+0)n
= xn
for x or n ≠ 0.