*Oh, the ambiguity**! *

**Introduction**

Most of us learned in high
school that 0^{0} 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): *x*^{0} = 1. But we also have Lim
(*y* à 0): 0* ^{y}* = 0. There is an obvious
discontinuity in the function

Your algebra professor, on the
other hand, may tell you that you can assume that 0^{0} = 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 0^{0} 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. Umm, 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 0^{0} 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 0^{0} 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 0^{0}.

**Exponentiation Defined as
Repeated Multiplication on N**

When you were first introduced
to exponents in elementary or high school, you probably started in the
exponents greater than or equal to two. After all, you need at least 2 numbers
to multiply. For all *a* in *N*, we have:

a^{2}=a.a

a^{3}=a.a.a=a^{2}.a

a^{4}=a.a.a.a=a^{3}.a

a^{5}=a.a.a.a.a=a^{4}.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. *a*^{2} = *a. a*

2. *a ^{b}*

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 0^{0}. Proof (194 lines)

This suggests that, in our
definition of exponentiation on N, we should simply leave 0^{0}
undefined. To this end, we can construct (i.e. prove the existence of) a unique
partial function
for exponentiation on *N*. Proof
(618 lines)

**The Laws of Exponents on N
for undefined 0^{0}**

We can define exponentiation on
*N* as follows:

1.
*a ^{b}* ε

2.
0^{1} = 0

3.
*a*^{0} = 1 (for a ≠ 0)

4.
*a ^{b}*

Using the above definition, we
can derive the 3 Laws of Exponents on *N*:

1.
The Product of Powers Rule: *a ^{b}*.

2.
The Power of a Power Rule: *(**a ^{b})^{c}*
=

3. The Power of a Product Rule: (*a.b*)* ^{c}*
=

Interestingly, these restrictions would *not* apply if 0^{0}
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 0^{0}
equal to 0 or 1? *Oh, the ambiguity!*