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): 0^y = 0. There is an obvious discontinuity in the function z = x^y 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. 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 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 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:

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. a^b+1 = a^b. 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)

The Laws of Exponents on N for undefined 0⁰

We can define exponentiation on N as follows:

1. a^b ε N (for a or b ≠ 0)

2. 0¹ = 0

3. a⁰ = 1 (for a ≠ 0)

4. a^b+1 = a^b. a (for a or b ≠ 0)

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

1. The Product of Powers Rule: a^b. a^c = a^b+c (for a ≠ 0 OR both b, c ≠ 0) Proof

2. The Power of a Power Rule: (a^b)^c = a^b.c (for a ≠ 0 OR both b, c ≠ 0) Proof

3. The Power of a Product Rule:  (a.b)^c = a^c. b^c (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!

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