The Battle Over Nothing: Why $0^0$ Keeps Mathematicians Up at Night

To the average person, mathematics is a realm of absolute rules. Two plus two is four; dividing by zero is a cardinal sin that breaks the universe. Yet, nestled quietly in textbooks and programming languages is a deceptively simple expression that manages to divide even the sharpest minds: $0^0$ (zero to the power of zero).

Is it $1$? Is it $0$? Or is it completely undefined? Depending on who you ask—a combinatorialist, an algebraist, or a calculus professor—you might get three different answers.

Let’s dive into the fascinating tug-of-war behind one of mathematics’ most stubborn paradoxes.

1. The Combinatorial Camp: Team $0^0 = 1$

If you ask anyone who works in combinatorics, set theory, or computer science, they will likely look at you with absolute confidence and say: $0^0 = 1$.

Why? Because exponentiation isn’t just repeated multiplication; it’s counting.

In set theory, $a^b$ represents the number of functions from a set of size $b$ to a set of size $a$.

• How many functions are there from a 3-element set to a 2-element set? $2^3 = 8$.
• How many functions are there from a 0-element set (the empty set, $emptyset$) to a 0-element set?

There is exactly one function possible: the empty function that maps nothing to nothing. Therefore, $0^0 = 1$.

Similarly, in algebra, an exponent represents repeated multiplication, and $0^0$ is an empty product. In mathematics, an empty product defaults to the multiplicative identity, which is $1$. This is why calculators, programming languages (like Python and C++), and binomial expansions (like $(x + y)^n$) rely on $0^0 = 1$ to keep formulas from breaking. Without this convention, calculus theorems like the Binomial Theorem would require frustrating asterisk exceptions.

2. The Calculus Camp: Team “It’s Undefined”

Step over into the calculus department, however, and you will hear a very different tune. Ask a calculus professor about $0^0$, and they will likely groan and call it an indeterminate form.