The Great Mathematical standoff: What on Earth is $0^0$?

At first glance, $0^0$ (zero to the power of zero) looks deceptively simple. It sits quietly on the page, tempting you to guess. Is it $0$? Is it $1$? Or is it a mathematical ghost that shouldn’t exist at all?

Depending on who you ask—a high school algebra teacher, a combinatorics expert, 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 little expressions.

1. The Argument for $1$: Algebra and Combinatorics

If you punch $0^0$ into most scientific calculators or programming languages, it will boldly spit out $1$. Why? Because in algebra, discrete math, and set theory, defining $0^0 = 1$ is not just convenient; it is essential for keeping the universe of math from breaking.

• The Rule of Exponents: We know that for any non-zero number, $a^1 / a^1 = a^{1-1} = a^0$. Since any number divided by itself is $1$, it follows that $a^0 = 1$. Even if $a = 0$, mathematicians prefer consistency.
• The Empty Set: In set theory, exponentiation $A^B$ represents the number of ways to map elements from set $B$ to set $A$. If both sets are empty (containing zero elements), there is exactly one way to do it—a mapping of absolute nothingness to nothingness. Thus, $0^0 = 1$.
• Polynomials and Series: Think about the polynomial expression $f(x) = x^0$. If you evaluate this at $x = 0$, you expect it to equal $1$ everywhere, including right at zero. Without $0^0 = 1$, formulas like the Binomial Theorem $(x+y)^n = sum binom{n}{k} x^{n-k} y^k$ would break down when terms evaluate to the zeroth power.

Key Takeaway: In discrete mathematics, $0^0$ is defined as $1$ as a matter of structural convenience and consistency.

2. The Argument for “Undefined”: The Calculus Rebellion

While algebraists love setting $0^0 = 1$, calculus professors break out in a cold sweat at the sight of it. In calculus, $0^0$ is classified as an indeterminate form.