Zero to the Power of Zero/Binomial Theorem

Example of Zero to the Power of Zero

Consider the real polynomial function:

$y = \paren {x + c}^n$

for $n \in \N, c \in \R$.

By the binomial theorem, $y$ contains a term of the form:

$\dbinom n n x^{n - n} c^n$

If we did not define $0^0 = 1$, $y$ would have a discontinuity at $x = 0$.

This would contradict Real Polynomial Function is Continuous.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text{F} \ (14)$