Is a single number a product?

Question

This may seem like a strange question, but I was wondering if a single number $a\in \mathbb{C}$ can be seen as a product under the mathematical definition? Put differently, I am wondering if $\prod_{i=1}^{1}a_i$ is a product.

Answer 1

Yes. We get to decide what technical terms mean and what notations mean, and it's convenient to define a product of any collection of numbers, including even the empty collection!

Just as an empty sum has value $0$ (the additive identity), an empty product has value $1$ (the multiplicative identity). For example, this motivates the factorial definitions $0! = 1$ and $1! = 1$. $$ \begin{array}{ll} A & \displaystyle{\prod_{a \in A} a} \\ \hline \{\} & 1 = 1 \\ \{1\} & 1 \cdot 1 = 1 \\ \{1, 2\} & 1 \cdot 1 \cdot 2 = 2 \\ \{1, 2, 3\} & 1 \cdot 1 \cdot 2 \cdot 3 = 6 \\ \qquad\vdots & \qquad\vdots \\ \{1, 2, \dots, n\} & 1 \cdot 1 \cdot 2 \cdots n = n! \end{array} $$