Natural Products
Why
We want to add repeatedly.
Definitiong result
For each natural number $m$, there exists a
function $p_m: \omega \to \omega $ which
satisfies
\[ p_m(0) = 0 \quad \text{ and } \quad p_m(\ssuc{n}) = \ssuc{(p_m(n))} + m \]
for every natural number $n$. The proof uses the recursion theorem (see Recursion Theorem).1
Let $m$ and $n$ be natural numbers. The value $p_m(n)$ is the product of $m$ with $n$.
Notation
We denote the product $p_m(n)$ by $m \cdot n$. We often drop the $\cdot $ and write $m \cdot n$ as $mn$.
Properties
The properties of products are direct applications of the principle of mathematical induction (see Natural Induction).2
Let $k$, $m$, and $n$ be natural numbers. Then
\[ (k \cdot m) \cdot n = k \cdot (m \cdot n). \]
Let $m$ and $n$ be natural numbers. Then
\[ m \cdot n = n \cdot m. \]