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Natural Sums
Needed by:
Integer Products
Natural Powers
Number of Set Products
Order and Arithmetic
Prime Numbers
Square Numbers
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Natural Products


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 $n$.


We denote the product $p_m(n)$ by $m \cdot n$. We often drop the $\cdot $ and write $m \cdot n$ as $mn$.


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

\[ m \cdot n = n \cdot m. \]

  1. Future editions will give the entire account. ↩︎
  2. Future editions will include the accounts. ↩︎
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