Definition:Multiplication/Natural Numbers
Definition
Let $\N$ be the natural numbers.
Multiplication on $\N$ is the basic operation $\times$ everyone is familiar with.
For example:
- $3 \times 4 = 12$
- $13 \times 7 = 91$
Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $\times$.
The same holds for any construction of $\N$ in an ambient theory.
Multiplication in terms of Addition
Let $+$ denote addition.
The binary operation $\times$ is recursively defined on $\N$ as follows:
- $\forall m, n \in \N: \begin{cases} m \times 0 & = 0 \\ m \times \left({n + 1}\right) & = m \times n + m \end{cases}$
This operation is called multiplication.
Equivalently, multiplication can be defined as:
- $\forall m, n \in \N: m \times n := +^n m$
where $+^n m$ denotes the $n$th power of $m$ under $+$.
Also defined as
Some treatments of Peano's axioms define the non-successor element (or primal element) to be $1$ and not $0$.
The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.
Under this $1$-based system, multiplication is consequently defined as follows:
- $\forall m, n \in P: \begin{cases} m \times 1 & = m \\ m \times \map s n & = m \times n + n \end{cases}$
or:
- $\forall m, n \in P: \begin{cases} 1 \times n & = n \\ \map s m + n & = m \times n + n \end{cases}$
Also see
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 4$: The natural numbers
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$