Definition:Natural Numbers
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Informal Definition
The natural numbers are the counting numbers.
The set of natural numbers is denoted $\N$:
- $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$.
This sequence is A001477 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
The set $\N \setminus \left\{{0}\right\}$ is denoted $\N_{>0}$:
- $\N_{>0} = \left\{{1, 2, 3, \ldots}\right\}$.
This sequence is A000027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
The set of natural numbers is one of the most important sets in mathematics.
Rigorous Definition
While everyday experience and familiarity with the natural numbers is more than sufficient to work with and understand the natural numbers, they can be defined purely in terms of sets and elements.
Consider the empty set $\varnothing = \left\{{ }\right\}$.
Define the counting measure, or "order," of this set as $0$.
Consider the set $S = \left\{{0}\right\} \ $. Define the counting measure of this set as $1$.
Since $\varnothing \subset \left\{{0}\right\} \ $, we define $0<1 \ $.
Suppose several symbols have been defined, each referring to the counting measure of a set in this fashion. Let $n \ $ be one of these symbols, such that $n>x \ $ if $x \ $ is any symbol already defined. Call the set of all symbols defined so far $N_n \ $. The next such symbol is defined as $n+1 = o\left({N_n}\right) \ $, where $o \ $ is the counting measure.
Any symbol which is so defined is called a natural number and the set of all such symbols is written $\N$.
As a Naturally Ordered Semigroup
The natural numbers can alternatively be defined as the archetypal naturally ordered semigroup.
Thus they can be shown to satisfy Peano's Axioms.
Hence the set of natural numbers can be generated directly from the Zermelo-Fraenkel Axioms.
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Also known as
First, note that some sources use a different style of letter from $\N$: you will find $N$, $\mathbf N$, etc. However, $\N$ is becoming more commonplace and universal nowadays.
The usual symbol for denoting $\left\{{1, 2, 3, \ldots}\right\}$ is $\N^*$, but the more explicit $\N_{>0}$ is standard on this site.
Some authors refer to $\left\{{0, 1, 2, 3, \ldots}\right\}$ as $\tilde {\N}$, and refer to $\left\{{1, 2, 3, \ldots}\right\}$ as $\N$.
Either is valid, and as long as it is clear which is which, it does not matter which is used. However, using $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ is a more modern approach, particularly in the field of computer science, where starting the count at zero is usual.
We also have the following notations which are sometimes used:
- $\N_0 = \left\{{0, 1, 2, 3, \ldots}\right\}$
- $\N_1 = \left\{{1, 2, 3, \ldots}\right\}$
However, you need to beware of confusing this notation with the use of $\N_n$ as the subset of the natural numbers:
- $\N_n = \left\{{0, 1, 2, \ldots, n-1}\right\}$
under which notational convention $\N_0 = \varnothing$ and $\N_1 = \left\{{0}\right\}$.
So it's as well to ensure you understand exactly which convention is being used.
The use of $\N$ or its variants is not universal, either.
Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951) uses $P = \left\{{1, 2, 3, \ldots}\right\}$.
This stems from the fact that Jacobson's presentation starts with Peano's axioms.
Paul R. Halmos: Naive Set Theory (1960) uses $\omega$.
Also see
- Results about natural numbers as an abstract algebraical concept can be found here.
Sources
- Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 4$
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 11$: Numbers
- W.E. Deskins: Abstract Algebra (1964): $\S 2.1$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 15$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.2$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(b)}$
