# Definition:Positive/Integer

## Definition

Informally, the **positive integers** are the set:

- $\Z_{\ge 0} = \set {0, 1, 2, 3, \ldots}$

As the set of integers $\Z$ is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus **positive** can be formally defined on $\Z$ as a relation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, **positive** can be defined directly as the relation specified as follows:

- The integer $z \in \Z: z = \eqclass {\tuple {a, b} } \boxminus$ is
**positive**if and only if $b \le a$.

The set of **positive integers** is denoted $\Z_{\ge 0}$.

An element of $\Z$ can be specifically indicated as being **positive** by prepending a $+$ sign:

- $+x := x \in \Z_{\ge 0}$.

### Ordering on Integers

### Definition 1

The integers are ordered on the relation $\le$ as follows:

- $\forall x, y \in \Z: x \le y$

- $\exists c \in P: x + c = y$

where $P$ is the set of positive integers.

That is, $x$ is **less than or equal** to $y$ if and only if $y - x$ is non-negative.

### Definition 2

The integers are ordered on the relation $\le$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

- $x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

Then:

- $x < y \iff x_1 + y_2 \le x_2 + y_1$

where:

- $+$ denotes natural number addition
- $\le$ denotes natural number ordering.

## Also known as

As there is often confusion as to whether or not $0$ is included in the set of **positive integers**, it may be preferable to refer to the set of **non-negative integers** instead.

The notation $\Z^+$ is common, but leaves it ambiguous as to whether $\Z_{>0}$ or $\Z_{\ge 0}$ is meant.

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 5$: The system of integers - 1964: J. Hunter:
*Number Theory*... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $1$. Introduction - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation