# Definition:Usual Ordering

## Definition

Let $X$ be one of the sets of numbers: $\N$, $\Z$, $\Q$, $\R$.

The **usual ordering** on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers.

### Natural Numbers

Let $\N$ denote the natural numbers.

The ordering on $\N$ is the relation $\le$ everyone is familiar with.

For example, we use it when we say:

- James has $6$ apples, which is more than Mary, who has $4$.

which can be symbolised as:

- $6 \ge 4$

### 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

The **usual ordering** is also known as the **natural ordering**.

## Also see

- Ordered Integral Domain is Totally Ordered Ring, indicating that the
**usual ordering**is a total ordering as required.

## Sources

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-3}$ Order properties of real numbers - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-2}$: Inequalities - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.21$