# Definition:Usual Ordering

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## 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

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.

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## 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.

- Complex Numbers cannot be Ordered Compatibly with Ring Structure, which demonstrates that $\C$ does not have a
**usual ordering**.

## 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 $2.21$ - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: $(2)$