# Definition:Divisor (Algebra)/Real Number

## Definition

Let $\R$ be the set of real numbers.

Let $x, y \in \R$.

Then **$x$ divides $y$** is defined as:

- $x \divides y \iff \exists t \in \Z: y = t \times x$

where $\Z$ is the set of integers.

That is, that $y$ is an integer multiple of $x$.

### Part

A more old-fashioned term for divisor is **part**:

In the words of Euclid:

(*The Elements*: Book $\text{V}$: Definition $1$)

## Terminology

Let $x \divides y$ denote that **$x$ divides $y$**.

Then the following terminology can be used:

- $x$ is a
**divisor**of $y$ - $y$ is a
**multiple**of $x$ - $y$ is
**divisible by**$x$.

In the field of Euclidean geometry, in particular:

- $x$
**measures**$y$.

To indicate that $x$ does *not* divide $y$, we write $x \nmid y$.

## Also known as

A **divisor** can also be referred to as a **factor**.

## Notation

The conventional notation for **$x$ is a divisor of $y$** is "$x \mid y$", but there is a growing trend to follow the notation "$x \divides y$", as espoused by Knuth etc.

From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: *Concrete Mathematics: A Foundation for Computer Science* (2nd ed.):

*The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.*

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \ $ or similar to denote non-divisibility.

## Also see

- Results about
**divisors**can be found here.