# Symbols:Number Theory

## Contents

## Divides, Divisor, Factor

- $x \mathop \backslash y$

This means "$x$ is a divisor (or factor) of $y$", or "$x$ divides $y$".

$\backslash$ is gaining in popularity over $\mid$, since many mathematicians are of the opinion that $\mid$ is overused, and hence confusing.

The $\LaTeX$ code for \(x \mathop \backslash y\) is `x \mathop \backslash y`

.

See Set Operations and Relations: Set Difference for an alternative definitions of this symbol.

## Does Not Divide, Is Not a Divisor or Factor

- $x \nmid y$

This means **$x$ is not a divisor of $y$**.

The $\LaTeX$ code for \(x \nmid y\) is `x \nmid y`

.

This symbol is preferable to $x \not \backslash y$ due to the somewhat confusing appearance of this symbol.

The $\LaTeX$ code for \(x \not \backslash y\) is `x \not \backslash y`

.

## Ceiling

- $\left\lceil{x}\right\rceil$

This represents the smallest integer greater than or equal to $x$. (See Definition:Ceiling Function).

The $\LaTeX$ code for \(\left \lceil{x}\right \rceil\) is `\left \lceil{x}\right \rceil`

.

Note that the `\left`

and `\right`

, as with other surrounding symbols such as parenthesis and brackets, adjust the size of the symbols as appropriate, for example with $\left\lceil {\dfrac x y} \right\rceil$. On $\mathsf{Pr} \infty \mathsf{fWiki}$, the delimiters `\left`

and `\right`

are a mandatory component of the house style.

## Floor

- $\left\lfloor{x}\right\rfloor$ or $\left[{x}\right]$

This represents the greatest integer less than or equal to $x$. (See Definition:Floor Function).

The $\LaTeX$ code for \(\left \lfloor{x}\right \rfloor\) is `\left \lfloor{x}\right \rfloor`

.

$\left\lfloor{x}\right\rfloor$ is gaining in popularity over the more traditional $\left[{x}\right]$, due to the already varied uses of square brackets.

Note that the `\left`

and `\right`

, as with other surrounding symbols such as parenthesis and brackets, adjust the size of the symbols as appropriate, for example with $\left\lfloor {\dfrac x y} \right\rfloor$. On $\mathsf{Pr} \infty \mathsf{fWiki}$, the delimiters `\left</tt> and <code>\right`

are a mandatory component of the house style.

## Coprime

- $x \perp y$

This denotes the statement that $x$ is coprime to $y$.

That is, that $\gcd \left\{{x, y}\right\} = 1$, where $\gcd$ denotes the greatest common divisor.

The $\LaTeX$ code for \(\perp\) is `\perp`

.

## Deprecated Symbols

### Divisor

- $x \mid y$

This means **$x$ is a divisor of $y$**.

$\mid$ has been (or is in the process of being) superseded by $\backslash$, which is becoming increasingly popular since many mathematicians are of the opinion that $|$ is overused, and hence a possible cause for confusion.

The $\LaTeX$ code for \(\mid\) is `\mid`

.

In the context of computer languages, $\mid$ is frequently called **pipe** from its use in Unix. This name is catching on in general mathematics.

### Floor, or Integral Part

- $\left[{x}\right]$

This represents the greatest integer less than or equal to $x$. (See Definition:Floor Function).

The $\LaTeX$ code for \(\left[{x}\right]\) is `\left[{x}\right]`

.

$\left[{x}\right]$ has been (or is in the process of being) superseded by $\left \lfloor{x}\right \rfloor$, due to the already widespread uses of square brackets.