# Symbols:Number Theory

## Divides, Divisor, Factor

$x \mathrel \backslash y$

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$\mid$ has been (or is in the process of being) superseded by $\backslash$, which is becoming increasingly popular since many mathematicians are of the opinion that $|$ is overused, and hence a possible cause for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