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.

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 $y$.

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

Ceiling

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

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

Note that the  and , 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, the delimiters  and   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).

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

Note that the  and , 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, the delimiters  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.

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$\mid$ has been (or is in the process of being) superseded by $\backslash$, which is becoming increasingly popular since many mathematicians are of the opinion that $|$ is overused, and hence a possible cause for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

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).

$\left[{x}\right]$ has been (or is in the process of being) superseded by $\left \lfloor{x}\right \rfloor$, due to the already widespread uses of square brackets.