# Symbols:Arithmetic and Algebra

## Contents

$+$

Plus, or added to. A binary operation on two numbers or variables.

Its $\LaTeX$ code is + .

See Set Operations and Relations and Abstract Algebra for alternative definitions of this symbol.

## Subtraction

$-$

Minus, or subtract. A binary operation on two numbers or variables.

Its $\LaTeX$ code is - .

See Set Operations and Relations and Logical Operators for alternative definitions of this symbol.

## Multiplication

### Times

$\times$

Times, or multiplied by. A binary operation on two numbers.

Usually used when numbers are involved (as opposed to letters) to avoid confusion with the use of $\ \cdot \$ which could be confused with the decimal point.

The symbol $\times$ is cumbersome in the context of algebra, and may be confused with the letter $x$.

It was invented by William Oughtred in his 1631 work Clavis Mathematicae.

Its $\LaTeX$ code is \times .

See Set Operations and Relations and Vector Algebra for alternative definitions of this symbol.

### Dot

$\cdot$

$x \cdot y$ means $x$ times $y$, or $x$ multiplied by $y$, a binary operation on two numbers.

Its $\LaTeX$ code is \cdot .

See Vector Algebra, Abstract Algebra and Logical Operators: Deprecated Symbols for alternative definitions of this symbol.

## Division

$\div$, $/$

Divided by. A binary operation on two numbers.

$x \div y$ and $x / y$ both mean $x$ divided by $y$, or $x \times y^{-1}$.

$x / y$ can also be rendered $\dfrac x y$ (and often is -- it tends to improve comprehension for complicated expressions).

$x \div y$ is rarely seen outside grade school.

Their $\LaTeX$ codes are as follows:

• The $\LaTeX$ code for $x \div y$ is x \div y .
• The $\LaTeX$ code for $x / y$ is x / y .
• The $\LaTeX$ code for $\dfrac {x} {y}$ is \dfrac {x} {y} .

## Plus and Minus

$\pm$

Plus or minus.

$a \pm b$ means $a + b$ or $a - b$, often seen when expressing the two solutions of a quadratic equation.

Its $\LaTeX$ code is \pm .

See Numerical Analysis for another definition of this symbol.

## Sum

$\displaystyle \sum$

$\displaystyle \sum_{k \mathop = a}^{n} x_k$ is the addition of the elements of the sequence $x_k$ for $k$ from $a$ to $n$ (inclusive).

The $\LaTeX$ code for $\displaystyle \sum_{k \mathop = a}^{n}$ is \displaystyle \sum_{k \mathop = a}^{n} .

## Product

$\displaystyle \prod$

$\displaystyle \prod_{k \mathop = a}^{n} x_k$ is the multiplication of the elements of the sequence $x_k$ for $k$ from $a$ to $n$ (inclusive).

The $\LaTeX$ code for $\displaystyle \prod_{k \mathop = a}^{n}$ is \displaystyle \prod_{k \mathop = a}^{n} .

## Absolute Value

$\left \vert{x}\right \vert$

The absolute value of the variable $x$, when $x \in \R$.

$\left \vert{x}\right \vert = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

The $\LaTeX$ code for $\left \vert{x}\right \vert$ is \left \vert{x}\right \vert .

See Set Operations and Relations, Complex Analysis and Abstract Algebra for alternative definitions of this symbol.

## Binomial Coefficent

$\displaystyle \binom n m$

The binomial coefficient, which specifies the number of ways you can choose $m$ objects from $n$ (all objects being distinct).

Interpreted as:

$\dbinom n m = \begin{cases} \dfrac {n!} {m! \left({n - m}\right)!} & : m \le n \\ 0 & : m > n \end{cases}$

The $\LaTeX$ code for $\dbinom {n} {m}$ is \dbinom {n} {m}  or \displaystyle {n} \choose {m}.

## Negation

$\not =, \not >, \not <, \not \ge, \not \le$

Negation. The above symbols all mean the opposite of the non struck through version of the symbol.

For example, $x \not = y$ means that $x$ is not equal to of $y$.

The $\LaTeX$ code for negation is \not followed by the code for whatever symbol you want to negate. For example, \not \ge will render $\not \ge$.

Note that several of the above relations also have their own $\LaTeX$ commands for their negations, for example \ne or \neq for \not =.