# Symbols:Arithmetic and Algebra

## Contents

## Addition

- $+$

**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

- $\pm$

$a \pm b$ 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 =`

.