# Symbols:Arithmetic and Algebra

## Symbols used in Arithmetic and Algebra

### Addition

- $+$

**Plus**, or **added to**.

A binary operation on two numbers or variables.

Its $\LaTeX$ code is `+`

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### Positive Quantity

- $+$

A unary operator prepended to a number to indicate that it is positive.

For example:

- $+5$

If a number does not have either $+$ or $-$ prepended, it is assumed to be positive by default.

The $\LaTeX$ code for \(+5\) is `+5`

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### Subtraction

- $-$

**Minus**, or **subtract**.

A binary operation on two numbers or variables.

Its $\LaTeX$ code is `-`

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### Negative Quantity

- $-$

A unary operator prepended to a number to indicate that it is negative.

For example:

- $-6$

The $\LaTeX$ code for \(-6\) is `-6`

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### Multiplication (Arithmetic)

- $\times$

**Times**, or **multiplied by**.

A binary operation on two numbers or variables.

Usually used when numbers are involved (as opposed to variables) 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 $x$.

Its $\LaTeX$ code is `\times`

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### Multiplication (Algebra)

- $\cdot$

$x \cdot y$ means **$x$ times $y$**, or **$x$ multiplied by $y$**.

A binary operation on two variables.

Usually used when variables are involved (as opposed to numbers) to avoid confusion with the use of $\times$ which could be confused with the symbol $x$ when used as a variable.

It is preferred that the symbol $\cdot$ is not used in arithmetic between numbers, as it can be confused with the decimal point.

Its $\LaTeX$ code is `\cdot`

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### Per Cent

- $\%$

$x \%$ means **$x$ hundredths.**

Hence $x \%$ has the same meaning as the fraction $\dfrac x {100}$

Its $\LaTeX$ code is `x \%`

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### Division

- $\div$, $/$

A binary operation on two numbers or variables.

$x \div y$ 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}`

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### Plus or 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`

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### Absolute Value

- $\size x$

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

$\size x = \begin {cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end {cases}$

The $\LaTeX$ code for \(\size x\) is `\size x`

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### Factorial

- $n!$

The **factorial of $n$** is defined inductively as:

- $n! = \begin {cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end {cases}$

The $\LaTeX$ code for \(n!\) is `n!`

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### Square Root

- $\sqrt n$

A **square root** of a number $n$ is a number $z$ such that $z$ squared equals $n$.

The $\LaTeX$ code for \(\sqrt n\) is `\sqrt n`

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### Binomial Coefficient

- $\dbinom n m$

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

Formally defined as:

- $\dbinom n m = \begin {cases} \dfrac {n!} {m! \, \paren {n - m}!} & : m \le n \\ 0 & : m > n \end {cases}$

The $\LaTeX$ code for \(\dbinom {n} {m}\) is `\dbinom {n} {m}`

or `\ds {n} \choose {m}`

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### Approximation

- $a \approx b$

An **approximation** is an estimate of a quantity.

It is usually the case that there exists some knowledge about the accuracy of the estimate.

The notation:

- $a \approx b$

indicates that $b$ is an **approximation** to $a$.

The $\LaTeX$ code for \(a \approx b\) is `a \approx b`

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### Proportion

- $\propto$

Two real variables $x$ and $y$ are **proportional** if and only if one is a constant multiple of the other:

- $\forall x, y \in \R: x \propto y \iff \exists k \in \R, k \ne 0: x = k y$

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

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