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 + .

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 .

Subtraction

$-$

Minus, or subtract.

A binary operation on two numbers or variables.

Its $\LaTeX$ code is - .

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 .

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 .

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 .

Per Cent

$\%$

$x \%$ means $x$ hundredths.

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

Its $\LaTeX$ code is x \% .

Division

$\div$, $/$

Divided by.

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} .

Plus or 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 .

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 .

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! .

Square Root

$\sqrt n$

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

Note that the overline is technically an example of a vinculum, enclosing the argument of $\surd$ in parenthesis.

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

$r$th Root

$\sqrt [r] x$

Let $x, y \in \R_{\ge 0}$ be positive real numbers.

Let $n \in \Z$ be an integer such that $n \ne 0$.

Then $y$ is the positive $n$th root of $x$ if and only if:

$y^n = x$

and we write:

$y = \sqrt[n] x$

Using the power notation, this can also be written:

$y = x^{1/n}$

Note that the overline is technically an example of a vinculum, enclosing the argument of $\surd$ in parenthesis.

The $\LaTeX$ code for \(\sqrt [r] x\) is \sqrt [r] x .

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}.

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 .

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 .

Ratio

$:$

Let $x$ and $y$ be quantities which have the same dimensions.

Let $\dfrac x y = \dfrac a b$ for two numbers $a$ and $b$.

Then the ratio of $x$ to $y$ is defined as:

$x : y = a : b$

It explicitly specifies how many times the first number contains the second.

The $\LaTeX$ code for \(x : y\) is x : y .