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
- $\div$, $/$
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
- $\pm$
$a \pm b$ 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 .
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 .