Some Useful Mathematical Notation

Stage: 5
We thought it would be useful to put together a page of commonly used notation that you might meet when studying higher mathematics.

The notation found below is by no means an exhaustive list, and if you have any suggestions for additions to the list, please get in touch.

Here are relation symbols:

$$
\begin{align}
= &\qquad a= b\mbox{ means that }a\mbox{ is equal to }b\\
\not= &\qquad a\not= b\mbox{ means that }a\mbox{ is not equal to }b\\
< &\qquad a< b\mbox{ means that }a\mbox{ is smaller than }b\\
> &\qquad a> b\mbox{ means that }a\mbox{ is larger than }b\\
\leq &\qquad a\leq b\mbox{ means that }a\mbox{ is smaller than or equal to }b\\
\geq &\qquad a\geq b\mbox{ means that }a\mbox{ is larger than or equal to }b\\
\ll &\qquad a\ll b\mbox{ means that }a\mbox{ is much smaller than }b\\
\gg &\qquad a\gg b\mbox{ means that }a\mbox{ is much larger than }b
\end{align}
$$

Here are symbols for different sets of numbers:

$$
\begin{align}
\emptyset &\qquad\mbox{empty set}\\
\mathbb{N} &\qquad\mbox{natural numbers}\\
\mathbb{Z} &\qquad\mbox{integers}\\
\mathbb{Q} &\qquad\mbox{rational numbers}\\
\mathbb{R} &\qquad\mbox{real numbers}\\
\mathbb{C} &\qquad\mbox{complex numbers}
\end{align}
$$


Here are symbols you might want to use in a proof:

$$
\begin{align}
\therefore &\qquad\mbox{therefore}\\
\square &\qquad\mbox{end of proof}\\
\# &\qquad\mbox{contradiction}\\
\forall &\qquad\mbox{for all}\\
\exists &\qquad\mbox{there exists}\\
\nexists &\qquad\mbox{there doesn't exist}\\
\Longrightarrow &\qquad\mbox{this implies}\\
\Longleftarrow &\qquad\mbox{is implies by}\\
\Longleftrightarrow &\qquad\mbox{equivalent to}
\end{align}
$$

Here are some symbols used in geometry:

$$
\begin{align}
\triangle ABC &\qquad\mbox{triangle }ABC\\
\angle ABC &\qquad\mbox{angle }ABC\\
{ }^\circ &\qquad\mbox{degree, e.g.   }30^\circ\\
\parallel &\qquad\mbox{parallel, e.g. parallel lines}\\
\perp &\qquad\mbox{perpendicular, e.g. perpendicular lines}\\
\cong &\qquad\mbox{congruent, e.g. congruent triangles}\\
\underline{a}\mbox{ or }\vec{a} &\qquad\mbox{vector}
\end{align}
$$

Here are symbols which are useful when working with complex numbers:

$$
\begin{align}
w^{\ast}\mbox{ or }\overline{w}&\qquad\mbox{the complex conjugate of }w\\
|w| &\qquad\mbox{the modulus of }w\\
\Re &\qquad\mbox{real part}\\
\Im &\qquad\mbox{imaginary part}
\end{align}
$$

Here are some other helpful symbols:

$$
\begin{align}
\propto &\qquad\mbox{proportional to}\\
\in & \qquad \mbox{contained in, e.g.   }n\in\mathbb{N}\mbox{ means that }n\mbox{ is a natural number}\\
\not\in & \qquad \mbox{not contained in}\\
\sum &\qquad\mbox{sum, e.g   }\sum_{k=1}^{10} k\mbox = 1 + 2+3+4+5+6+7+8+9+10\\
\prod &\qquad\mbox{product, e.g   }\prod_{k=1}^5 k\mbox = 1 \times 2 \times 3\times 4\times 5\\
n! &\qquad n\mbox{ factorial,   }n!=\prod_{k=1}^n k\\
\int &\qquad\mbox{integral, e.g.   }\int_0^1 x^2\;\mathrm{d}x\\
\frac{\mathrm{d}}{\mathrm{dx}} &\qquad\mbox{differentiate with respect to }x\\
\lim_{x\to a} &\qquad\mbox{limit as }x\mbox{ tends to }a\\
\infty &\qquad\mbox{Infinity}
\end{align}
$$