There are lots of different ways of writing maths that are equivalent and can be used interchangeably. This can be confusing if your lecturer uses a different notation than you are used to.

The reason that there are different ways to say the same thing in maths is because mathematical notation has been developed over many thousands of years by many different people.

In English ‘big’ (from Middle English) and ‘large’ (from Latin) mean the same thing. There are similar examples in maths.

The most important thing to remember is that choice of notation is often cultural or simply personal preference, and therefore it is okay to ask for clarification.

Numbers and calculations

Division and multiplication

Sometimes we think of \(\frac{a}{b}\) as a ‘fraction’ and \(a\div b\) as a ‘calculation’, but they are actually interchangeable.

\[\begin{equation} \frac{2}{5} = 2\div 5 = 2/5 \end{equation} \]

When working with numbers we often use a multiplication sign \(\times\), but this is usually dropped when working algebraically as it looks a bit like the letter x.

If you are working with vectors then then it is important to distinguish between the cross product \(a \times b\) and the dot product \(a \cdot b\), but with scalars (e.g. straightforward numbers), they are interchangeable.

\[\begin{equation} a \times b = a \cdot b = ab = a(b) = (a)b = (a)(b) \end{equation} \]

Dots and commas

Different cultures tend to use dots and commas in different ways, if you aren’t sure please just ask your lecturer to clarify their notation.

A decimal point may be shown in several ways:

\[\begin{array}{ll} 2 \frac{3}{10} & = 2 \cdot 3 \\ & = 2.3 \\ & = 2,3 \end{array} \]

but a dot can also be used as multiplication

\[\begin{array}{ll} 2 \times 3 & = 2 \cdot 3 \\ & = 2.3 \\ & = 2(3) \end{array} \]

Commas and dots might also be used for separating groups of a thousand

\[\begin{array}{ll} 2145600 & = 2 145 600 \\ & = 2,145,600 \\ & = 2.145.600 \end{array} \]

Calculus

Maths grammar

\[\begin{array}{r|l} \text{Calculus (noun)} & \text{The branch of maths that includes differentiation} \\ \text{Differentiation (noun)}& \text{The process of finding the derivative}\\ \text{Differential (noun)} & \text{The change in the variable; the differential of $x$ is $dx$}\\ \text{Derivative (noun)} & \frac{dy}{dx} = \text{The rate of change of the function}\\ \text{Differentiate (verb)} & \text{The act of finding the derivative; you differentiate the function $y$ to find $\frac{dy}{dx}$}\\ \text{Derive (verb)} & \text{Red herring! Whilst this sounds like the verb form of derivative it isn't! Use 'differentiate' instead.} \end{array}\]

Equivalent ways of showing differentiation

\[\begin{array}{ll} \textbf{function} & f(x)= x^2 & y= x^2 & y = x^2 & u^2 & v = z^2 & s = t^2 & f(x) = x^2 \\ & \color{orange}{\downarrow} & \color{blue}{\downarrow} & \color{green}{\downarrow} & \color{red}{\downarrow} & \color{purple}{\downarrow} & \color{olive}{\downarrow} & \color{pink}{\downarrow} \\ \textbf{derivative} & f'(x)= 2x & \frac{dy}{dx}=2x & y' = 2x & \frac{d(u^2)}{du}=2u & \frac{d}{dz}v = 2z & \dot{s} = 2t & D(f) = 2x \\ & \color{orange}{\downarrow} & \color{blue}{\downarrow} & \color{green}{\downarrow} & \color{red}{\downarrow} & \color{purple}{\downarrow} & \color{olive}{\downarrow} & \color{pink}{\downarrow}\\ \textbf{second derivative} & f''(x) = 2 & \frac{d^2y}{dx^2}=2 & y'' = 2 & \frac{d^2(u^2)}{du^2}=2 & \frac{d^2}{dz^2}v = 2 & \ddot{s} = 2 & D^2(f) = 2 \end{array} \]

Letters that look like d but mean different things

\[\begin{array}{lll} \Delta{x} & \text{"delta x"} & \text{the change in x} \\ \delta{x} & \text{"delta x"} & \text{the small change in x} \\ d{x} & \text{"the differential"} & \text{the change in x when it approaches zero} \\ \partial{x} & \text{"the partial differential"} & \text{the change in x when it approcahes zero} \\ d(x,y) & \text{"distance between x and y"} & \text{the distance function (metric spaces and topology)} \end{array}\]

Exponentials

In the expression \(a^b\), \(b\) can be called the ‘power’ the ‘exponent’ or the ‘index’ (indices is the plural of index).

In the expression \(e^b\), \(e\) is a number that might be called ‘Euler’s number’ (which is pronounced ‘oy-luh’).

The expression \(e^x\) is a function, and can also be written in function notation:

\[\begin{equation} e^x = \text{exp}(x) = \text{exp}x \end{equation}\]

The choice of notation might be used to emphasise that it is a function, or perhaps just to make it easier to read:

\[\begin{equation} e^{\frac{2x^2}{3 - x}} = \text{exp}\big({\frac{2x^2}{3 - x}}\big) \end{equation}\]

Vectors

There are so many different notations that can be used for vectors, it can be very confusing.

Vectors are often written in lowercase, and vertices are often uppercase.

You can use a single letter to show a vector, or split it into its components.

\[\begin{array}{lllll} \vec{OA} & = \textbf{a} & = \underset{^\sim}{a} & = \vec{a} & = \underline{a}\\ = (a_1, a_2) & = (a_1 \space a_2) & = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} & = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} &\\ = a_1 \textbf{i} + a_2 \textbf{j} & = a_1 \hat{i} + a_2 \hat{j} & \end{array}\]

\(\hat{a}\) is usually reserved for the ‘unit vector in the direction of \(\vec{a}\)’

Watch out for pointy brackets though, as they are not used to show something is a vector, but for an operation involving two vectors (the inner product)

\[\begin{array}{lll} \mathbf{a} = (a_1,a_2) & \mathbf{b} = (b_1,b_2) & \text{are vectors}\\ \langle \mathbf{a}, \mathbf{b} \rangle && \text{is the inner product of the two vectors} \end{array}\]

Matrices

Matrices are usually named with a letter, which is written in bold font. You can use curved or square brackets for a matrix, or sometimes even no bracket at all.

Another word for (curved) brackets is ‘parentheses’ and {curly} brackets are sometimes called ‘braces’.

\[\begin{equation} \textbf{A} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} = \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \end{equation}\]

As a general rule matrices are written with uppercase letters,\(A\), and elements of a matrix are denoted with lower case \(a\). Each element of the matrix can be written with a subscript, \(a_{ij}\), that tells you where it is located:

\[\begin{equation} \textbf{A} = \begin{pmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{pmatrix} \end{equation}\]

Curly brackets are not generally used for matrices, but because they are used for set notation you could write a matrix using curly brackets:

\[\begin{equation} \textbf{A} = \{a_{ij}\} \end{equation}\]

Straight lines

Straight vertical lines are used for a property of the matrix, rather than the matrix itself. Watch out - some lecturers’ handwriting makes it tricky to tell what style of line is intended! If in doubt, just ask for clarification.

Single lines show the determinant of a matrix (telling you something about the scale factor when using the matrix to enlarge a vector).

\[\begin{equation} det(\textbf{A}) = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} \ne \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \end{equation}\]

Double lines are for the norm of a matrix (telling you something about the size of the elements).

\[\begin{equation} norm(\textbf{A}) = \begin{Vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{Vmatrix} \ne \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \end{equation}\]

If you are working with numbers or vectors then single and double lines are often used interchangeably to mean ‘distance from the origin’ or ‘magnitude of the vector’.

\[\begin{equation} \Vert -3 \Vert = \vert -3 \vert = abs(-3) = 3 \\ \Vert (3,4) \Vert = \vert (3,4) \vert = \sqrt{3^2 + 4^2} = 5 \\ \end{equation}\]

Statistics

Random variables

In general uppercase letters are used to name random variables and lowercase letters are for the values those variables.

For example

\[\begin{array}{ll} X & \text{possible values obtained when rolling a dice} \\ x & \text{the value after a single roll } \\ X = \{x\} & \text{the set of possible values} \\ & = \{1,2,3,4,5,6\} \\ x_1, x_2, x_3 & \text{the results obtained from three rolls of the dice} \\ x_i & \text{the value of one of the rolls of the dice} \\ P(X = x) & \text{the probability that the value rolled is a specific value} \\ P(X = 2) & \text{the probability that a 2 is rolled} \end{array}\]

Summary statistics

\[\begin{array}{l\l} \mu & \text{'mew'} &\text{population mean}\\ \mu_{X} & \text{'mew X'} &\text{population mean of X}\\ \bar{x} & \text{'x bar'} & \text{sample mean} \\ \sigma & \text{'sigma'} &\text{standard deviation}\\ \sigma_{X} & \text{'sigma X'} &\text{standard deviation of X}\\ s && \text{sample standard deviation} \\ \rho & \text{'ro'} & \text{product moment correlation coefficient of population} \\ r_x && \text{product moment correlation coefficient of sample} \end{array}\]

Dictionary

General notation

\[\begin{array}{ll} = & \text{equal to} \\ \neq & \text{not equal to} \\ \approx & \text{is approximately equal to} \\ \infty & \text{infinity} \\ \propto & \text{is proportional to} \\ \therefore & \text{therefore} \\ \because & \text{because} \\ < & \text{less than} \\ > & \text{greater than} \\ \le & \text{less than or equal to} \\ \ge & \text{greater than or equal to} \\ \implies & \text{implies} \\ {:} & \text{such that} \\ \forall & \text{for all} \end{array}\]

Set notation

\[\begin{array}{ll} \in & \text{is an element of} \\ \notin & \text{is not an element of} \\ \subset & \text{is a subset of} \\ \emptyset & \text{the empty set} \\ A' & \text{the complement of} A \\ \mathbb{N} & \text{the natural numbers} \\ \mathbb{Z} & \text{the integers} \\ \mathbb{R} & \text{the real numbers} \\ \mathbb{Q} & \text{the rational numbers} \\ \mathbb{C} & \text{the complex numbers} \\ \cup & \text{union} \\ \cap & \text{intersection} \\ \end{array}\]

Series notation

\[\begin{array}{ll} \sum & \text{sum of} \\ \sum_{i=1}^{n} a_n & = a_1 + a_2 + ... + a_n\\ \prod & \text{product of} \\ \prod_{i=1}^{n} a_n & a_1 \times a_2 \times ... \times a_n \\ n! & \text{n factorial} \\ & =n \times (n-1) \times... x 2 x 1 \\ \binom{n}{k} & \text{n choose r (the binomial coefficient)} \\ {}^{n}C_{k} & \text{n choose r (the binomial coefficient)} \end{array}\]

Function notation

\[\begin{array}{ll} f: x \mapsto y & \text{the function f maps x onto y} \\ f^{-1} & \text{the inverse of f} \\ \underset{x \rightarrow \infty}{lim} f(x) & \text{the limit of f(x) as x tends to infinity} \end{array}\]