Mathematics is a Language

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Read First: This page features many symbols and several might be new. You may want to note each down and label accordingly, but you also could just read this page and come back when you need to, whatever suits you.

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So you’re studying A-Level Maths, but you don’t speak the language?

Remember, maths is a language. Often referred to as the universal language. It is the language of logic, rules and systems. It is the language in which the universe is written (according to our friend Galileo).

So if we are to speak with the universe, we must learn the words and grammar it uses. I’ll describe the new symbols you need to know below in English, which hopefully gives you a chance at both translating the mysteries of our existence and easing through your exams.

What you need to know

You don’t have to memorise these all right now (come back and look when you are confused). We will add more symbols as we go and build a final list in the summary.

Logic symbols

Symbol	Meaning	Example
$=$	Is equal to (conditional equality, true for specific values or a calculated result)	$x = -3, x = -5$ can be true in the context of $x^2 + 8x + 15 = 0$ but isn’t always the value of $x$
$\equiv$	Identically equal to (is always equal, true for all variable values)	$\sin^2(x) + \cos^2(x) \equiv 1$ is an identity which is true for all values of $x$
$\neq$	Is not equal to (used both for values in a context and for facts that are always true)	$f(x) = \dfrac{1}{x}, x \neq 0$ so $x$ isn’t equal to $0$ in this context. Or $1 \neq 2$ is a universal fact.
$\approx$	Is approximately equal to	$\pi \approx 3.14159$
$\Rightarrow$	Implies / Leads to	$x > 0 \Rightarrow x > -10$ . The first condition implies the second.
$\Leftarrow$	Is implied by / Follows from	$x = 2 \Leftarrow x + 8 = 10$ . The second condition implies the first.
$\Leftrightarrow$	If and only if (iff) / Is equivalent to	The implies logic needs to go both ways now. The test is if $a \Leftrightarrow b$ then $a \Rightarrow b$ and $a \Leftarrow b$ . E.g. $5x > 15 \Leftrightarrow x > 3$ .
$\therefore$	Therefore	$3x = 12 \therefore x = 4$
$\because$	Because	$x = 4 \because 3x = 12$
$\forall$	For all / For every	$\forall$ even number $n$ there exists a $k$ where $2k = n$ and $k$ is an integer
$\exists$	There exists	$\exists x \text{ such that } x^2 - x - 1 = 0$

Note: You’ll also see longer arrows, $\implies$ and $\iff$ . They mean exactly the same as $\Rightarrow$ and $\Leftrightarrow$ . The last two symbols, $\forall$ and $\exists$ , aren’t on the Edexcel notation list, but they’re very common to see.

Set Notation

Symbol	Meaning	Example
$\in$	Is an element of	$3 \in \{1, 2, 3 \}$
$\notin$	Is not an element of	$0.5 \notin \{1, 2, 3\}$
$\{x_1, x_2, \dots\}$	The set with elements $x_1, x_2, \dots$	$\{2, 4, 6, \dots\}$ is the set of positive even numbers
$\{x : \dots\}$	The set of all $x$ such that … (set builder notation)	$\{x : x > 5\}$ is every number greater than $5$ , and $\{x \in \mathbb{R} : x \neq 0\}$ is every real number except $0$
$\emptyset$	The empty set	$\{1, 2\} \cap \{3, 4\} = \emptyset$
$\mathbb{N}$	The set of natural numbers, $\{1, 2, 3, \dots\}$ (the counting numbers)	$5 \in \mathbb{N}$
$\mathbb{Z}$	The set of integers, $\{0, \pm 1, \pm 2, \pm 3, \dots\}$	$-7 \in \mathbb{Z}$
$\mathbb{R}$	The set of real numbers (all rational and irrational numbers)	$\pi \in \mathbb{R}$ and $\sqrt{2} \in \mathbb{R}$
$\mathbb{Q}$	The set of rational numbers, $\left\{\dfrac{p}{q} : p \in \mathbb{Z}, \; q \in \mathbb{Z}^+\right\}$ (fractions)	$\dfrac{3}{4} \in \mathbb{Q}$ and $0.25 \in \mathbb{Q}$
$\cup$	Union (all elements in either set or both)	$\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$
$\cap$	Intersection (elements common to both sets)	$\{1, 2, 3\} \cap \{2, 3, 4\} = \{2, 3\}$

Note: You’ll also see $\mathbb{Z}^+$ , the positive integers $\{1, 2, 3, \dots\}$ , and $\mathbb{Z}^+_0$ , the non-negative integers $\{0, 1, 2, 3, \dots\}$ . So why have $\mathbb{Z}^+$ when it’s the same set as $\mathbb{N}$ ? Mathematicians don’t agree on whether $0$ is a natural number, and many books write $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ . $\mathbb{Z}^+$ and $\mathbb{Z}^+_0$ leave no doubt about whether $0$ is included.

Operators & Values

Symbol	Meaning	Example
$\propto$	Is proportional to	$y \propto x^2 \Rightarrow y = kx^2$
$\sum$	Sum of	$\sum_{r=1}^{4} r = 1 + 2 + 3 + 4 = 10$
$\vert{}x\vert{}$	The modulus (absolute value) of $x$	$\lvert -5 \rvert = 5$ and $\lvert 5 \rvert = 5$
$!$	Factorial	$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
$\infty$	Infinity	As $x \to \infty$ , $\dfrac{1}{x}$ tends towards $0$
$-\infty$	Negative infinity	As $x \to -\infty$ , $\dfrac{1}{x}$ tends towards $0$

A Magic Word

One of the most powerful words in the mathematical language is one you will recognise: the humble unassuming “Let”. So let me use it in an example…

Whenever it is helpful in maths we can say " $\text{Let } a = b$ " and substitute the new one in.

So more likely something like " $\text{Let } a + 3 = y$ " so that we can eliminate $a + 3$ out of an equation by switching it out for $y$ .

Example: Solve $x^4 - 5x^2 + 4 = 0$

We’re going to use the fact that $x^4 = (x^2)^2$ to identify that if we substitute $u$ in for $x^2$ then we simplify the equation.

\begin{aligned} \text{Rewrite equation:} &\quad (x^2)^2 - 5(x^2) + 4 = 0 \\ \text{Let } u = x^2: &\quad u^2 - 5u + 4 = 0 \\ \text{Solve for } u: &\quad (u - 1)(u - 4) = 0 \implies u = 1, \, 4 \\ \text{Substitute back:} &\quad x^2 = 1 \implies x = \pm 1 \quad \text{and} \quad x^2 = 4 \implies x = \pm 2 \end{aligned}

Why should you learn this language?

If you’re studying maths, it is useful to be fluent in these symbols, so that you can translate them without thinking…

\text{we like } \pi \because \textbf{it tastes good} \iff \text{it is well-cooked } \therefore \text{chefs} \in \text{good salary group}

I regularly want to use them outside of maths for concise communication. It really does all combine and click into feeling like a language of logic at some point.

It’s helpful for you to start to recognise these symbols and what they mean, but we’ll fill in how to use them and really understand them over the whole course. The next two sections help to fill in communicating with these symbols and the important new number sets.