How to Write Maths

In mathematics, each symbol holds significant and precise meaning, and it’s much easier to write something objectively wrong, whereas in words we have more wiggle room due to this subjectivity malarkey.

First Step: Enjoy it

Professional mathematicians often talk about the pleasure of writing maths out by hand. You’ll see many academics working on blackboards. You can watch this video about the chalk mathematicians import from Asia, which is how modern greats like Terence Tao like to work. Others love a nice mechanical pencil and a good quality rubber.

My grandparents gave me a fancy 4-colour pen and a mechanical pencil which I became very attached to across A-Level & University for my notes. I also love writing on my iPad, which has infinite zoom and panning for rough work.

My point is that taking your time and learning to enjoy the process of laying out your thinking can lead to more appreciation of why people like the practice. Arranging your thinking and logic can become a deliberate and mindful process: here each marking you add has significance.

Avoid writing nonsense

Example: It’s surprisingly common to see working a bit like this:

Sometimes students use $=$ to mean something like “and next”, but the above claims all those terms are equal, so $11 = 9 = 3$.

Or sometimes students spread their calculations across the page, but the examiner can only give marks for what they can follow.

To avoid this, there are two important rules to follow:

• We write maths downwards, lining equals signs in a column, so each line is clearly equal to the one before.
• We write one statement per line so that the reader can isolate each statement as separate.

So the above example becomes:

Here we perform operations to each side maintaining the balance of equality, and we never write anything contradictory. Plus it’s easier to follow.

Note: Make your letters easy to read, too. Write a curly $x$ so it can’t be mistaken for a multiplication sign $\times$, and put a bar through your $z$ so it isn’t mistaken for a $2$. A $y$ is lovely: it’s always clear what it is.

Another valid example is where you avoid repeating writing one side of the equation which stays the same. So the left-hand side here is unchanging, but by leaving it blank we imply this to the reader. What is important here is that we line up the equals so it follows through.

Definition: The above shows why it is nice to be able to refer to either side of the equation independently. We call the two sides of an equals sign the left-hand side (LHS) and the right-hand side (RHS).

Key Example: Show that $(x + 1)^2 - 1 = x(x + 2)$.

Both sides simplify to $x^2 + 2x$, so $\text{LHS} = \text{RHS}$.

Note: I quite like the symbol $\stackrel{?}{=}$ to show we’re not sure whether both sides are equal and we need to check. It’s not on the Edexcel notation list but it may come up in this course for examples/questions. I think it’s pretty self-explanatory though.

How do we connect our sensible maths together?

The symbols from section 1 are how we tell the examiner how one line connects to the next:

• $\Rightarrow$ when one line leads to the next. Common in working and chaining together steps, worth getting used to.
• $\Leftrightarrow$ when it works both ways, which matters when you need to show no solutions were lost or gained. Mostly seen in proofs (but you could use it more widely).
• $\therefore$ to state the result you’ve reached. Helpful to show the examiner where your conclusion is.

Example: A triangle has sides $5$, $12$ and $13$. Is it right-angled?

Pythagoras’ theorem works both ways. For a triangle whose longest side is $c$:

So we check the two shorter sides against the longest:

Note: We usually assume that if a triangle is right-angled, we can use Pythagoras’ theorem. But as the relationship is “if and only if”, and implies both ways, if we can show that Pythagoras’ theorem is true for the sides, then we’ve also shown that the triangle is right-angled.

Steps that work both ways

Think of an equation as a balance: to keep it level, whatever we do to one side we do to the other. Some steps can always be undone: adding or subtracting the same thing, or multiplying or dividing by the same non-zero number. Because we can undo them, the equation before and after the step have exactly the same solutions, which we write with $\Leftrightarrow$:

Steps That Only Work One Way

Other steps can’t be undone. Squaring is the classic one:

But $x^2 = 9$ doesn’t lead back to $x = 3$, because $x = -3$ works too. The arrow only points one way.

Multiplying both sides by $0$ is the most extreme case: $1 = 2 \implies 0 = 0$. The new line is true, but it tells us nothing, and we can never get back.

Does your maths read sensibly?

Every line of working should be a true statement, and the examiner should be able to see why it follows from the line before.

A good habit is learning how to verbalise your working, swapping out the symbols for words, checking if any line makes a false statement.

When in doubt, remember to check your answers by putting them back into the original equation.

Very Important Note: Did you know that one of the most famous nonsense poets responsible for Alice in Wonderland was also a mathematics lecturer at Oxford University? Perhaps when you spend your day being so careful with making sense, in the evenings you like to cut loose with some “the slithy toves did gyre and gimble in the wabe”.

So before every new line, ask: Does this make sense? Can the reader follow my logic?