How to write maths

The rules below are from Kevin Houston’s How To Write Mathematics, with a few additions from me in italics.

• Write in sentences. Write in sentences. And once more to really hammer it home: Write in sentences.

• Explain what you are doing – keep the reader informed. For example, if you use the sum-of-roots rule to find a third root of a cubic equation, write something like:

Call the third root a. Sum of roots = 5 = a+2+i+2−i = a+4, so a=1.

When you get to the answer, tell the reader about it. So: “Area = 78” rather than just “78”. “Equation of tangent is ty = x + at2” rather than just “ty = x + at2“.

• Every sentence should have a verb. Usually, in A level maths, the verbs are: =, ⇒, |, or > or another inequality symbol. Use = properly. = means equals.

Never write something like 3×(1−2i)×(1+2i) = 15 = b = −30. It is not true that 15 = −30. Instead:

Product of roots = −b/2 = 3×(1−2i)×(1+2i) = 15
So b = −30

Define your symbols. If you are going to use the symbols a±bi for the third and fourth roots of a quartic, write:

Call the other roots a±bi

Don’t leave the reader to guess what you mean by a and b. Never use the same letter in the same calculation to mean two different things.

Always write each line below the previous one – not off to one side. Leave plenty of space between lines and on the right-hand side of the page

Each line of working should follow logically from the previous one.

Do big, well-labelled diagrams with a ruler. Try to get all your working onto a single page. (Start each problem on a new page.

Tegan makes the valid argument that with a multi-part problem, having a big diagram may contradict getting all the working onto a single page. In that case, compromise. Maybe start the problem on a left-hand page, so you can continue it to the right-hand page and still have all your working visible. (Don’t worry about leaving a page blank. It’s ok).

Be concise. In Further Maths, unlike GCSE, it’s not good to “show all your working”, though it is good to explain it. Having your working on a single page makes it easier for the reader to follow it, and easier for you to avoid mistakes caused by copying information wrongly from one page to another.

• We tend not to use decimal approximations of numbers in pure mathematics… If the final answer is sin 7, then leave it as that rather than say 0.656986598. In pure maths, write sin (π/6) as ½, not 0.5, and certainly sin (π/3) as √32, not 0.866. The reader seeing 0.5 won’t know whether you mean “exactly ½” or “0.5 to one decimal place”, and the difference between (exactly) √3/2 and (approximately) 0.866 may be important.

Houston summarises:

Have mercy on the reader. Do not make it difficult for them – particularly someone marking your work… Sorting through a jumble of symbols is likely to frustrate and annoy any assessor – not a good recipe for obtaining good marks.

“How to write maths” notes (odt)

“How to write maths” notes (pdf)

Worksheet on how to write maths (odt)

Worksheet on how to write maths (pdf)

“Writing maths” homework (pdf)

“Writing maths” homework (odt)