The Elements of Mathematical Style (Version 0.1)

This is a guide for A-level students on how to look like you know what you’re doing. Some of it’s common sense, some of it’s arbitrary (I’m willing to argue the toss before over-ruling you). Please feel free to add ideas, suggestions and criticism as comments!

If you leave your twitter handle in comments, I’ll give you credit when I revise the post.

I’m already grateful to @cardiffmaths, @christianp, @mrgpg, @reflectivemaths and @sxpmaths for their wrong-headed arguments and suggestions ;o)

Mathematical style in algebra

  • When writing a complicated product, the usual form is (constants)(variables)(functions), e.g. $\frac{7}{2}x^2(x-1) \sin(x+4)$

Mathematical style in differentiation

  • Prefer $\diff yx$ to $y’$ unless you have a compelling reason not to((At least until you’re doing second order ODEs)). The ‘prime’ notation cost British science about 100 years of development after Newton.
  • Use the correct letters in your derivatives. If you’re differentiating $A$ with respect to $r$, call it $\diff AR$, not $\diff yx$
  • It’s called a derivative, not a differential.
  • With a second derivative, the 2 goes early on top and late on the bottom: $\diffn 2yx$.
  • With product, quotient and chain rule, explicitly write out what you’re calling $u$, $v$ etc.

Mathematical style in footwear

  • Don’t wear socks with sandals, or vice-versa.

Mathematical style in fractions and decimals

  • Always prefer fractions to decimals unless the question calls for decimals.
  • Mixed fractions like $3\frac{1}{2}$ are an abomination. Use top-heavy fractions like $\frac{7}{2}$ instead.
  • Avoid oblique fractions, especially when multiplied by anything else, e.g. $1/2x$. Use either $\frac{2}{3}x$ or (possibly) $\frac{2x}{3}$.
  • Avoid stacking fractions. Instead of $\frac{\frac{a}{b}}{\frac{c}{d}}$, write $\frac{a}{b} \div \frac{c}{d}$ or – better yet – $\frac{a}{b} \frac{d}{c}$.
  • Where possible, leave fractions in factorised form. Prefer $\frac{(x+2)(x+3)}{(x+1)(x+6)}$ to $\frac{x^2 + 5x + 6}{x^2 + 7x + 6}$.

Mathematical style in functions

  • Use brackets with all functions. Even if the question says $\sin 2x$, write $\sin(2x)$.
  • Modulus signs ($\left| x \right|$) count as brackets.

Mathematical style in handwriting

  • Avoid multiplication signs if possible. Using a $\cdot$ is best if there’s any chance of confusion with an $x$.
  • Write $x$ as two back-to-back semicircles.
  • Cross your $z$s so as not to confuse them with 2s.
  • Be careful not to confuse S with 5.

Mathematical style in integration

  • Always include $\d x$ or similar in your integrals. It seems petty now, but when you do substitution in C4, it makes much more sense if you’re used to seeing $\d x$.
  • With substitution, always write down your substitution and its derivative.
  • With parts, always write down $u$, $\diff vx$ and everything else; if you need to do a second stage, call your variables $U$ and $V$.
  • Write limits in front to the right of an integral sign ($ \int^b_a$) and after brackets ($[\frac{x^4}{4}]^b_a$).
  • If you have cause to combine constants or otherwise change them, be sure to rename them. Don’t have $c$ meaning different things in different places.

Mathematical style in proofs

  • Never move elements from one side of a proof to the other.
  • State what you’re trying to prove, and any assumptions you’re making.
  • After finishing a proof, write a Halmos tombstone ($\blacksquare$).

Mathematical style in sketching graphs and diagrams

  • If you have to ask whether your graph is big enough, the answer is ‘no’.
  • Labels are much more important than accuracy.
  • Label all important points and lines:
    • Where the curves meet axes
    • Turning points
    • Horizontal and vertical asymptotes.
  • Saying what your axes represent is more important than measuring them.

Mathematical style in straight lines

  • $(y – y_1) = m(x-x_1)$ is the only game in town.
  • Give lines in the form $ax + by + c = 0$ unless you’re told otherwise.

Mathematical style in surds

  • Avoid square roots on the bottom of fractions1; rationalise the denominator where possible.
  • In calculus, always turn roots into fractional powers as soon as possible.

Mathematical style in trigonometry

  • Prefer $\arctan$ and similar to $\tan^{-1}$.

Mathematical style in vectors

  • Underline all vector variables, wavily if possible.
  • Write vector components as a column. Avoid ${\bi, \bj, \bk}$ notation.
  • Draw a force as an arrow with a solid head, acceleration as an arrow with a double head and velocity as an arrow with a single head.

* Edited 13/12/13 for formatting.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. For now, at least – as you progress, you may learn when you can ignore this rule. Follow it until then. []


One comment on “The Elements of Mathematical Style (Version 0.1)

  • Chase Turner

    Each of the sciences has its own “isms” to convey commonly understood formulas, constants, or embodying hints about usage of greek (and other) symbolic representations. For example, Mathematicians understanding an implied distinction between lower case “rho” versus upper case “rho” in Mathematics is not the same in Physics. Whereas “Pi” is a universal constant in both sciences.
    What I am seeking are external references to resources akin to an “Elements of Style” guide — that illustrate usage examples of greek symbols (both lower AND Upper — to know they don’t even look the same) in a number of different sciences. Comparison and contrast of those examples will help educate students as to when they can expect the greek symbol is indeed a constant, versus “here it has a different meaning”, versus “here, it is just a place holder only”. Finally, advice as to when it is okay to wander off from regular symbolic representations and use your own. (I say this after a colleague showed me a Math PhD thesis where the choice of symbols from the UTF-8 character set was … terrible. Yes, the meanings were free of of conceptual collisions with long-established symbolic usage, but it was visually hard to process .. because the selected symbolic characters from that UTF-8 were unfamiliar to my eyes).

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