_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 fractions ((For now, at least - as you progress, you may learn when you can ignore this rule. Follow it until then.)); 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.