## Syntactic blu-tack and the value of inequalities

The syllabus for one of the first-year courses on which I teach begins with a paragraph or so of worthy but uninspiring topics from Standard Grade and Higher: fractions; indices; logs and exponentials; quadratic equations; and so on. Lurking among these is the innocuous phrase “solving inequalities”. When I inherited the course, this was generally passed over in about twenty minutes at the end of a lecture, and there was a gentleman’s agreement that it was unlikely to appear in the exam. Since then, I’ve confirmed my lack of gentility by gradually increasing the amount of time spent on this topic, because I’ve come to feel that inequalities are among the most useful problems that students can tackle with a basic mathematical toolkit.

The reason for this is that — unlike solving elementary equations, which can too easily be reduced to voodoo symbol-manipulation rules — solving inequalities sometimes requires thought. Consider the difference between solving the equation

$\dfrac{x-1}{x+1} = 0$

and solving the inequality

$\dfrac{x-1}{x+1} > 0$.

The commonest trap that students fall into is simply to treat the inequality as an equation with the = sign written differently, and to proceed

$\dfrac{x-1}{x+1} > 0 \implies x-1 > 0 \implies x > 1.$

A more sophisticated student may spot the implicit multiplication by $x+1$, and identify the basic logical steps:

$\dfrac{x-1}{x+1} > 0$, so either $x-1>0$ and $x+1>0$, or $x-1<0$ and $x+1<0$.

Thus x may either be higher than 1 or lower than -1.

At this point, further temptations lie in ambush. One is to write the final line as “Thus $-1 > x > 1$”. Another — not peculiar to inequalities — is to handle the implication signs carelessly, forgetting that the point of a solution process is to be able to run the logic in both directions. A lot of students are bewildered to discover that the statement

$x > 1 \implies \dfrac{x-1}{x+1} > 0$

is valid, but the statement

$\dfrac{x-1}{x+1} > 0 \implies x > 1$

is not. To properly understand this, they must properly understand why the implies sign is more than just a fancy equals sign. Understanding this seems to me to be worth a little pain.

Another temptation, and one for which professional teachers like me are substantially responsible, can be illustrated by a fictional but representative example of a student’s working:

$\dfrac{x-1}{x+1} > 0 \iff x-1>0$ and $x+1>0$, $x-1<0$ and $x+1<0$

$\iff x>1, x<-1$

$\iff x \in (-\infty,-1),(1,\infty).$

The idea is clearly there, but instead of being held together by logical connectors such as “or” and “and”, or set operators like $\cup$ and $\cap$, the algebra is blu-tacked together with the universal soft connector: the comma.

I’ve nothing against commas, in their place. (I’ve nothing against blu-tack, either: in fact like most office workers I hoard it dragonishly and put it to all sorts of unlikely uses.) The trouble is that at present in the English language commas aren’t content to be confined to their place. They’ve colonised the former territory of the semicolon; they’ve seized the strongholds of the colon with barely a shot fired in anger; and under the banner of the comma-splice they’ve possessed themselves of a good deal of lebensraum that once belonged to the full stop. The consequence in each case has been an infuriating loss of precision: instead of clause A complementing clause B, implying clause B or being distinct from clause B, it now sits sheepishly in the vicinity of clause B with a comma slouching on its elbows between them and responding to all grammatical interrogation with “yeah, whatever”. (There are also the insurgent commas that have started appearing between noun phrases and their verbs: “Grammar maven N. C. Dominie, said that…” I know these only represent a failure to balance a parenthesis properly, but — brrr.)

Hopefully the point is clear despite my tangled metaphors: the comma is too common in current English usage, and it is too frequently associated with sloppy thought. Its ubiquity in English, though, is as nothing compared with its ubiquity in mathematical usage, even among people who really should know better.

How frequently, as a maths teacher, have I (or have you) written something on the whiteboard like the following?

$(x-1)(x+1) = 0 \iff x = -1, 1$.

When we did this, did we pause to point out to the students the ambiguity, or did we leave them thinking that “$x=-1 \ \mathrm{or}\ x=1$”, “$x=-1\ \mathrm{and}\ x=1$” and “$x=-1 \ \mathrm{and} \ 1$” were all somehow equivalent and meaningful statements? Did we try to explain to them how not to confuse the statements “the roots of the quadratic are -1 and 1”, “the solutions of the quadratic equation are $x=-1 \ \mathrm{and} \ x=1$”, and “the solution of the quadratic equation is “$x=-1 \ \mathrm{or} \ x=1$””? Like hell we did.

Similarly, how often — as teachers or as professional paper-blackeners — have we written something like this?

$f(x) = \left\{ \begin{array}{l} 0, \quad x < 0, x > 1\\ 1, \quad \mathrm{otherwise.} \end{array} \right.$

The comma now acts, apparently, as a substitute for “if” as well as as a substitute for “and”. Of course we can figure out what is really meant without too much cognitive overheating — but I seem to remember that mathematical notation was supposed to exist so the reader didn’t have to figure out what we really meant.

We bewail, and quite rightly, the fact that students emerge from too many of our courses able to perform all kinds of ritual manipulations  but unable to identify logical fallacies at the level of “A implies B therefore B implies A”. There are plenty of reasons for this, but I suspect that one of them is that we’re reluctant either to give up our own sloppy notational habits or to spend time in class working on the kind of exercises that expose their inadequacy.

I faintly dread the stage of term — only a week or two away now — when inequalities appear. I’d far rather bluster through them and get on to the showier bits of the course. (Roll up! Roll up! See the Binomial Coefficient! Marvel at the Gallery of Trigonometric Identities! Watch in Awe as our Partial Fractions integrate Rational Functions you Never Thought were Integrable!) I dread in particular the mixture of confusion and contempt with which at least half the class will respond to what they see as my entirely unnecessary pedantry. Unfortunately, I’ve had to admit to myself that all this is cowardice.

A question to all my fellow slovens. Mathematics is supposed to be about building the most robust structures of thought the human brain has ever constructed. What sort of cowboys are we if we let our students think these structures can be held together with blu-tack?

To end on a more positive note, here’s a little obstacle course that your — and my — students might benefit from spending a little time on. If you don’t believe me, try it and see.

Q1a. Solve the equation $(x+1)(x-1) = 0$.
Q1b. Solve the inequality $(x+1)(x-1) > 0$.
Q1c. Solve the inequality $(x+1)(x-1) < 0$.

(There are probably several ways to do this, but I’d encourage them to sketch the graph of the quadratic in each case.)

Q2a. Solve the equation $(x+1)^2 = 0$.
Q2b. Solve the inequality $(x+1)^2 > 0$.
Q2c. Solve the inequality $(x+1)^2 < 0$.
Q2d. Solve the inequality $(x+1)^2 < 1$.

Thank you. Anyone who got all those right first time may stay after the show and clean the erasers.