Writing mathematics

Whenever you try to communicate mathematical statements and ideas, you must start speaking and writing the language of mathematics. Exams and homework are also supposed to show the teacher how you think, whether you understand the concepts, and whether you've put enough work into mastering more mechanical skills. Here are a few pointers which should help you write better papers, and - hopefully - get better grades.

Be precise and clear

Like many other points I will try to make, this one is almost as important in everyday speech as in writing mathematics. But what may be understandable when talking to a friend ("hand me that doodad over there") can be unacceptable in mathematics ("that variable is kind of small, so there").

Imagine that you are giving directions to someone you don't know, and that your life depends on it. You would speak clearly, or write in block letters, use basic terms, and you'd rely on well-defined, universal concepts rather than on notions specific to your culture or environment.

It is much the same with mathematics. When you use words, think about their meaning. When you use symbols, make sure they will mean the same to both you and the professor. This is easy in math, because almost everything you will be writing about has a precise, agreed upon definition. A "continuous function", a "Riemann integral", or an "open interval" mean the same thing to everyone with mathematical training.

Let's end with my favorite quote from a posting by Herman Rubin: "Mathematics must be written so that it is impossible to misunderstand, not merely so that it is possible to understand."

Premises, premises...

Mathematics is all about implications, statements of the form "assuming that X, we have Y", or "if X then Y", or "since X is true, Y must hold", or "because we know X, Y follows". An implication has a premise - X, and a conclusion - Y. Some well-known rule of reasoning or a theorem must "fit" the particular situation for the implication to be accepted.

For example, "1/(a - 4) is positive since a is a real number greater than 4" is a perfectly valid statement; it follows from basic properties of inequalities (we must have a - 4 > 0, and reciprocals of positive numbers are positive). There is not much of a chance that the professor will mark this with a big red WHY?, unless you are taking an upper level number theory course in which you are supposed to derive such simple properties from even more basic principles.

But in a Calculus course a statement "1/(a - 4) is positive because a is a real number" will be rejected as a `non sequitur': the conclusion does not follow from the premise. It may very well be that in the context of the problem a is indeed greater than four, and that the conclusion is true, but you have not explained this clearly!

Use correct "grammar" and "punctuation"

As with spoken language, math has certain ground rules which have to be followed when you form sentences.

For example, a relation such as "=" can only be placed between two quantities which have a chance of being equal (e.g. when solving an equation), or when you are asserting that the quantities are in fact equal. One of the most common problems we see in student papers is the lack of respect for equality and other relations.

Consider this example: "x2 - 1 = 0 = x = 1, -1" and look closely at all the equalities separately. This is unintelligible babbling. It can be fixed by using symbols, or simply short English words and phrases to indicate what is happening: "if x2 - 1 = 0, then x2 = 1, so that x = 1 or x = -1".

We often use relations "chained together" to make a point; for example: x > y - 1 > 5 - 1 > 4, so that x must be greater than 4. This is perfectly OK as long as the relation is `transitive' (as > happens to be). But mixing disparate relations in this fashion can lead to nonsense: x > y - 1 = 4 < x + 2 > 3 doesn't really say anything and resembles a banana split with an anchovy topping.

Be accurate

More than with other sciences, in mathematics "what you see is what you get". There are few hidden meanings, little should be left unsaid, and there isn't too much room for interpretation or context. Equality means equality, 5 means 5, the Riemann integral is a Riemann integral whether the person reading what you wrote is a Romanian, a freemason, or a vegetarian. That's because we have precise definitions of things. This is also what makes mathematics difficult: it isn't enough to be "close" - you must be right on the money for your argument to count.

So if you write x = 1.4, you'd better mean 1.4, and not the square root of two! When you are told to find an integral of a function and you compute its derivative instead, don't be surprised if you don't get any credit. If you write about an isosceles triangle but your reasoning relies on it being equilateral, you won't get many points...

This insistence on accuracy is not some type of malice, or the professor's way of having fun; mathematics simply cannot exist without the rigor of meaning exactly what you say, and your papers cannot be fairly judged in any other manner. So don't get frustrated or angry, don't say that "I had it almost right". Just make sure you get it absolutely right on the next test.

Be concise, if you can

After all this you may think that you should write 2-page justifications for everything you do. Not so; on the contrary! It takes very little to put your mathematical thinking down on paper. All you need is some commonly used symbols and a handful of words such as "if ... then ...", "therefore", "hence", "because", "since", "so", "this means", "that's why". In fact, most mathematicians prefer to see that rather than read long essays which use common English, because the terse mathematical way of expressing things is usually more precise and less distracting.

If you can, write only as little as necessary to make the meaning very clear. It isn't easy to judge exactly "how much is enough", but you can try the following experiment: imagine that you have a little brother in the class, that he is a few weeks behind in his studies, that you are explaining the problem to him, and that he wants to know why you are doing whatever you are doing.

But if you have doubts, it is generally better to say more than to say too little. You'll make your teacher's life a bit harder, and you might come across as a stickler for unnecessary detail, but you aren't likely to get marked down for that.

Please send comments, suggestions, corrections to behr@math.niu.edu.