10.4 Common Pitfalls

Mathematical writing can fail even when the main idea is correct. The problem is often not the mathematics itself, but the way the mathematics is presented. A reader may lose the argument because a term was not defined, a step was skipped, or a symbol was used in two different ways.

This section describes common mistakes and how to avoid them.

Using terms before defining them

A reader should know what every important word means before it is used.

For example, suppose a text says:

Let $G$ be connected. Then every two vertices of $G$ are joined by a path.

This may be confusing if the reader has not yet seen the words “connected,” “vertex,” or “path.” A better version defines the terms first:

A graph is connected if every two vertices are joined by a path. Let $G$ be a connected graph.

Definitions prepare the reader for the argument.

Hiding the assumptions

A theorem must state its assumptions clearly. Without assumptions, the claim may be false.

For example:

The square root of $x^2$ is $x$.

This is not always true for real numbers. If $x = -3$, then

not $-3$.

A clearer statement is:

If $x \geq 0$, then

The assumption $x \geq 0$ matters.

Skipping important steps

Writers often skip steps because the result feels obvious to them. For a beginner, the missing step may be exactly the hard part.

For example:

Since $a$ and $b$ are even, $a+b$ is even.

This may be true, but it does not explain why. A better version writes:

Since $a$ is even, $a = 2m$ for some integer $m$. Since $b$ is even, $b = 2n$ for some integer $n$. Therefore,

Since $m+n$ is an integer, $a+b$ is even.

The proof becomes easier to check.

Reusing symbols carelessly

A symbol should have one meaning in a given argument.

For example, if $n$ first means an integer, do not later use $n$ to mean the number of elements in a set unless the change is clearly stated. Reusing symbols without warning forces the reader to guess.

Good writing keeps notation stable. If a new object is needed, introduce a new symbol.

Writing too many symbols

Symbols can make mathematics precise, but too many symbols can make a simple idea hard to read.

For example:

This is correct, but for a beginner it may be easier to write:

If $a$ and $b$ are even integers, then $a+b$ is even.

Use symbols when they help. Use words when they explain.

Confusing examples with proof

Examples can support understanding, but they do not prove a general statement.

For example, checking

and

shows that some sums of even numbers are even. It does not prove that every sum of two even numbers is even.

To prove the general statement, we need an argument that works for all even integers.

Claiming something is obvious

Words like “obvious,” “clearly,” and “trivially” can hide important reasoning. They may also make a beginner feel that they missed something simple.

Instead of writing:

Clearly, $a+b$ is even.

write the reason:

Since $a+b = 2(m+n)$ and $m+n$ is an integer, $a+b$ is even.

A short explanation is usually better than a claim of obviousness.

A practical habit

After writing a proof or explanation, read it as if you are seeing the topic for the first time. Check whether every term is defined, every assumption is visible, and every important step has a reason.

Good mathematical writing reduces guessing. It helps the reader follow the argument instead of reconstructing it.