Clarity

Why clarity matters in proofs

Clarity in a proof means that a careful reader can:

• Follow every step without guessing what you meant.
• See why each step is valid.
• See how each step helps reach the conclusion.

A proof can be logically correct but still unclear. In this chapter, the focus is not on what counts as a valid proof (that belongs to other chapters), but on how to present a proof so that its logic is easy to understand.

Clarity has three main aspects:

1. Clear language.
2. Clear logical flow.
3. Clear use of notation and structure.

We will look at each of these with simple guidelines and examples.

Clear language

State what you are doing

Tell the reader what the goal is and what method you are using. For example:

• “We want to prove that $n^2$ is even.”
• “We will use a direct proof.”
• “We proceed by contradiction.”
• “We prove this by induction on $n$.”

You do not need to explain the methods themselves here (that belongs to proof technique chapters); just signal which one you are using so the reader knows what to expect.

Unclear:

• “Consider $n^2$.” (What are we trying to show?)

Clear:

• “We will show that if $n$ is even, then $n^2$ is even.”

Use complete sentences

Mathematics is not just symbols; it is mathematical sentences. Avoid writing a chain of formulas without explanation.

Unclear:
$$
n \text{ even} \Rightarrow n = 2k \Rightarrow n^2 = 4k^2 \Rightarrow \text{even} \Rightarrow \square
$$

Clear:
“Assume $n$ is even. Then there exists an integer $k$ such that $n = 2k$. Squaring both sides gives
$$
n^2 = (2k)^2 = 4k^2.
$$
Since $4k^2 = 2(2k^2)$ is twice an integer, $n^2$ is even.”

The mathematical expressions support the sentences, rather than replace them.

Avoid ambiguous words

Some everyday words are vague in a proof. Prefer precise phrases.

Less clear:

• “Obviously $n^2$ is even.”
• “It is clear that $A$ is a subset of $B$.”
• “It is trivial that $x > 0$.”

More clear:

• “Since $n = 2k$ for some integer $k$, we have $n^2 = 4k^2 = 2(2k^2)$, so $n^2$ is even.”
• “If $x \in A$, then by definition of $A$ we have $x \in B$, so $A \subseteq B$.”

Words like “obvious”, “clear”, “trivial” should be avoided, especially when you are learning. They often hide steps that actually need explanation.

Use consistent terms

Once you choose a term, stick to it in that proof.

Confusing:

• “Let $f$ be a function. This mapping $g$ …” (Are $f$ and $g$ the same?)

Clear:

• “Let $f$ be a function. We now define a new function $g$ by …”

If you say “integer” at the start, do not suddenly switch to “whole number” in the same proof unless you explain that you are using them as synonyms.

Clear logical flow

Make assumptions explicit

Every proof starts from some assumptions (the hypotheses of the theorem, or definitions you are using). State them clearly.

Unclear:

• “Then $n$ is even, so $n = 2k$.” (Where did “$n$ is even” come from?)

Clear:

• “Let $n$ be an even integer. Then, by definition of even, there exists an integer $k$ such that $n = 2k$.”

Clarity improves if you signal when you are using a definition or a previous result:

• “By definition of prime, …”
• “Using the theorem proved above, …”
• “By the hypothesis, …”

Justify non-obvious steps

Not every step needs a long explanation, but steps that use a definition, theorem, or subtle argument should be justified.

Unclear:

• “So $x^2 - 4x + 4 \ge 0$ for all $x$.”

Clear:

• “We have $x^2 - 4x + 4 = (x - 2)^2$, so $x^2 - 4x + 4 \ge 0$ for all real $x$.”

The justification does not always need a separate sentence, but it should be visible:

• “Since $(x - 2)^2 \ge 0$ for all real $x$, it follows that $x^2 - 4x + 4 \ge 0$ for all real $x$.”

Show the direction of reasoning

When proving an “if–then” statement
$$
\text{If } P \text{ then } Q,
$$
make it clear which is the assumption ($P$) and which is the conclusion ($Q$).

Clear pattern for direct proof:

• “Assume $P$. … Therefore $Q$.”

For example:
“Assume $n$ is odd. Then $n = 2k + 1$ for some integer $k$. … Hence $n^2$ is odd.”

Clear pattern for proof by contradiction:

• “Assume, for contradiction, that $\neg Q$ while still assuming $P$. … This leads to a contradiction. Therefore $Q$ must be true.”

The chapter on proof techniques explains when to use these methods; here the key is to state the structure of the reasoning out loud.

Avoid “backwards proof” mistakes

It is useful to plan a proof by working backwards from what you want to prove, but the written proof must go forward in logic. A common source of confusion is a chain like:

Unclear:
$$
n^2 \text{ even} \Rightarrow n^2 = 2k \Rightarrow n = \sqrt{2k} \Rightarrow n \text{ even}.
$$

This mixes planning and proving. Instead, either:

• Use a direct forward proof: start from $n$ even and deduce $n^2$ even, or
• Use proof by contrapositive, but then clearly state you are doing so.

Clear (direct):
“Assume $n$ is even, so $n = 2k$ for some integer $k$. Then $n^2 = 4k^2 = 2(2k^2)$, so $n^2$ is even.”

Clarity improves when every arrow $ \Rightarrow $ in your written proof represents a valid implication that goes from known information to a new conclusion.

Clear notation

Define symbols before using them

Every new symbol should be introduced.

Unclear:

• “Then $S$ is nonempty and $m = \min S$.”

Clear:

• “Let $S$ be the set of positive integers greater than 10. This set is nonempty, so it has a least element; denote this element by $m$.”

Introduce variables with phrases like:

• “Let $n$ be an integer.”
• “For each real number $x$, define …”
• “Fix $\varepsilon > 0$.”

If a symbol is standard and you are sure your intended audience knows it (for example, $+$, $=$), you do not need to define it. But anything problem-specific should be explained.

Use notation consistently

Do not change the meaning of a symbol inside a single proof.

Confusing:

• “Let $n$ be an integer. Then $n = 2n_1$ for some integer $n_1$. Hence $n_1^2 > n$.”

Here $n$ first means an integer, then it is reused in the subscript $n_1$ in an unclear way.

Clear:

• “Let $n$ be an integer. Then $n = 2k$ for some integer $k$. Hence $k^2 > n$.”

Each symbol should refer to only one mathematical object at a time in a proof.

Avoid cluttered expressions

Very long expressions inside a single line or equation can be hard to read. Break them into steps.

Cluttered:
$$
\frac{(a+b)^2 - (a-b)^2}{(a+b)^2 + (a-b)^2} = \frac{a^2 + 2ab + b^2 - (a^2 - 2ab + b^2)}{a^2 + 2ab + b^2 + a^2 - 2ab + b^2} = \frac{4ab}{2a^2 + 2b^2} = \frac{2ab}{a^2 + b^2}.
$$

Clearer:
“We compute:
$$
\begin{aligned}
(a+b)^2 &= a^2 + 2ab + b^2, \\
(a-b)^2 &= a^2 - 2ab + b^2.
\end{aligned}
$$
Then
$$
\begin{aligned}
(a+b)^2 - (a-b)^2 &= (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab, \\
(a+b)^2 + (a-b)^2 &= (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2.
\end{aligned}
$$
Therefore
$$
\frac{(a+b)^2 - (a-b)^2}{(a+b)^2 + (a-b)^2}
= \frac{4ab}{2a^2 + 2b^2}
= \frac{2ab}{a^2 + b^2}.
$$”

Clear structure and layout

Use paragraphs and line breaks

A proof often has a few logical stages. Separate them into paragraphs.

For example, a typical proof might have:

1. A paragraph introducing the goal and assumptions.
2. One or more paragraphs doing the main argument.
3. A final paragraph or sentence stating the conclusion.

Unstructured:
“Let $n$ be an even integer so $n = 2k$ for some integer $k$ so $n^2 = 4k^2 = 2(2k^2)$ and so $n^2$ is even and so the result holds.”

Structured:
“Let $n$ be an even integer. Then there exists an integer $k$ such that $n = 2k$.

Squaring both sides gives
$$
n^2 = (2k)^2 = 4k^2 = 2(2k^2).
$$
Since $2k^2$ is an integer, we see that $n^2$ is even. This completes the proof.”

Even simple white space can make a proof much easier to read.

Signal the beginning and end of proofs

In many texts, proofs are labeled and marked. For example:

“Proof. … This completes the proof.”

or

“Proof. … $\square$”

At this stage, what matters is that:

• The reader can see where the proof starts.
• The reader can tell when you are done.

A final sentence like “Therefore, the statement holds.” or “Hence the theorem is proved.” is often helpful.

Use connecting words

Words and phrases that connect sentences help the reader follow the logic:

• “Thus”, “therefore”, “hence” (indicate a conclusion from previous steps).
• “Because”, “since” (indicate a reason).
• “On the other hand”, “however” (indicate contrast or a different case).
• “Now”, “next”, “finally” (indicate steps in a sequence).
• “Case 1”, “Case 2” (indicate a case analysis).

Example:

“First, we show that $f$ is increasing. Next, we show that $f$ is bounded above. Finally, we use these two facts to prove that $f$ has a maximum.”

These words are not required, but they greatly improve readability.

Handling cases and sub-arguments clearly

Many proofs involve splitting into cases or proving helpful sub-results. Clarity requires that each piece is clearly labeled and closed.

Case analysis

When dividing a proof into cases, clearly label them and make sure you cover all possibilities.

For example:

“We consider two cases:

Case 1: $n$ is even. …

Case 2: $n$ is odd. …

In both cases, we conclude that $n^2 \equiv 0$ or $1 \pmod{4}$. Therefore, for every integer $n$, we have $n^2 \equiv 0$ or $1 \pmod{4}.$”

A common mistake is forgetting to say how the cases together imply the final statement. A short sentence like “Combining both cases, we see that …” makes the connection explicit.

Sub-arguments and lemmas

Sometimes it is clearer to separate a complicated part as a small sub-result.

For example:

“We first prove a helpful claim.

Claim: If $x > 0$, then $x + \frac{1}{x} \ge 2$.

[Short proof of claim.]

Now we return to the main proof. Using the claim with $x = a$, we obtain …”

Even if you do not use the word “lemma” or “claim”, separating the intermediate step and announcing when you are using it helps the reader.

Common clarity mistakes and how to fix them

Here are some typical issues that make proofs hard to read, along with clearer alternatives.

Jumping steps

Problem:
“Thus $n^2 - n$ is even, so $n$ is odd.”

To a beginner, this might not be obvious.

Clearer:
“Since $n^2 - n = n(n-1)$ is the product of two consecutive integers, one of them is even. Therefore, $n^2 - n$ is even. This implies that $n$ and $n-1$ cannot both be odd; hence $n$ must be odd.”

The exact argument might vary, but the important part is to fill in the reasoning that connects the statements.

Using results without saying so

Problem:
“Then $f$ is continuous, so $f$ is differentiable.”

If this uses a known theorem (which might even be false, as written here), it should be signaled.

Clearer:
“By Theorem X (stated earlier), continuity of $f$ on $[a,b]$ together with [extra conditions] implies that $f$ is differentiable on $(a,b)$.”

In an elementary context, at least write something like “By the theorem proved above, …” so the reader knows you are relying on a previous result, not inventing a new argument.

Mixing formal and informal styles awkwardly

Problem:
“Assume $n$ is even, so $n = 2k$ blah blah $n^2$ is even.”

Better:
“Assume $n$ is even. Then $n = 2k$ for some integer $k$. It follows that
$$
n^2 = (2k)^2 = 4k^2 = 2(2k^2).
$$
Since $2k^2$ is an integer, $n^2$ is even.”

Using full sentences and placing formulas where they support the text creates a more readable argument.

Practicing clarity

Clarity is a skill you improve with practice. Here are practical habits:

• After writing a proof, reread it as if you were another student who does not already know the answer. Ask:
• Can I see where the proof starts and ends?
• Is every symbol introduced?
• Are non-obvious steps justified?
• Is it clear which assumptions are used and where?
• Try to explain your proof out loud to someone else. If you stumble over a part, that part likely needs clearer writing.
• Compare your proof to a well-written solution (for example, from a textbook) and notice differences in structure and wording, not just the ideas.

Good proofs are not only correct; they are also easy to follow. Focusing on clarity will make your proofs more convincing to others and will also deepen your own understanding of the mathematics involved.