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MathidiaMathematics Academy
Mathematical research methods
Lesson 70 of 75
Review

Mathematical exposition

Objective. Write a rigorous, readable mathematical report.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Mathematical exposition is the craft of writing a rigorous yet readable report: stating definitions before they are used, presenting theorems with clearly labelled hypotheses and conclusions, and giving proofs that justify every non-trivial step while motivating the reader with brief remarks on why each step is taken. For example, a well-written exposition of the intermediate value theorem first defines continuity precisely, states the theorem's hypotheses (continuous function on a closed interval with opposite-signed endpoints), and only then gives the proof, rather than diving into the proof before the reader knows what is being assumed. Good exposition also signposts the overall proof strategy in a sentence before the technical details begin.

Worked example

In mathematical exposition, which ordering of a report's sections best supports a reader following a proof of a new theorem?

Options
  • Proof, then theorem statement, then definitions
  • Definitions, then theorem statement with hypotheses, then proof
  • Proof only, with definitions left implicit
  • Theorem statement with no listed hypotheses, then proof
  1. 1Consider what a reader needs before they can follow a proof's steps.
  2. 2Presenting the proof before definitions forces the reader to guess undefined terms.
  3. 3Giving definitions first, then the theorem with its hypotheses, then the proof lets the reader verify each step against known meanings.
  4. 4This ordering, definitions then theorem then proof, is the structure that best supports a reader.

Answer: Definitions, then theorem statement with hypotheses, then proof

Concept mastery

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Common mistake. A common mistake in mathematical exposition is using a term or symbol in a proof before it has been defined, forcing the reader to guess its meaning from context rather than following a clearly ordered argument.

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