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MathidiaMathematics Academy
A Level synthesis
Lesson 74 of 75
Challenge

Proof and communication

Objective. Write concise, logically complete advanced solutions.

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

Learn: the key idea

Proof and communication at this level means writing advanced mathematical arguments that are both logically complete, with each step justified and following necessarily from the last, and concise, avoiding unnecessary repetition or unexplained jumps. Good practice includes stating which proof method is used (direct proof, proof by contradiction, or proof by exhaustion), defining any variables introduced, and ending with a clear concluding statement that restates what has been shown. For example, proving √2 is irrational by contradiction requires explicitly assuming the opposite, deriving a contradiction about parity, and concluding the original statement must be true.

Worked example

Prove, using clear logical communication, that the sum of any two consecutive integers is odd.

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Illustrated card showing ?
  1. 1Let the two consecutive integers be n and n + 1, for integer n
  2. 2Their sum is n + (n + 1) = 2n + 1
  3. 32n is always even, so 2n + 1 is one more than an even number
  4. 4Hence 2n + 1 is odd, proving the sum of two consecutive integers is always odd

Answer: Proved: the sum of two consecutive integers, 2n + 1, is always odd

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Common mistake. Students write correct working but no concluding sentence, leaving the reader to infer what has been proved. Always finish a proof with an explicit statement, such as "hence the result is proved", tied to the original claim.

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