Mathematical Communication Math Example 2

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Example 2

medium
Write a complete, well-structured mathematical proof that 'if nn is an even integer, then n2n^2 is divisible by 4.'

Solution

  1. 1
    Claim: If nn is even, then 4∣n24 \mid n^2.
  2. 2
    Proof: Assume nn is even. By definition, n=2kn = 2k for some integer kk.
  3. 3
    Then n2=(2k)2=4k2n^2 = (2k)^2 = 4k^2.
  4. 4
    Since k2k^2 is an integer, n2=4k2n^2 = 4k^2 is divisible by 4. □\square

Answer

n2=4k2, hence 4∣n2□n^2 = 4k^2, \text{ hence } 4 \mid n^2 \quad \square
A well-communicated proof states the claim, declares all variables, states all assumptions, carries out each step with a reason, and marks the end with □\square. This structure makes the argument readable and checkable.

About Mathematical Communication

Mathematical communication is the clear expression of definitions, reasoning, notation, and conclusions.

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