Mathematical Communication Math Example 1

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Example 1

easy
Rewrite the following unclear statement into precise mathematical language: 'Adding two things and squaring is not the same as squaring them and adding.'

Solution

  1. 1
    Identify the claim: (a+b)2≠a2+b2(a+b)^2 \ne a^2+b^2 in general.
  2. 2
    Precise statement: 'For real numbers aa and bb, (a+b)2=a2+2ab+b2(a+b)^2 = a^2+2ab+b^2, which equals a2+b2a^2+b^2 only when ab=0ab=0, i.e., when a=0a=0 or b=0b=0.'
  3. 3
    Counterexample demonstrating the point: a=1,b=2a=1, b=2: (1+2)2=9(1+2)^2=9 but 12+22=51^2+2^2=5. 9≠59 \ne 5.

Answer

(a+b)2=a2+2ab+b2≠a2+b2 in general (equality iff ab=0)(a+b)^2 = a^2+2ab+b^2 \ne a^2+b^2 \text{ in general (equality iff } ab=0\text{)}
Precise mathematical communication avoids vague words ('things') and quantifies clearly. Writing 'for all real numbers a,ba, b' and giving the exact expansion leaves no room for misinterpretation.

About Mathematical Communication

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

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