Mathematical Communication Math Example 3

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Example 3

easy
Identify what is wrong with the following mathematical communication and rewrite it correctly: 'It's clear that x2≥0x^2 \ge 0.'

Solution

  1. 1
    Problem 1: 'It's clear' is not mathematical justification — it asserts without reasoning and may alienate readers who don't find it obvious.
  2. 2
    Problem 2: The domain of xx is not stated.
  3. 3
    Rewrite: 'For any real number xx, x2≥0x^2 \ge 0. This follows because x2=x⋅xx^2 = x \cdot x is a product of xx with itself; when x≥0x \ge 0, the product is non-negative; when x<0x < 0, the product of two negative numbers is positive.'

Answer

∀x∈R,  x2≥0 (with justification, not ’it’s clear’)\forall x \in \mathbb{R},\;x^2 \ge 0 \text{ (with justification, not 'it's clear')}
Good mathematical communication never says 'obviously' or 'it's clear' as a substitute for reasoning. Every claim should be justified, and the domain of each variable should be stated explicitly.

About Mathematical Communication

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

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Example 2 medium

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Example 4 medium

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