Error Analysis Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
A student writes: 'a2+b2=a+b\sqrt{a^2+b^2} = a+b.' Identify the error and find a numerical counterexample.

Solution

  1. 1
    Identify the error: the student incorrectly 'distributed' the square root over addition. x+y≠x+y\sqrt{x+y} \ne \sqrt{x}+\sqrt{y} in general.
  2. 2
    Counterexample: a=3,b=4a=3, b=4. LHS: 9+16=25=5\sqrt{9+16}=\sqrt{25}=5. RHS: 3+4=73+4=7. 5≠75 \ne 7.
  3. 3
    Correct statement: a2+b2≤∣a∣+∣b∣\sqrt{a^2+b^2} \le |a|+|b| (triangle inequality), with equality only when a=0a=0 or b=0b=0.

Answer

a2+b2≠a+b in general; counterexample: a=3,b=4 gives 5≠7\sqrt{a^2+b^2} \ne a+b \text{ in general; counterexample: } a=3, b=4\text{ gives } 5 \ne 7
This error (distributing roots over sums) is one of the most common algebraic mistakes. Error analysis — identifying what went wrong and why — is essential for building correct mathematical habits.

About Error Analysis

The systematic study of how errors arise in calculations or models, how large they are, and how they propagate through subsequent steps.

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More Error Analysis Examples

Example 2 medium

A student 'proves': '[formula] for all [formula]' by checking [formula]. Identify the error in this

Example 3 easy

Find the error: a student solves [formula] by writing [formula], then [formula]. What went wrong?

Example 4 medium

A student cancels incorrectly: [formula]. Identify and correct the error.

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