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[parent] freshman's dream error (Example)

The name “freshman's dream theorem” comes from the fact that people who are unfamiliar with mathematics commonly make the error of distributing exponents over addition and/or subtraction, typically when working in fields of characteristic zero. An example is the equation $ (x+y)^2=x^2+y^2$ for $ x,y \in \mathbb{R}$. The equation is incorrect unless $ x=0$ or $ y=0$. By no means does the exponent need to be a natural number or an integer for this error to occur. An example of this is the equation $ \sqrt{x+y}=\sqrt{x}+\sqrt{y}$ for $ x,y \in \mathbb{R}$ with $ x \ge 0$ and $ y \ge 0$. This equation can be rewritten using the exponent $ \frac{1}{2}$, and again, the equation is incorrect unless $ x=0$ or $ y=0$.

An easy way to explain to someone who is under the impression that exponents distribute over addition and/or subtraction is to provide a simple counterexample. For instance, when $ x=3$ and $ y=4$, we have:

\begin{displaymath}\begin{array}{ccccccc} (x+y)^2 &=& (3+4)^2 &=& 7^2 &=& 49 \ \ x^2+y^2 &=& 3^2+4^2 &=& 9+16 &=& 25 \end{array}\end{displaymath}

On the other hand, the freshman's dream theorem yields some instances in which exponents can be distributed over addition and/or subtraction.



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Cross-references: freshman's dream, counterexample, integer, natural number, equation, characteristic, fields, subtraction, addition, exponents
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This is version 4 of freshman's dream error, born on 2006-07-29, modified 2007-05-30.
Object id is 8192, canonical name is FreshmansDreamError.
Accessed 2021 times total.

Classification:
AMS MSC: 97D70 (Mathematics education :: Education and instruction in mathematics :: Diagnosis, analysis and remediation of learning difficulties and student errors)
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