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单词 FreshmansDreamError
释义

freshman’s dream error


The name “freshman’s dream theoremMathworldPlanetmath” comes from the fact that people who are unfamiliar with mathematics commonly make the error of distributing exponents over additionPlanetmathPlanetmath and/or subtraction, typically when working in fields of characteristic zero. An example is the equation (x+y)2=x2+y2 for x,y. The equation is incorrect unless x=0 or y=0. By no means does the exponent need to be a natural numberMathworldPlanetmath or an integer for this error to occur. An example of this is the equation x+y=x+y for x,y with x0 and y0. This equation can be rewritten using the exponent 12, 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 counterexample. For instance, when x=3 and y=4, we have:

(x+y)2=(3+4)2=72=49x2+y2=32+42=9+16=25

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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更新时间:2025/5/4 22:53:41