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:


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

Title freshman’s dream error
Canonical name FreshmansDreamError
Date of creation 2013-03-22 16:07:23
Last modified on 2013-03-22 16:07:23
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Example
Classification msc 97D70