freshman’s dream error

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 $x\\geq 0$ and $y\\geq 0$ . 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 counterexample. For instance, when $x=3$ and $y=4$ , we have:

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

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

单词	FreshmansDreamError
释义	freshman’s dream error 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 $x\\geq 0$ and $y\\geq 0$ . 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 counterexample. For instance, when $x=3$ and $y=4$ , we have: $\\begin{array}[]{ccccccc}(x+y)^{2}&=&(3+4)^{2}&=&7^{2}&=&49\\\\\\\\x^{2}+y^{2}&=&3^{2}+4^{2}&=&9+16&=&25\\end{array}$ 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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