FreshmansDreamError 2006-07-29 08:14:31 2007-05-30 04:56:22 Example freshman's dream % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \PMlinkescapetext{simple} counterexample. For instance, when $x=3$ and $y=4$, we have: \begin{center} $\begin{array}{ccccccc} (x+y)^2 &=& (3+4)^2 &=& 7^2 &=& 49 \\ \\ x^2+y^2 &=& 3^2+4^2 &=& 9+16 &=& 25 \end{array}$ \end{center} On the other hand, the freshman's dream theorem yields some instances in which exponents can be distributed over addition and/or subtraction.