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'freshman's dream error'
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| Title of object: |
freshman's dream error |
| Canonical Name: |
FreshmansDreamError |
| Type: |
Example |
| Created on: |
2006-07-29 08:14:31 |
| Modified on: |
2006-07-29 08:14:31 |
| Classification: |
msc:97D70 |
Preamble:
% 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
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Content:
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 that are not 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 $y \ge 0$, which can be rewritten using the exponent $\displaystyle \frac{1}{2}$. 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. |
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