This is the first in a series of “false proofs.” Despite their falsity, they will be part of the Proof Gallery. The reason for putting them there is that often times a false proof gives insight into the nature of the problem domain. We will be careful to choose problems which do so.
Problem: Show 1 = 2.
“Solution”: Let 
. Then 
, and 
. Factoring gives us 
. Canceling both sides, we have 
, but remember that 
, so 
. Since 
is nonzero, we may divide both sides to obtain 
, as desired.
Explanation: This statement, had we actually proved it, would imply that all numbers are equal, since subtracting 1 from both sides gives 
and hence 
for all real numbers 
. Obviously this is ridiculous.
Digging into the algebraic mess, we see that the division by 
is invalid, because 
and hence 
.
Division by zero, although meaningless, is nevertheless interesting to think about. Much advanced mathematics deals with it on a very deep and fundamental level, either by extending the number system to include such values as 
(which still gives rise to other problems, such as 
and 
), or by sidestepping the problem by inventing “pseudo” operations (linear algebra) and limiting calculations (calculus).
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This reminds me of a video by Minute Physics: http://www.youtube.com/watch?v=kIq5CZlg8Rg
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Par for the course as far as mathematics in physics goes 🙂 Especially when it comes to sequences and series.
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My favorite appendix from Charles Seife’s Zero: The Biography of a Dangerous Idea is his proof that Winston Churchill is a carrot. His hilarious and otherwise irrefutable logic is based on this false proof. Love it.
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Ha! This gave me a good laugh. Here’s a link for everyone else: http://mcs.mines.edu/Research/k12-partnership/students/christerkarlsson/Documents/Churchill.pdf
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