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I've been across this on the web:

\begin{align} \frac{0}{0} & = \frac{100-100}{100-100} \\ & = \frac{10^2-10^2}{10(10-10)} \\ & = \frac{(10-10)(10+10)}{(10)(10-10)} \\ & = \frac{(10+10)}{10} \\ & = \frac{2}{1} \\ & = 2 \end{align}

Of course this is wrong but I can't tell where exactly since there is no obvious (to me at least) atrocity... Is this in the first line $ \frac{0}{0} = \frac{100-100}{100-100}$ ?

Peter Smith
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Thomas Ayoub
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    It's cancelling $10-10$ (dividing above and below the line with $0$). – orion Aug 27 '15 at 13:01
  • By the way, you could just as well say $0/0=1$ (cancelling directly) or $0/0=(0\cdot 62389)/(0\cdot 32)=62389/32$. You are just masking the obvious mistake of division by zero, by calling your zero something else. – orion Aug 27 '15 at 13:02
  • It's also the very first expression: 0/0 doesn't mean anything, so you can't manipulate it. – Mankind Aug 27 '15 at 13:03
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    This is the "proof" why $0/0$ is undefined : if you rewrite it as $(100-100)/(50-50)$ wou will get $4$ ... – Mauro ALLEGRANZA Aug 27 '15 at 13:05

3 Answers3

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I'm going to rewrite this proof by dividing by all powers of ten and what not.

The proof essentially goes:

$$ \frac{0}{0}=\frac{2\cdot 0}{1\cdot 0}=\frac{2}{1}$$

The problem is in the first line, when you write $\frac{0}{0}$, which is undefined. Of course you could define it, but then it would be equal to every fraction since $$\frac{a}{b}=\frac{c}{d}$$ if $ad=bc$, and if $a=b=0$, then this is always true, since for any $c$ and $d$, $0\cdot d=0\cdot c$.

But then all fractions are equal to each other, so there is only really one fraction: $\frac{0}{0}$. This seems a lot less useful than the system we had before.

jgon
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You cannot divide any number by $0$! In abstract algebra there are constructed algebraic structures where things similar to "dividing by zero" can be performed (zero divisors).

A new branch of mathematics is the nonstandard analysis in which with numbers which are tending to $0$ can be performed any calculations (these are infinitesimal numbers; it is in the set of hyperreal numbers).

kryomaxim
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You cannot remove the factor $$ \frac {10-10}{10-10} $$

and assume that the rest is $0/0$.

johannesvalks
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