Where's the foolish part ? Prove that 0/0 = 2

Question

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}

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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}$ ?

Answer 1

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.

Answer 2

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

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

Answer 3

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).