## Friday, 11 December 2009

### If Zero by Zero were defined....

If zero by zero were defined a whole lot of crazy things would start to happen.
Just as an example I quote this section of Wikipedia.

With the following assumptions:
\begin{align} 0\times 1 &= 0 \\ 0\times 2 &= 0. \end{align}
The following must be true:
$0\times 1 = 0\times 2.\,$
Dividing by zero gives:
$\textstyle \frac{0}{0}\times 1 = \frac{0}{0}\times 2.$
Simplified, yields:
$1 = 2.\,$
The fallacy is the implicit assumption that dividing by 0 is a legitimate operation with 0/0 = 1.
Although most people would probably recognize the above "proof" as fallacious, the same argument can be presented in a way that makes it harder to spot the error. For example, consider the following equations:
\begin{align} (x-x)x &= 0 = x^2-x^2 \\ (x-x)x &= (x-x)(x+x) \end{align}
Dividing by x − x gives:
$x = x+x\,$
and dividing by x gives:
$1 = 2.\,$
Therefore it would  be possible to prove that 1 = 2 and 2=3. In fact it would be within our capability to prove that any number in the number system is equal to any other number. If this were true then everything would be exactly the the same.It is therefore obvious that division by zero is undefined. If zero by zero were defined all the buildings and bridges would tumble and crumble. All calculations would be rendered useless. You wouldn't be able to tell the difference between an elephant and an ant. If zero by zero were defined, there would be no difference between you and me, no difference between big and small, no difference between hot and cold. The universe would be astoundingly uniform. There would be no stars or galaxies or beautiful nebulae. No black holes, no Quasars, no interstellar dust. The universe would be just one big boring entity.