Why Not Define $0/0$ To Be $0$?

Question

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but don't know how convincing it is:

$1=\dfrac{1}{1}=\dfrac{1}{1}+0=\dfrac{1}{1}+\dfrac{0}{0}=\dfrac{1\times 0}{1\times 0}+\dfrac{0\times 1}{0\times 1}=\dfrac{0}{0}+\dfrac{0}{0}=0+0=0$, a contradiction.

Is there any better explanation why not to define $\dfrac{0}{0}$ to be $0$ (or any other number)?

Thanks.

[1] http://en.wikipedia.org/wiki/Division_by_zero#In_algebra

Answer 1

If you define $\frac00$ to be $0$, then you either have to abolish some other basic rules of arithmetic or accept the following argument: Since $3\times 0=0$, divide both sides by $0$, thereby cancelling the $0$ factor on the left and leaving $3=\frac00=0$. Neither alternative looks desirable to me.

Answer 2

Note: The below answer does not answer properly why we cannot define $\frac 0 0=0$ but only shows that it can be approached from different directions to get different limits. It is left on here as this is the answer I had been taught in school and is easy to understand but should be understood to be inadequate.

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from the right: $$\lim_{x\rightarrow0}\frac xy = 0$$ from the left: $$\lim_{x\rightarrow0}\frac xy = 0$$ if y is not 0. However it is also true that.

from the right: $$\lim_{y\rightarrow0}\frac xy = \infty$$ from the left: $$\lim_{y\rightarrow0}\frac xy = -\infty$$ if x is not 0

$$\lim_{y\rightarrow0}\frac yy = 1$$

These widely different answers means we can't define 0/0 easily.

Answer 3

You can define $0/0$ if you want, but that would be missing the point. One considers quotients as an operation. So what you want to be able is to have the multiplicative inverse of $0$, which can be easily seen to not make sense.

The idea of considering the fraction $p/q$ is to be able to think about it as $p\times1/q$. But, what arithmetic can you do with $0/0$?

Answer 4

With this welter of lengthy and diverse answers, you must be thoroughly confused by now; so I apologize for adding yet another answer to the pile:$$\dfrac00 \text{is meaningless}.$$That is all you need to know. If anyone challenges this, ask him then what it does mean, and why. You will easily be able to show that, using his own terms, a different meaning can be deduced.

Answer 5

Informally, $0/0$ is indeterminate, so it would be wrong to give it a value. It would be like o items out of 0 things. Is it none of the divisor or all of it, or maybe halfway? Since there's nothing to compare, all of these could be seen as valid viewpoints, so you cannot assign it a specific value.

Answer 6

Problematic expressions like $\frac{0}{0}$ (and many others) were considered in detail by Cauchy in his Cours d'Analyse. The term "indeterminate form" itself was coined by his would-be student Moigno.

To Cauchy what the expression $\frac{0}{0}$ meant was the process of determining the nature of the ratio of two infinitesimals, say $\alpha$ and $\beta$. Here the problem is that the ratio $\frac{\alpha}{\beta}$ can itself be infinitesimal, or infinite, or neither. For example, if $\beta=\alpha^2$ then the ratio is infinite. If $\beta=\sqrt{\alpha}$ then the ratio is infinitesimal. If $\beta=\alpha$ then the ratio is appreciable (not infinitesimal).

From this point of view it is clear that one cannot assign any definite value to $\frac{0}{0}$ without possessing additional information about the numerator and denominator.