User:Barnaby dawson/zero divided by zero

In mathematics $$0/0$$ is an indeterminate form. This means that it is not a valid expression and has no determined value.

However if a function F is given by an expression in terms of x and that expression gives us $$0/0$$ for some value of x=X then there is often a continuous function that agrees with F everywhere and is defined at the point x=X also. For instance if you have the equation $$y=x/x$$ then it is defined everywhere apart from 0 then we can find a continuous function that agrees with F everywhere but is defined at 0 namely $$y=1$$.

A good method for finding such functions is to take the limit of the value of F as x approaches X. This will only work if the limit is unique (is the same when approaching from any direction).

Naive arguments to give this expression a value
There are several naive reasons which may be given for considering this expression to have some definite value:


 * Anything divided by itself is 1.     Hence $$0/0=1$$
 * Zero divided by anything is 0.       Hence $$0/0=0$$
 * Anything divided by zero is infinity. Hence $$0/0=\infty$$

Unfortunately the first two statements (in this form) are rules of thumb only. The third although often mentioned is technically incorrect. It should read that anything divided by zero is undefined. The first two rules correctly stated are:


 * Any none zero number divided by itself is 1.
 * Any number multiplied by zero is 0.

It should be noted that $$\infty$$ is not regarded as a number (this is why we don't need to qualify the second point with "any none zero".

There are other arguements used to 'prove'0/0' has a value. Most are based on mathematical fallacies.

Why not just give it a value and be done?
Sometimes in mathematics something has not been defined yet and may not make much obvious sence but yet can be usefuly defined. There are many examples of this in modern mathematics:


 * $$-1$$ was once considered to have no meaning "You can't have -1 cakes". However to aid computation it was found to be useful as a concept and few now doubt its existence as a mathematical entity.
 * $$\sqrt -1$$ was once considered to be undefined. Many people argued that it was just a imaginery construct to solve the cubic and shouldn't be considered real.  This is the origin of the terms imaginary and real.  However it was found that a whole new beautiful world of complex numbers opened up if you did allow them.
 * The logarithm function when used on the positive real numbers (naively) can be thought to be a fairly nice function. However if we consider complex numbers then the logarithm of any given number has an infinite number of values.  These values are fairly well behaved ($$log e^3=3+2i\pi n$$ where n is an integer) and so it is easy to deal with the complexities involved by either Riemann surfaces or by picking a certain branch (mathematics) of the logarithm function (choose one of the values arbitrarily and stick with it).
 * The factorial is a function represented by an explanation mark. We have $$1!=1,\ 2!=2*1,\ 3!=3*2*1$$ and $$n!=n*(n-1)!$$.  This was considered to only be meaningful with integers (what is (1/2)! ?).  However Legendre found a function called the gamma function which agrees with factorial at the integers but is also defined on the positive real numbers (it can be defined at all numbers bar the negative integers by analytic continuation).

So given these examples is it not possible that $$0/0$$ can be generalised?

Unfortunately not in any realistic way. Whereas in the previous examples the values an expression could take were limited in some reasonable way this is not the case with $$0/0$$. To show this we should first note that any reasonable definition would require that the following properties:


 * Any expression derived from a continuous function should evaluate in a way that keeps that function continuous.
 * A continuous function does not have any sudden jumps in its value (The function taking the value -1 if x 0 is not continuous).
 * If more than one value is necessary in different circumstances then those values should be limited in some way.
 * For the logarithm (of a real number) you could say that there is only one value which is real.

If we take $$y=c*(x/x)$$ for any value of c. We can easily see that this equation is the same as $$y=c$$. However for any value of c we could still get $$0/0$$ to make the equation continuous we must allow $$0/0$$ to take the value c. However this is true for any value of c.  Hence the values given are not limited in any realistic way. We cannot satisfy both properties above sensibly. This is why mathematicians decided to leave this expression undefined.