Is Division by Zero Really Impossible?

[esalkin]

x/0 = ?

When I was in grade school, learning about multiplication and division, the teacher told us that you can't divide by zero. When I asked her why not she looked stumped (I don't think math was her best subject ). She thought a second then replied, with that old standby of adult/child communications, "because that is the rule."

I have since learned the reasons behind "the rule"; however; I still ask the question: "why not." Since x*0 = 0 it stands to reason that x/0 has an answer also. (light has no problem with x/0.) The fact that our limited math skills cannot provide an adequate answer does not mean that it does not exist. Think of all the mathematical questions that were impossible to answer before the Calculus was discovered. The day that someone can solve for the equation x/0 is the the same day we figure out "Everything."

 

[selfAdjoint]

You say that x times 0 = 0 has an answer, but what in particular is that answer? Is it 5? well 5 solves the equation because 5 X 0 = 0 all right. Is it π? Yes π solves the equation. In fact every number solves the equation. This is the basic property of zero: if a X n = a for every n then a = 0. Now try to evaluate 1/0, look for a numbeer n such that n X 0 = 1. But we just showed that for every number n, n X 0 = 0. So there isn't any number that is equal to 1/0, or 2/0 or anything divided by zero. That's why we don't use division by zero, because there isn't any numerical val;ue it could have..

 

[quartodeciman]

Karl Weierstrass in the nineteenth century added -∞ and +∞ to the standard open set of real numbers in an attempt to close it. -∞ was supposed to be -1/0 and +∞ was supposed to be +1/0. But things broke down after that. For instance, -∞ + 1 = -∞ and +∞ + 1 = ∞. So certain common postulates for real numbers can't be applied when these new elements are involved, and combinations like +∞ + -∞ become unmanageable. So these so-called extended real numbers retain only a symbolic value of tendency and can't really be treated like bonafide real numbers.

Lots of people try to justify division by 0 in a variety of ways, but the energy spent on it might better be spent finding ways to avoid doing this at all. Introduction of infinitesimal and infinite numbers in a consistent way might help. Two schemes for doing this are nonstandard real hypernumbers and surreal numbers. These schemes and their variations are almost becoming respectable, but are not widely used yet. In a sense, it is better to master number systems without things like infinitesimals and infinites, then take them them up later. But I don't know what the future may bring.

 

[phoenixthoth]

why is this here under "theory of everything" and not in the general math part?

 

[esalkin]

Originally posted by selfAdjoint
That's why we don't use division by zero, because there isn't any numerical value it could have..

That is exactly my point. Our current math can't handle it. Think of all the 'unsolvable' problems that could be solved if (or when) we discover that math than can handle it.

Originally posted by phoenixthoth
why is this here under "theory of everything" and not in the general math part?

Yes, the 'how-to' of division-by-zero should be discussed in the math forums. My point here is that if we assume that x/0 gives a value or set of values, then 'everything' falls into place. Of course, if a Neanderthal had assumed that 'E=something' it would not have done him any good but then again he just might gone on to discover mass and the speed of light.

 

[selfAdjoint]

I'll move the thread to general math.

 

[Hurkyl]

Division by zero is undefined in ordinary arithmetic because its cons greatly outweigh its benefits. In general the concept simply isn't useful.

When it is useful, then we use it, in rigorously defined ways. One of my favorite examples arises when studying inversions in geometry; we add to the plane a point at infinity, to which an inversion sends a point when we have to divide by 0. This is great for studying inversions because it neatly wraps up a lot of minor details. (For a more arithmetic example, the mobius functions from complex analysis are very closely related to geometric inversions)

However, as mentioned, the point at infinity is, in general, not useful. For instance, with the point at infinity, it is no longer true that "Through any two distinct points there exists a unique line", we can no longer define what it means for a point on a line to be between two other points on a line, and we cannot always compute the length of a line segment.


The point is, the "normal" mathematical constructs are extremely nice things. Once you start breaking them (such as by defining division by zero), they start to lose their nice properties. In general, the loss greatly outweights the benefit, so such ideas are only useful in particular situations.

 

[hey.like]

Not complete the zero

It isn't complete accept about zero.naturely it is rule. some time. the rule isn't complete , but to use is good enough. If the interest enough about zero, its means are very enough. but in some idle time please!

 

[uart]

Originally posted by esalkin
light has no problem with x/0.

[?] Hmmm you might have to explain that one, I have no idea what you are talking about with light there.



Ok if you want to have some sort of definition of division by zero then a candidate could be something like $$x/0 = \pm \infty$$ for all finite non zero x. But note the $$+\infty$$ does not neccessarily correspond to +ive values of x and visa versa. So for example $$5/0$$ would still be $$\pm \infty$$

Now with such as wide margin of error as $$\pm \infty$$, even for a given fixed x like 5 then do you still really think it makes sense to have division by zero ?

BTW. Take a look at the graph of $$y=1/x$$ right at the origin and tell me what value or range of values it takes there.