Have we 'discovered' how to divide by zero?

Factual Questions
21

Like anything else, you can construct number systems in which this has solutions.

Useful and interesting ones which are at least somewhat close in spirit to more conventional number systems, even. For example, the affinely extended real line or real projective line, in which the various infinities satisfy this equation. And, the straightforward co-inductive type of “natural numbers” in a programming language like Haskell would contain solutions to equations like this.

Or, you can go to Topologist route and take the trivial ring with one element equal to both 0 and 1. That element satisfies this equation as well, if thoroughly boringly.

22

Well, not necessarily. This proof might work perfectly well and one might just deny that y*(x/y) is always equal to y. (Why call it division then? I don’t know, but the sort of people who feel a compulsion to come up with some meaning for x/0 may want to do so). Or, as Topologist suggested, though you were no doubt implicitly excluding this scorched earth solution, you might just bite the ultimate bullet and agree that everything is equal to 0.

23

Ah, I see now that you were posting this as an equation which might motivate a desire to divide 1 by 0, rather than, as I had originally thought, posting it as an analogy to illustrate that some equations are better off left without invented solutions. Very well then; carry on.

24

I wouldn’t say it “breaks the rules of math”. You just have to create an entirely different system of math. A good analogy might be fictitious alternate history novels. You could, of course, create an internally consistent story based on the assumption that the Axis won WW2, and you wouldn’t have to break the rules of language to do so. But of course such a story would be merely a curiosity… you wouldn’t try to base present-day foreign policy on that fictitious history.

25

Well, if you insist that reciprocation should be continuous at zero, we might settle on a number to be the limit of this getting bigger and bigger. E.g., in the real projective line, division by zero is defined whenever the numerator is nonzero, and the answer is unsigned infinity (a single value which is the limit of both 0, 1, 2, … and 0, -1, -2, …). Of course, something normally expected breaks down, and, in this case, the problem is that multiplication isn’t always defined; specifically, 0*unsigned infinity is undefined. Also, the system lacks a linear ordering; at least, it can’t be made an ordered ring. But some people are happy enough with that; it seems to be a system which models some useful concepts, anyway.

26

Another example would be a number system where, instead of using a number line, you use a number circle with circumference 1 (one might call this “real numbers modulo 1”, though I don’t know if that terminology is used).

And further on Topologist’s further on me, there’s a common type of algebraic structure called a “group”, and it’s very handy if the operation you’re dealing with and the set you’re applying the operation to form a group. For example, the real numbers form a group under the familiar sort of addition operation. Well, a ring has two operations on it, one called “addition” and one called “multiplication” (which may or may not resemble the familiar operations by those names). The entire ring is a group under the addition operation. And if you find the additive identity (which has to exist, since that’s one of the properties which makes it a group) and remove it from the ring, then what’s left forms a group under the multiplication operation. The rational numbers, the real numbers, and the complex numbers are all rings that you may be familiar with, and they all work this way.

OK, so that’s addition and multiplication. What about subtraction and division? Well, for every member of a group, there’s another member (or maybe the same one) which is the inverse of that member (this is another property required of groups). To subtract a number, you instead add the inverse of the number, and to divide by a number, you multiply by the inverse of the number. But remember, we removed zero (the additive identity) from our set before we turned it into a group under multiplication, so zero wasn’t there when we were handing out inverses. You can’t divide by zero, since to do so you’d need to multiply by the inverse of zero, and zero doesn’t have an inverse, nor even a reason to have one.

27

Rings don’t necessarily have multiplicative inverses for nonzero elements; the natural numbers and the integers form rings too. I assume you meant “field”.

In which case,
Ah, alright. Although, in a ring/field, one does usually define multiplication by zero (with an outcome which must be zero). It’s a bit odd to look at something like -1a + 1a and say you can’t collect the terms, you know? But I see what you’re saying: the only reason we take the perspective that a number should have a multiplicative inverse is because things in the multiplicative group should have a multiplicative inverse. But the multiplicative group can (and will) be defined to exclude 0, so 0 needn’t have a multiplicative inverse.

Although, in a way, this is just a fancy way of saying “0 doesn’t get to have a multiplicative inverse, by arbitrary fiat”. Of course, it’s not arbitrary, but I’m not sure this sheds any light on it.

28

Floating Point maths on a computer returns NaN (Not a Number) for division by zero. Dr James Anderson, from the University of Reading’s computer science department has proposed *nullity * as a term for NaN. This would sit outside the conventional number line.

I think nullity is bollocks, frankly. Programmers should handle the divide by zero case before the program (and the airliner) crashes, and mathematicians don’t really need the concept.

Si

29

The guy is a nutcase. ‘nullity’ doesn’t solve anything - it’s not as if you can just plough on ahead and plug it into the next set of calculations - it’s just an unnecessary label for the same dead end.

30

So we move from

Divide by zero error - restart Universe

to

You have achieved Nullity - Reality makes no sense from this point on.

:wink:

Si

31

Quite - it’s like if we replace the word ‘broken’ with something else, we’ll be able to break things and still pretend they’re intact. Utter wibble.

32

On the other hand, if I have 5 pies and share them with my 1/2 friend (I don’t know him that well), he’ll get ten pies. If I then take five pies back from him, we have five pies each and everyone’s a winner! I could even do it again!

33

It should probably be noted that, as anyone who has studied Calculus should know, the concept of limits is a way around the “you can’t divide by 0” restriction in certain situations. A function like 8x/x (to take a really simple example) can’t be evaluated when x = 0 because that would involve division by 0, but it does have a limit as x approaches 0.

34

I wouldn’t quite put it like that. In this case, the limit is simply describing the behavior of a particular function as the numerator and denominator both approach zero. This is quite distinct from actually assigning a particular value to the discrete value zero divided by the discrete value zero. Nowhere in calculus do you assign a value to 0/0; you only recognize it as an indeterminate form requiring more investigation.

35

FWIW, this is how I always make it make sense to me when I loose track of why dividing by zero is a no-no. Except I just use 1 pie. And it’s a cheesecake, not a pie.

And the solution winds up with me eating it to console myself for not having any friends.

36

In the world of electrical engineering, specifically signals and systems, division by very, very small numbers is common, and results in what is known as an impulse. Most people relate to these as voltage or current spikes. An impulse is a signal with a very large amplitude over an extremely small time frame, and the power generated is the area of the impulse.

This is the closest I can come in my mind to a practical illustration of why division by 0 is officially undefined. You can get extremely large numbers that are not infinite, but get into the neighborhood.

The theoretical impulse is much cleaner than what can be generated in real life, BTW. Real life spikes have measureable time periods and amplitudes, and generate harmonics that are even easier to characterise.

37

OK, I’ll assume I meant “field”, too, then. Like I said, I’m not overly conversant with the terminology.

And the removal of the zero isn’t entirely arbitrary, since by just doing that, one ends up with a very elegant sort of mathematical structure, of the same sort as the mathematical structure one gets for addition. It seems a small price to pay.

38

IIRC floating point processors on a computer can return positive infinity for positive x divided by zero, and negative for negative, and they can also return positive zero for positive anything divided by positive infinity and so forth. They generate exceptions too, but it’s up to the programmer to decide whether to handle the exception as an error. This is perfectly handy, for example if you are an electrical engineer calculating the parallel resistance of several circuits one of which is open. This view of division by zero was standard in Gauss’ day, but more recent generations have backed away from such cavalier treatment.

Please, anybody point out if I don’t RC.

39

Okay. While aspects of this have been hinted at already, it’s time for me to incur the wrath of the math wonks by presenting what I once immodestly called Bullwinkle’s Axiom:

x / 0 = unsigned infinity

Simple, isn’t it? Here’s my explanation, such as it is:

What happens when you divide a certain number by an increasingly smaller dividend? The quotient gets larger. Consider:

4 / 0.2 = 20
4 / 0.02 = 200
4 / 0.002 = 2000

And so forth. Now it seems to me that if you carry this process to its logical extreme, you wind up having to conclude that 4 / 0.000… would be a really huge number; in fact, infinity.

To put it another way: zero and unsigned infinity are reciprocals.

About the term “unsigned infinity”: I use this because the result of division by zero could be either “negative infinity” or “positive infinity” (if you use negative dividends in the example above, you get negative quotients). I sometimes suspect that whoever declared division by zero impossible was simply freaked out by the fact that it could produce two answers.

Another way to illustrate Bullwinkle’s Axiom:

Consider your basic Cartesian Graph (you know, the thing with the x and y axes). A horizontal line plotted on such a graph has zero slope. As a line tilts away from the horizonal its slope increases (positively or negatively, depending on which way you’re going). As your line gets closer and closer to vertical, the slope gets hugher and hugher. And when the line is vertical it is then said to have - no slope. Huh? Using my previous logic, one would think the thing would have infinite slope (crunch the numbers and you’ll see what I mean). The catch, of course, is that a vertical line describes both infinitely positive slope and infinitely negative slope - the same paradox that apparently led to division by zero being declared immoral, illegal and fattening.

Well, there you have it. x / 0 = unsigned infinity because zero and unsigned infinity are reciprocals of each other. Not the easiest thing to describe. All I can do is suggest you play with the numbers until you achieve enlightenment.

(One final stray thought: it seems appropiate that zero, which is neither positive nor negative, should have a reciprocal that is both.)
Now excuse me while I duck - I know what’s coming from the math wizzes. . . .

40

I’ll be damned. I haven’t seen any such thing, but it sure is nice to see some corroboration for “Bullwinkle’s Axiom”. (You posted while I was pecking out my missive, so I missed it.)