Divide by zero

I know there's enough of you math munchkins out there to address this. While I do a lot of grunts in the trenches math every day, I don't frequent the frontiers. So I must humbly ask the answer to this question.

What if, instead of division by zero being undefined ("we won't go there") it was defined as equalling infinity?

I'm not claiming some revelation or anything, I just want one of you who can speak to the subject to let me understand.

Our regular math (i.e., quantifiable small numbers) wouldn't change. Carpenters would still be able to get a set of stairs put up as before.

What cosmological cataclysm would this change in assuumptions entail?

Maybe not equal to infinity, but why not equal to zero? If X*0=0, then why does not X/0=0? Is it because X*Y can never equal X/Y, assuming the same value for X and Y in each equation?

------------------
The overwhelming majority of people have more than the average (mean) number of legs. -- E. Grebenik

First, you might want to know how division is defined in the first place. Roughly speaking, we define division by its relation to multiplication: division is the inverse (or opposite) of multiplication. For example, the reason we say that 10/5 = 2 is BECAUSE 2*5 = 10...multiplication is the "main event", and division simply undoes multiplication. Now, apply this to division by zero. Suppose we say that 2/0 = infinity. Fine, then by defintion, we are saying that 0 * infinity = 2. Now, suppose we say that 3/0 = infinity...uh oh, 0 * infinity = 3! Which is it? Hence, the term "undefined"--it can't be defined in a way which is consistent for all problems. (The same argument holds for divsion by zero not equaling 0 either: 2/0 = 0 means that 0*0 =2).
What we do have to express the idea that dividing by something very small should result in something very large is the concept of a limit (remember that from freshman calc?). So, we can say that as the denominator of a fraction APPROACHES zero, the value approaches infinity. So, we do "go there" -- we just never reach where we're going!

(oh, and hi! i've been lurking here for a couple weeks now waiting for a good math question to jump in on :) )

The reason, I think, is pretty straight forward. If you're going to divide, you have to divide by SOMETHING. Zero, by definition, is nothing. Therefore, division by zero is not actually division.

------------------
"The day after tomorrow is the third day of the rest of your life." -George Carlin

Welcome on board, lynne. Gee, you made my answer look kinda cheesy.

Where's a good music topic when you need one.

------------------
"The day after tomorrow is the third day of the rest of your life." -George Carlin

Lynne, your definition is wonderful:
>>First, you might want to know how division is defined in the first place.<<

However, good as it is, can I offer a dumbed down one, from my second grade teacher?

If you have four apples and you divide them between two people, each gets two; if between three people, each gets one and one-third. If you divide four apples "among" one person, that person gets all the apples. If there are NO people to divide the apples among, you don't divide them at all, you put them back in the bowl, or whatever.

To summarize: when there are no people among whom to divide the apples, then you are NOT DOING DIVISION. If you say "What about the bowl?" That's dividing by one-- one bowl.

x/0 just isn't possible.



------------------
------------
--Rowan
Shopping is still cheaper than therapy. --my Aunt Franny

Nice job Lynne. I jumped in here with a ready answer, only to find that you had beat me to it. There's getting to be too many bright people here!

The problem is even deeper than that. It can be proven that if x/0 is defined as being equal to anything at all -- you can even call it "foo", and try to treat it as an extension to the number system of the same kind as imaginaries -- then math breaks down altogether, and 1=2. The only way for math to work is to declare that x/0 is always illegal.

------------------
John W. Kennedy
"Compact is becoming contract; man only earns and pays."
-- Charles Williams

Things are even more interesting when you get into calculus. Using L'Hopital's rule, you can get equations that look like: 0/0 = 3.

The result can really be any number; it depends on the value of 0. (Don't even get me started on dividing infinity by infinity...)

Lynne's answer was great. I'd just add one little thing. When you do problems where you are mathematically modeling the real world, and division by zero comes up, it actually means (almost always) that your model doesn't work, i.e there is something about the behavior of reality that forbids the situation that gave you division by zero. Saying that division by zero "is forbidden" cannot be defined away without losing this important aspect of mathematics, even if you could put it on a logical footing. Which you can't for the reason Lynne showed.

This is an important conceptual distinction between 1/0 and lim(x->0) of 1/x. The former gives you problems, the latter need not.

>>(Don't even get me started on dividing infinity by infinity...)

That's an easy one, isn't it? Any number divided by itself is 1.

Infinity/Infinity = 1.

I'm a math god.

Actually, Athena, that isn't necessarily the case. Based on L'Hopital's rule, infinity/infinity could be 1, could be something else entirely. Actually, the more proper way to phrase that would be that as the numerator and denominator approach infinity, the answer could approach 1, or it could approach some other value. L'Hopital's rule concerns itself with 3 scenarios that I remember off the top of my head, 10 years after Calculus II (someone fill me in if I miss anything). They are:

(N/D) --> ? as N --> 0, D --> 0
(N/D) --> ? as N --> infinity, D --> infinity
1^X --> ? as X --> infinity

In each of the above cases, N, D, and X may be variables or expressions. Someone who's a little more fresh on this, please correct me if necessary (as if I have to tell anyone here to do THAT!).

Great, now we're going to have to use a message board forum to teach people how to take derivatives.

------------------
"I wept because I had no shoes, then I met a man with no feet. So I took his shoes" - Dave Barry

Wouldn't you be a math godess, Athena?

Actually, L'hopital's rules aside, there are different sizes of infinity. Counterintuitive, I know, but check out G. Cantor's set theoretic notions of transfinite numbers. Kind of neat, and completely useless, as far as I know.

Strainger, you're right on target with L'Hopital's rule. The two basic forms are

(N/D) --> ? as N --> 0, D --> 0
(N/D) --> ? as N --> infinity, D --> infinity

and there are a few other forms (like that 1^x) that can be rearranged into the two basic ones (hey, anybody want a refresher in logarithmic differentiation? yechh.)


That whole idea of approaching (as in limits) is absolutely critical in both theory and application. Infinity is a way to symbolize a very useful mathematical concept (the idea of growing without bounds), but it isn't a number, and you have to be very careful to not act like it is(trying to cancel infinities and so on)...or at least, I do, since I actually use/teach this stuff...the rest of you needn't stay awake nights worrying that you've improperly used infinity! (More than likely, infinity will NOT come seeking revenge and/or monetary compensation).

How's this for a tag line?
"The good Christian should beware of mathematicians and all those who make
empty prophecies. The danger already exists that mathematicians have made
a covenant with the devil to darken the spirit and confine man in the
bonds of Hell." -- Saint Augustine

In defense of St. Augustine, the "mathematicians" he was talking about were bozos similar to the current "Bible code" bozos.

------------------
John W. Kennedy
"Compact is becoming contract; man only earns and pays."
-- Charles Williams

Hey, hey, hey! Y'all should just shut up about this L'Hopital's rule and limits and stuff. Don't you know that calculus is useless in the real world????? :)

Seems to me the confusion is that people equate 'zero' with a thing. It's no-thing or nothing. You know, like nothing there.

Thanks to lynne, et al; sometimes this board works the way it's envisioned - I got an answer that works. I like handy's comment. With a math that quantifies things between zero and infinity (check the monkeys thread), zero is when we're fresh out of quantities.

Regards

Athena: I said not to start on the infinity thing. Now we've got math geeks all over the place, and you know how hard it is to get them out of the upholstery. =B^)

AuraSeer

Like that. We're embedded now.

Is it because X*Y can never equal X/Y, assuming the same value for X and Y in each equation? What if X=1 and Y=1, doesn't X*Y=X/Y

Or, for that matter, X=anything and Y=1.

I just did a quick look through the posts, and here's my theory... I'm not a mathematician, so don't get annoyed or anything.

The reciprocal of infinity has to be 0 because one over an infinitely large number would be so small that it would equal 0, just as the repeating decimal .999 would be equal to 1.
If the reciprocal of infinity is 0, wouldn't it follow that the reciprocal of 0 is infinity?

------------------
"[He] beat his fist down upon the table and hurt his hand and became so
further enraged... that he beat his fist down upon the table even harder and
hurt his hand some more." -- Joseph Heller's Catch-22

The reciprocal of infinity is actually a limit, and not a discrete computation. The reciprocal of zero is , he he, undefined. Sorry, you can't get around this. Ahhh, what pleasure it must be to be undefined? But, anyway, there are limits where the denominator approaches zero, these expressions go to infinity.

By the way, I'd like to say that Lynne has just a great way of explaining this stuff. I wish I had a teacher like that.



------------------
¾È ³ç, ÁÖ µ¿ ÀÏ

Oh yeah, as for the x*y can never equal x/y:

Setting x*y = x/y gives us

y^2 = 1.

y = 1 or -1 and x can be any real number.

I'll get out of the upholstery now. :)

------------------
¾È ³ç, ÁÖ µ¿ ÀÏ

Sorry, one more thing. As for Calculus being useless, I agree. But then, so is the Mona Lisa. :)



------------------
¾È ³ç, ÁÖ µ¿ ÀÏ

<< The reciprocal of infinity has to be 0 because one over an infinitely
large number would be so small that it would equal 0, just as the
repeating decimal .999 would be equal to 1. >>

I dunno how many times some people need to be told something before it penetrates that thick skull. If you look at a sequence of numbers, 1/N, where N gets larger and larger and larger, you can see that the fraction 1/N gets smaller and smaller and smaller, ever closer to 0... approaching zero as a limit, but NEVER equaling zero, for the reasons lynne gave.

And zero is not simply "no-thing." Zero is the additive identity under the group operation of addition in the number system. Sheeesh.

WiseOldMan said:
What if X=1 and Y=1, doesn't X*Y=X/Y

What about this:

If x=2 and y=2 then


x + y = x * y = xy


Deep, eh?

This might be a bit late, but:
L'hospital rule is not applicable when either the N or D or N' or D' (the derivative) is undefined, ie, 0 or infinity.
Math is a rigid subject. It NEVER fails. If it does (as in division by zero), disallow the opertaion and you are fine.

Veera, would you care to explain L'Hopital's rule to the audience please? I think I may just have to s#!tcan my Calculus textbook, along with all the other ones around the office.

I hope I will get it right THIS time:
Assume you have N(x)/D(x), where N and D are dependent on x.
If you want N(a)/D(a), where a is a constant, we just calculate N(a) and D(a). BUT if N(a) and D(a) is singular, the ratio has no meaning.
If all you want is just the limit of the ratio as x tends to a, the L'Hospital rule says the limit is N'(a)/D'(a). If N'(a) or D'(a) is again singular, oops!!
Was that enuf? Basically, division by zero is undefined because its nonsense.
Now, someone please reply to my mints and impotency query. That has been playing on my mind more that /0!!!

Strainger, you don't need to shitcan the calc book; you had it right. I think what veera1's getting at is that L'hopital's rule doesn't apply when either the numerator or the denominator but not both tend toward zero or infinity. For example, we can't (and don't) apply it in the case of say 3/x as x goes to zero, so it can't be used to answer the question "what's 3 over 0"? It does apply to something like 3x/x as x approaches zero, which is one of the forms (0/0) you stated. Also, if you should happen to get N'/D', and still have both heading towards 0/0 or inf/inf, you can apply L'Hopital's rule again.

(Not sure what provoked the resurrectuion of this topic, but what the hell...)

I was just being sarcastic about shitcanning the Calculus book, Lynne. Veera just sounded like he was way out in left field and I chose to be a smart-ass about it. I do remember L'Hopital's rule pretty well. Occasionally, you run into an instance of N(x)/D(x) -> 0/0 or infinity/infinty no matter how many derivatives you take, in which case, IIRC, you call it "undefined" and move on to the next homework problem.

By the way Veera, given the number of mints you eat per day, I'd say that you are probably the best experiment regarding the mints/impotency question. If your Mr. Happy still works right, I'd say your friend was B.S.ing you and that mints do not cause impotency. Seeing Dennis Franz's butt on NYPD blue on the other hand ...

(my apologies for answering this in the wrong thread)

...but MUCH more entertaining...perhaps we should rephrase this as "does attempting to divide by zero cause impotency?" I claim only if you do it excessively.

Lynne, so now I know what those Calculus teachers were trying to do to me! Academic saltpeter! Farging bastages.

When you divie a number by 1, you get the number. If you divide the number by .1, you get a higher number (10 times higher), if you divide it by .01, the result is even higher, and so on. The closer you get to 0, the closer you get to infinite...so why doesn't x/0=infinite?
Well, here's what I think...try the same thing, only divide by negative numbers. The closer you get to 0, the lower the number gets, and the closer it gets to "negative infinite" (Okay, so there is no such thing as negative infinite, but there isn't such a thing as positive infinite either.)
Since 0 is neither positive nor negative, then we can say that x/0 = x/-0. If x/0 = infinite, then x/0 would also equal negative infinite, which would make it impossible to do.
There is one exception, though. It is possible to divide by zero in one condition. 0/0. Zero divided by itself has the solution of all real numbers, because for any number, if you multiply it by zero again, the answer is zero!
I have another point that is kinda irrelevant, but still wierd to think about. What is infinite divided by infinite? The first answer that comes to your mind is 1, but is that the answer? If you take any positive number and multiply it by infinite, the answer is infinite, so would the answer to infinite / infinite be any real number greater than 0? Geh. I've got a headache now.

0/0 and infinity/infinity are also forbidden, and for the same reason. If they are allowed, then you can prove that 1=2.

0*infinity and several other cases are forbidden, as well.

------------------
John W. Kennedy
"Compact is becoming contract; man only earns and pays."
-- Charles Williams

Oh, for the love of God, people. Please refer to the posts by AuraSeer posted 06-10-99 02:30 PM, me posted 06-10-99 04:30 PM , and lynne posted 06-10-99 05:19 PM. If you're really interested in more details about Messr. L'Hopital's wonderful rule, I can detail it even further.