This has always bugged me, but I've come to accept it just like the rest of us. Division by zero is "undefined." You have to love math. Why is it that there CAN be an answer for multiplying by zero, but not dividing? Just HOW do you multiply by ZERO in the literal sense? It makes no more sense than DIVIDING by zero. If we can use such a wild notion as multiplying by zero to come up with fantastic science, why not dividing by zero as well? With all these great minds, can't someone come up with a definition that works? :blink:

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I had the impression that dividing zero by zero is undefined, but dividing a non-zero number by zero puts you in the realm of the transfinite.

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The reason why it is undefined is as follows:
Any value for x that you chose ( provided x is a real number ) will satisfy that equation. Therefore, when considering what value to assign "0/0" we are left only with the ability to chose a value arbitrarily. In other words, no value for 0/0 makes anymore sense than any other. This is what is referred to as "Indeterminate"
Now, consider:
In this case, we have a real difficulty, because for any finite x, 0*x = 0 which is ( for obviously important reasons ) not 3! So if you wanted to say that 3/0 was "10" ( for example ), you would be saying that 0*10 = 3. So clearly, 3/0 can't be any finite number.
Additionally, consider the function f(x) = 1/x, for |x| very small ( | | is absolute value, or magnitude ), in other words, x is very close to 0. If x is negative and close to 0, then f(x) is negative and very large ( in a magnitude sense ). If x is positive and close to 0, then f(x) is positive and very large. This would indicate that simply saying "1/0 is infinity" has its own problems, namely: it could be positive or negative infinity, and we have no reason to chose one over the other. For this reason ( namely, 0x = 3 having no solutions in the finite, and having serious difficulties in the transfinite ), we simply say "Division by 0 is undefined", which can be thought of as saying: "Any choice for it's value is just wrong".

It may interest you to know, however, that there are certain situations in which division by 0, and infinity become semi-acceptable to talk about, but unforunately, this is not the case with the normal situation ( such as working with "Real Numbers", for example )

Hope this clarifies the issue for you,

-joeboo

Antoniseb;

Is that in a textbook somewhere? It makes a little more sense, but still lacks complete definition. You can still multiply 0 by 0 with definition, but not divide 0 by 0. Is there any proof, logic or analogies that supports the transfinite definition? Could there be a geometric representation to this?

Some may find this to be a stupid question, but the real stupid questions are the ones that we DON'T ask! There could be some real implications to the relativity of space-time here for all we know. :blink:

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Great things can be accomplished by those who are willing to defy the belief that it can't be done.
~ 400poundgorilla

Yes, it does! What are those situations?

BTW Joeboo... thanks, it did help a bit to hear what I've been taught, and never fully understood. I knew that there was some rationalization for it in terms of variables, but either I really don't comprehend it, or some part of it goes against my grain of logic. Maybe I'm trying to make something 3-dimensional out of 2-dimentional math, I'm not sure, but it's still puzzling. As a fan of Cartesian geometry, I picture this definition of a point on the xy plane as having zero length or heigth, yet it can be an infinite line (positive OR negative) in the z axis. It's a relative way of looking at it that is of special interest to me. Somehow I wonder how the varible instances can be turned into 3 dimensions and then make sense of a completely different nature. Would the same math then still work? :blink:

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<span style='color:green'>&quot;The boldness of asking deep questions may require unforseen flexibility if we are to accept the answers&quot;
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Great things can be accomplished by those who are willing to defy the belief that it can't be done.
~ 400poundgorilla

In much the same way you can sqrt(-1) and get a non-real number it is used in math, actually it is commonly used in quantum physics.

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In the theory of equations, we have this nifty rule that in an equation in one variable of degree n (highest power of the variable is n), there are exactly n (not necessarily distinct) solutions. If you allow division by 0, this rule is completely trashed-- division of an equation by 0 reduces its information content to zilch-- nothing is solvable. The numbers we are most comfortable with possess a very comforting property-- that of unique factorization. Except for the order of the factors, all integers can be factored in exactly one way, and we use this property to develop general solutions to equations with integer coefficients. To tell it short, if division by 0 is allowed, each integer has an indefinitely large number of factorizations, and no unique solutions are possible to the equations.

Even without division by 0, there exist certain types of number whose factorization is not unique-- the people who study these algebras are very.. strange, and very busy. It makes number theory very difficult to prove anything with, and even makes a self-evident property like equality difficult to prove.

The devil we know is not very frightening. Steve

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Pick one.

If I recall correctly, there's also some neat stuff on limits, so that in some cases what 'looks like' 0/0 (for example) - in the limit - is well defined. In (at least some of) these cases 'division by 0' may 'make sense'.

More generally, 'math' is a formal system, so why certain things 'make sense' and others don't is entirely determined by the axioms of the piece of math you are working with.

Glad I could help explain.

First, the "division by zero" issue. I perhaps should have clarified this point. Never is an actual division by zero appropriate. However, there are situations in which we can nearly do so. This is closely related to the notion of "Limits". If you've had a calculus course, you're undoubtedly familiar with them. In the event you haven't, I'll provide a very simplified ( and not-quite correct ) description. Think of a limit as an action which analyzes the behaviour of a function ( or sequence ) as you get "close" to a point. Consider the functions f(x) = sin( 1/x ) and g(x) = x*f(x) = x*sin( 1/x ) for small x ( when I say small, I mean small in magnitude ). As x gets really close to 0, 1/x begins to "cover the real line" at a very rapid pace. In fact, 1/x covers all numbers larger than 1 "at the same rate" as y = x covers the numbers between 0 and 1. To conceptualize the graph of f(x), think of the sine curve, and take the entire graph of sin(x) on the right side of the vertical line x = 1, and cram that 'backwards' into the space between x=0 and x=1. This is precisely what f(x)=sin( 1/x ) looks like near 0. From this, it is apparent that f(x) oscillates back and forth from -1 to 1 infinitely often between 0 and any positive number. This is reinforcement for the fact that 1/0 is bad mojo, it makes seemingly normal functions behave VERY badly. However, consider g(x) = x*f(x). As x gets close to 0, g(x) will still oscillate infinitely many times, but now, instead of going between -1 and 1, it will oscillate between -x and x. Therefore, as x gets smaller, so does g(x). This is not the case with f(x). Essentially, what we are saying here is:
Note that in neither the case of f(x) or g(x) do we consider the value at x = 0 to have any meaning. In fact, we simply say "f and g are not defined at 0". However, while f doesn't seem to follow any regular behavior close to zero, g(x) is getting really small. And this is what a limit is. As x approaches 0, g(x) also gets close to 0. Therefore, we'd say: "The limit as x goes to 0 of g(x) is 0"
So we didn't divide by 0 ( in the sin(1/x) ) but in the case of g(x), we were able to consider the way the function behaves at 0 ( I italicize at because the precise concept requires a bit more detail than is necessary here ).
There are also other situations where 'division by zero' seemingly pops up. This is in a field called complex analysis where there are special classes of functions defined on regions of the complex plane. The functions have some very nice properties, namely, the ability to evaluate these functions by looking at the singularities of other functions ( a singularity is a place at which a function is not defined, ie 1/z has a singularity at z = 0 ). Specifically, if f(z) is one of these special functions ( called holomorphic ), then if I know the value of f(z) on every point of a circle in the complex plane, and f(z) is defined everywhere in that circle ( accurately speaking, I should say "Holomorphic in the circle", but that's not very helpful ), then I can determine the value of f(p) for any p inside the circle simply by looking at the function g(z) = f(z)/(z-p) on that circle! Notice how g(z) isn't even defined at z = p, but somehow, it tells me everything I need to know about f at the point p. So while actually dividing by zero is bad, looking at cases where you're almost dividing by zero can be very very good.

Now, regarding infinity. Again, limits pop up. First, a caution: Whenever someone says "infinity", unless it's very clear what they are talking about, you should seek clarification. Infinity is a term that gets thrown around a lot, but has very different meanings in very different circumstances. As an example of this, we'd say that the size of the rational numbers are infinite, and the size of the real numbers is infinite. But they are not the same size!! The reals are, in a way, infinitely bigger than than the rationals. Additionally, in an example I'm about to give, you'll see me use the terms "negative infinity", "positive infinity" and "infinity" ( -inf, +inf, and inf for short ), but they will all represent different things, some you might be familiar with. In any case, if you take to heart anything I've written, remember this: Infinity is, unfortunately, an overused and underdefined term, so make sure you know what "infinity" someone refers to when they use it. Anyway ....
Consider the real numbers, aka "The Real Line", 1 Dimensional Euclidean Space, or R.
When one says "positive infinity", with regards to R, one usually means: The limit of f(x)=x as x increases without bound", or in other words, something that is greater than any real number. When someone says "negative infinity" with regard to R, one usually means: "The limit of f(x) as x decreases without bound", or something that is less than any real number. It is crucial at this point to point at that +inf and -inf are NOT part of the reals, they are instead an abstract concept which is used mostly for notational purposes. They allow us to say things like, "The limit as x goes to -inf of e^x is 0" However, there is a notion of infinity which possesses an important value when combined with the real numbers.
Consider the function f(x) = 1/x^2. As x gets close to 0, whether we approach from the positive or negative side of 0, f(x) gets bigger and bigger and bigger. In fact, it increases without bound. Similarly, g(x) = -1/x^2 as x approaches 0, g(x) gets more and more negative, decreasing without bound. Both f and g behave very nicely for every real number with the exception of x = 0. As x approaches 1, g(x) approaches g(1) = -1. As x approaches 10, f(x) approaches f(10) = 0.01. This is a good thing ( namely, we call this "continuous" ). However, near 0 things seem to break down. This is where "infinity" comes in. Thru a process called "compactification" ( compactification means "to make compact" and compact is a property of a (topological) space which basically describes how tightly packed a space is ). Since the real numbers have no greatest or least element ( recall +inf and -inf not in R ), it's not 'compact', we can't really grab all the real numbers at once ( I'm abusing the language a great deal to avoid describing the concept of compactness in more detail ) . However, if we include a point "At Infinity", then we can grab a bunch of numbers around, say 0, and a bunch of numbers "Around Infinity" and we'll get all the reals! This is compactification.
To better visualize what is happening: Imagine if you will , the real line drawn vertically in the plane ( say the line x = 1. ) Additionally, imagine the unit circle with the point (-1,0) removed. Now, try to envision the line being wrapped towards the circle, but shrinking as it bends, such that when you're done wrapping it:
This is essentially the inverse of a Stereographic Projection. Now, simply add 1 point to the plane, at (-1,0) and call it "Infinity". That's essentially what's going on; you're gluing the ends of the real numbers to a point outside the real numbers. So, we have the "Compactification of the Reals", which is commonly known as "The ( projectively ) extended real number system", and denoted by R*. So why do we care? Because now, in R*, functions like f(x) = 1/x^2 and g(x) = -1/x^2, while not defined at 0, no longer behave badly at 0, because they approach "Infinity". Typically, what one might do in this case is define a compound function such as, "f(x) = 1/x^2 for x not 0, and f(0) = 'Infinity' " ( note this is in R* not R ). You might be thinking: "but -1/x^2 goes to negative infinity!", but this is not the case in R*, and this goes back to what I was saying earlier about infinity. However, if you wanted to, you could compactify the reals by adding the points "-Inf" and "+Inf" to the "Left and Right ends of the Real Line". This also is compactification of the real line, and it is called "The (Affinely) Extended Real Numbers" and is denoted R-bar.

Anyway, I'm sure this is way more than you wanted to know, and I apologize for the simplistic manner in which I tried to present everything, but brevity was the goal. When accuracy was sacrificed -and it was painful to do so- , I tried my best to put down a note indicating the exception. If you have any more questions, I'd love to try and answer them, but I'll need to be more brief next time around )

-joeboo

ps - I happened to notice some questions you posted in this thread regarding the center of the universe, and this provides a good light to view it in. Go back to that business with the circle and the line in the plane. Remember when you had wrapped the line around the circle, prior to adding in "Infinity"? Well, in an effort to explain why there is no center, consider the following: You can see the entire circle, but remember it's really the real line. Even though 0 is conveniently in between the positive and negative reals, realze that -1000 is just as conveniently placed in between all numbers less than -10000 and all numbers greater than -10000. Now, when you consider this on the "punctured circle", you see that even though 0 looks like a good place for a center, as soon as you "Compactify the reals" by adding the Infinity in, it becomes even more apparent that the decision to call 0 the center is arbitrary. Also, you can't consider the point (0,0) the center, because there is no such analogue in the real numbers. The fact that you see it at all is because we "embedded the reals into the plane" ( or in more general terms, we embedded the space into a higher dimensional space ). Finally, even tho you can see the size of the punctured circle, ( and measure it's circumference from a higher dimension ), it is still infinite. If you don't see this, or want to know why this "embedding nonsense" is important ( or what it even means ) I'll try to explain it. In any case ... it's time to sleep =/

( edit: typo )

I don't find multiplication by zero to be wild notation at all. If you sell bags of potatoes for five Euros each, and you sell zero bags today, what were your revenues?

That's multiplication by zero. Now let's see if we can divide by it. Suppose you sold zero bags of potatoes, and your revenues were zero Euros. From this information, can you divide zero by zero to calculate what price you must be charging?

Yes, I think that is the perfect way to put it. Real examples often make things easier to understand. Or for example, if you have zero cakes, how much does it cost? Obviously zero. Now, if you want to cut a cake into zero pieces, how big will each piece be?

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No.

Division by zero is just plain old undefined. It doesn't matter what is in the numerator.

"Transfinite" generally refers to either a means of proof based on well-ordering of arbitrary sets (transfinite induction) or else to the cardinal or ordinal numbers. Even in the setting of cardinal and ordinal numbers you cannot divide by zero.

About the closes that you can come is to look at something like Y/X and consider the limit as X tends to zero. That may have meaning. It may grow without bound, in which case it is sometimes said to tend to infinity. But the actualy expression Y/0 is meaningless.

The problem is that 1/x has the property that x * 1/x = 1. That is what division actually means. And there is no number that is a multiplicative inverse for 0. That is because 0*x = (0 + 0)*x = 0*x + 0*x and subtracting 0*x from both sides you get 0*x=0. So there is nothing that you can multiply times 0 and get 1.

Yes, but all of the mathematics that most people have ever seen, including most physicists is based on the basic axioms that permit ordinary arithmetic. That is the Zermelo Fraenkel plus choice system, or what is often seen as the Peano Axioms. http://en.wikipedia.org/wiki/Zermelo...kel_set_theory

They do not permit division by zero under any circumstances.

It is possible to talk about things like Y/X and take limits in a meanigful way in some circumstances in which the limits of Y and X separately tend to zero. That is case for instance with (sin x)/x which tends to 1 as x tends to 0, but that is not division by zero. You probably dealt with this one using L'Hopital's rule in an introductory calculus class.

Mathematics is just not that arbtrary.

Indeed 0/0 is undefined for good reason.

Something like: Lim x->0 x is 0, Lim x->0 xy is also 0, so Lim x->0 xy/x (apparently 0/0) is y, so 0/0 could (apparently) be any value at all.

Does that make sense?

Thread necromancy alert -- 4+ years.

No. Because you have made zero, you could have charged any price at all for a sack of potatoes. Any price would give you the same amount of revenue.

Fred

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If it's at all helpful at this point..

Origin of universe is humanly incomprehensible

One should be able to see it arithmetically right away as

1: 1 = 1
1: 0.1 = 10
1: 0.01 = 100
1: 0.001 = 1000

So, the lesser the denominator the greater is the result of the operation. You approach to zero and the result tends to infinity. And infinity is undefined, right? Simple.

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Infinity is the right-hand limit of 1/x as x->0+. Infinity is practically useful, and therefore better defined than n/0. If you were to say, "The left-hand limit is -infinity, the right-hand limit is infinity, therefore the value at zero is undefined", then I would understand what you mean.

No. Infinity is not undefined. It is not a real number. There are in fact different sizes of infinity -- see the theory of cardinal and ordinal numbers.

But 1/0 is simply undefined. It is not infinity. It is not anything. If 1/0 were infinity, what would 2/0 be ?

The point is that division has very clear meaning, and it implies the existence of a multiplicative inverse for whatever numbers (real numbers) are involved. Infinity is not one of them.

This is really not a debatable issue. That is how mathematics works.

You can DEFINE 1/0 to be infinity if you like, but you will have discarded the rest of mathematics if you do that.

In the theory of measure and integration is is often conveneint to work with the extended real numbers, which are the reals with +/- infinity "tacked on", and rules for addition and multiplication about as you would expect but even there 1/0 remains undefined, as is 1/(infinity).

My bold, and quite so. A math teacher in high school summed it up for me in nonmathematical language by saying that you could create a mathematics that includes dividing by zero, but it won't be useful for anything.

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Sorry, I didn't catch that.

That's my point

I wouldn't say infinity is undefined. You just can't expect it to behave like the numbers do. It wasn't brought up properly.

I've certainly seen applications for such systems, but you have to recognize, if you allow this sort of operation, there is a price to pay.

I've seen various totally rigourous systems of mathematics that include conventions about what 0 times infinity is, or things like that. You just have to recognize that infinity can't be manipulated like a number, and make sure the additional rules you introduce to manipulate infinity are consistent with themselves and the existing rules.

Zero times infinity is perfectly OK within the usual cardinal and ordinal number systems. It is zero. It doesn't matter which "infinity" you are talking about (and yes, in the cardinal and ordinal systems there are different sizes of infinity and it makes perfect sense). This is absolutely consistent with the Zermelo Fraenkel, plus choice, axioms for set theory. There is no problem whatever with this. You can see a simple explanatin of the cardinals and ordinals in Halmos's little book Naive Set Theory.

It has NOTHING to do with 1/0, which is simply undefined and meaningless.

This is common usage in, for example, measure theory. It is not universal.

As per my post, I agree with that.

Of course it does, capitalization notwithstanding. If you define infinity one way and apply one set of rules to it, this constrains by logical consistency your ability to apply other sets of rules to it.

It is defined in the Riemann sphere, where zero times infinity is undefined.

Actually, cardinal and ordinal numbers have little to do with the extended real numbers which are commonly used in measure theory. The use of the extended reals in measure theory is nothing more and nothing less than a notational convenience.


Sorry, but that is simply incorrect. Cardinal and ordinal arithmetic is quite logical, and it permits the operations of addition and multiplication. It has absolutely nothing to do with any sort of definition of 1/0.



Yes, you can make that convention, and it works topologically. But the "point at infinity" in the Riemann sphere is not an infinite cardinal number or anything other than the point added to form the one-point-compactification of the plane. It has nothing to do with the usual notion of infinity as applied to the definition of infinite sets, for instance. The Riemann sphere is a useful manifold, but it is not a number system. This is a rather bogus example, since it is based on notation and nothing more. As I said earlier, you can DEFINE 1/0 to be infinity. In the case of the Riemann sphere that is exactly what is done, but in doing that you lose other things and it does not carry over to other situations. The Riemann sphere is not a field, it is not a ring, it is just a comples manifold.

This is a bogus example, and brings no light to the topic being discussed.

There is one situation in which you can divide, not by zero, but by "infinitely small numbers"--the field of surreal numbers (it's on Wikipedia) of John H. Conway (the Game of Life guy). It's pretty...surreal in that there is no set of all surreal numbers, but it includes all real numbers, every cardinal number including the infinite ones, something much like the ordinals (but using different rules of arithmetic), and lots more different kinds of infinity, and their reciprocals which are infinitesimals. Still, division by zero in the surreal field is forbidden.

A modification of this (some guy name Robinson I think) leads to Freshman calculus, with all the tricks of dividing by infintessimals to get derivatives--tricks invented by Newton--for which he was criticized because, though the methods just happened to produce the right answers, were not very rigorous. Weierstrass invented the modern definition of the limit to make it rigorous, and Robinson figured out how to make Newton's manipulation of infintessimals rigorous, showing why it worked (i.e. it didn't just accidentally work, there was something to it).

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The guy that you are thinking of is Abraham Robinson who wrote the book Non-standard Analysis (Princeton University Press). He uses ultrafilters to construct a system in which there are infinitesimals, and can develop calculus using that methodology. this is a recognized branch of analysis, but there are relatively few specialists. We had one in the department when I was in graduate school.

Paul Halmos was known for opposing non-standard analysis, but as I recall he ran into trouble when a theorem relating to invariant subspaces of operators on Hilbert spaces was proved using non-standard methods and he was unable to construct a standard proof.

And division by zero is still undefined in Robinson's hyperreals.

Many extensions of the reals can be constructed, of varying interest and usefulness, but if they remain fields, then they have a zero, division by which is undefined.

completely and utterly correct

The same observation applies to rings, where some elements may be invertible, but zero never is.

6/2 = 3 since 3 X 2 = 6

If 6/0 = b, then 0 X b = 6

This cannot be true since 0 X b = 0 for any value of b. Secondly, 6 cannot have an infinite number of whole number factors that one can divide 6 by and get zero for an answer. Thus, division by zero is undefined.