Some of the commenters to yesterday's post raised some interesting questions on the subject of dividing by zero. So interesting, in fact, that I felt the subject deserved another post.

My SciBling, revere, of Effect Measure: writes the following:

OK, I shouldn't jump in here because I'm an epidemiologist and not a mathematician, but, what the hell. All I can do is be wrong (which I am used to).Some algebraists do permit division by zero, but only in the case 0/0. Thus, Rotman in Advanced Modern Algebra, Revised Printing, p. 121, has this definition:

Def.: Let a and b be elements of a commutative ring R, Then a divides b in R (or a is a divisor of b or b is a multiple of a), denoted a|b, if there exists an element c in R with b = c*a.

As an extreme example, if 0|a, then a= 0*b for some b in R. Since 0*b=0, however, we must have a=0. Thus, 0|a iff a=0.

I haven't read the piece in question, so this isn't a comment on whether Anderson's use makes sense or not. But the real line is certainly a commutative ring; in fact it's an integral domain (and of course a field).

The reason I say this is definitional is that I believe most mathematicians would say that the divisor in an integral domain, D, must be in D/{0}, whereas Rotman isn't making this restriction, and perhaps he is unusual. This doesn't affect the cancellation requirement for a domain, I don't think, since that still requires the common factor to be non-zero.

There is a saying, show me a nitpicker at age 5 and I'll show you an epidemiologist at age 35. And I'm a lot older than that.

For readers who may not be up on their abstract algebra, allow me to clarify a few terms.

Think about the integers; positive, negative and zero. There are certain sorts of arithmetic operations that are permissible on the integers and certain other ones that are not. For example, we can add two integers to obtain another integer, or we can multiply two integers to obtain another integer. We also have an additive identity element, namely 0. In other words, there is an integer with the property that when it is added to any integer, it has no effect (that is x+0=x for any integer x). There is also a multiplicative identity, namely 1. That is, x times 1=x for any integer x. Furthermore, addition and multiplication are both commutative operations, meaning that x+y = y+x and xy = yx. They also satisfy the associative property, which I don't feel like defining right now.

What else? Well, addition and multiplication interact via the distributive property. This says that a(x+y) = ax + ay.

But there is one important sense in which addition and multiplication are different. With addition, every integer has an inverse. That means that for any integer x, there is another integer y with the property that x+y = 0. For example, if x=5, then y=-5. As a result, we have another operation, called subtraction, that can be viewed as the inverse operation to addition.

Multiplication doesn't have that. If x is an integer other than 1 or -1, you will not be able to find an integer y with the property that xy = 1. Consequently, division is not a well defined operation on the integers. In some cases you can divide. You can say that 30 divided by 10 is 3, for example. But most of the time the result of dividing one integer by another is something that is not an integer.

We make use of many of these properties when we solve standard algebraic equations in middle school or high school. Consider the equation:

x

^{2}-5x+6 = 0.

In elementary algebra we learn to solve such equations by factoring them:

(x-2)(x-3) = 0.

Factoring in this way is only possible because the integers come equipped with the distributive property. The next step is to argue that since we are multiplying together two integers and obtaining zero as a result, it must be true that one of the factors is equal to zero. This leads us to the equations x-2 = 0 and x-3 = 0, which have the solutions x=2 and x=3.

Sets that are like the integers in the sense of having two commutative and associative operations, equipped with identity elements, and with the property that the product of two non-zero things is always non-zero is referred to as an “integral domain”. Literally, a domain that is like the integers. If we remove the assumption about non-zero things, we are left with a “commutative ring with identity”. If we do not require that multiplication have an identity element that we have a “commutative ring.” And if we don't require that multiplication be commutative we have, simply, “a ring”

Another example of an integral domain would be the collection of all polynomials with integer coefficients. You can add and multiply polynomials, and they satisfy all the familiar rules of algebra.

On the other hand, consider the set of all two by two matrices with integer entries. Such matrices can be added and multiplied at will, but matrix multiplication is not commutative. In general, if A and B are two matrices then AB is not the same matrix as BA.

Even worse, it is possible to multiply two non-zero matrices together and end up with the zero matrix (that is, the matrix all of whose entries are zero). So if you are trying to solve a quadratic equation with matrix coefficients, ye olde factoring will not work.

If you remember some linear algebra, however, you will recall that there is a multiplicative identity for matrix multiplication. We can summarize the foregoing by saying that the set of all two by two matrices with integer entries are a “ring with identity.”

Part of a typical abstract algebra class is devoted to studying many different sorts of objects that come equipped with binary operations, and trying to decide precisely what properties they satisfy. You see, the attitude in abstract algebra is that it doesn't matter what sorts of objects the symbols actually represent. (That's the “abstract” part). All that matters is the properties they possess. (That's the “algebra” part).

Now let's return to revere's comment. He quotes Rotman defining the notion of one element dividing another. He is working in an arbitrary commutative ring, meaning that multiplication is commutative but does not necessarily have an identity and may have “zero-divisors” (meaning that the product of two non-zero things might be zero.) He then says that a divides b if there is another ring element c with the property that ab = c.

This conforms to our usual notions of divisibility. For example, we say that 6 divides 30 because there is another integer, namely 5, with the property that 6 x 5 = 30.

Revere points out that this seems to leave open the possibility of dividing by 0. Rotman's definition allows us to say that 0 divides 0, since if we let x represent any other ring element we have that 0 = x0. Doesn't this imply that it is meaningful to talk about 0 divided by 0?

No, it does not.

To see why, return for a moment to subtraction. When working in the integers, we are accustomed to thinking of subtraction as an operation separate from addition. There's addition on the one hand, and subtraction on the other.

But that's not really true. Subtraction is defined entirely in terms of addition. When we write x-y we mean “add to x the additive inverse of y.&rdquo. In the integers, for example, we can say that 7-5 = 7+(-5). In an arbitrary ring, it is assumed that every element has an additive inverse. “Subtract b from a” is then a short hand way of saying “determine the additive inverse of b and add it to a.”

Likewise for multiplication and division. Division is not an operation separate from multiplication. Rather, division is defined entirely in terms of multiplication. When we write something like x/y, that is shorthand for “multiply x by the multiplicative inverse of y.” If y is not the sort of thing that has a multiplicative inverse, then it is mere gibberish to write x/y. (Just like if the only numbers you know about are the positive integers, then it is gibberish to talk about something like 2-5.)

In particular, the binary operation “division” is only meaningful in an environment where we have multiplicative inverses. In the integers, for example, most elements do not have multiplicative inverses. In an arbitrary commutative ring there is no assumption that an arbitrary element has a multiplicative inverse. Even worse, in an arbitrary commutative ring there is no assumption that there is a multiplicative identity element, and without such an element it is meaningless to talk about multiplicative inverses in the first place.

To sum up, we need to distinguish two different phrases. The first is the phrase “a divides b.&rdquo. The second is “divide a by b.” These are two different statements. The first phrase is meaningful in any ring, and is defined entirely in terms of multiplication. The second refers to a particular binary operation that is not defined for a general ring.

So it is perfectly correct to say that 0 divides 0. But it is not correct to divide by zero. As it happens, many books on number theory and abstract algebra define “x divides y” in such a way that x is not allowed to be 0, but this is not strictly necessary. No harm is done by leaving open the possibility that x = 0.

But let's go a little farther, In high school algebra we generally do not limit ourselves to the integers when we are solving equations. Typically we allow the rational numbers as well. By a rational number we mean one that is expressible as the ratio of two integers. In the rational numbers, every element (except zero) has a multiplicative inverse. The integers themselves sit inside the rational numbers. Consequently, in this enlarged universe the nonzero integers do have multiplicative inverses. For example, we can say that the multiplicative inverse of 5 is 1/5, because 5 x 1/5 = 1. The fraction 1/5 is a rational number, but it is not an integer.

If we start from an integral domain and add the assumption that every nonzero element has a multiplicative inverse, the resulting object is called a “field.” The set of rational numbers are a field, as are the real numbers and the complex numbers. It can be proved that it is possible to imbed any integral domain within some field. In other words, you can simply make up a bunch of symbols that serve as multiplicative inverses for the elements of your integral domain, and include them in your set in such a way that you do not mess up any of the other important algebraic properties.

So let us suppose we are working in a field. Now it is meaningful to write a/b. It means multiply a by the multiplicative inverse of b. As long as b is the sort of thing that has a multiplicative inverse, we have a meaningful expression.

But here's the catch. Even in a field, the element zero does not have a multiplicative inverse. Look at the definition of a field in any abstract algebra textbook, and you will find that a field is a commutative ring with identity in which every nonzero element has a multiplicative inverse. (As an aside, this definition implies that the product of nonzero things is always nonzero. Proving that is a typical homework exercise in abstract algebra classes.)

And the reason 0 does not have a multiplicative inverse is precisely the one I mentioned in yesterday's post. For 0 to have a multiplicative inverse, we would have to be able to solve the equation 0x = 1. That's simply what the term “multiplicative inverse” means. The reason 5 and 1/5 are multiplicative inverses, for example, is that 5 x (1/5) = 1. But no matter what field you are working in, it is always true that 0x = 0.

So there you go. Zero does not have a multiplicative inverse. In any ring. Period. Consequently, when you write the expression x/0, you are writing gibberish.

Of course, people write big books on ring theory, and there's plenty more to say. But I think I will stop here for now. There were many other interesting comments to yesterday's post, but I will save them for a future blog entry.

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Jason wrote:This depends somewhat on the convention one adopts about the status of the single element of the trivial ring, { 0 }. Some texts (e.g. my ancient copy of MacLane and Birkhoff's

Algebra) consider this to be a ring with a multiplicative identity, in which case 0 is its own multiplicative inverse in that ring. Other texts require the multiplicative identity (if it exists) to be different from 0. Presumably such texts would consider the 0 of the trivial ring as having no multiplicative inverse. According to Wikipedia there are also texts which define the term "ring" in a way which excludes the trivial ring.Jason wrote:JK wrote:Indeed. I would disagree with Jason here. I don't think his condition for "x/y" to be meaningful is sufficiently general. If x and y are elements of a commutative ring (e.g. the integers) and the equation y × z = x has a unique solution for z in that ring, then it seems perfectly reasonable to me to use the expression "x/y" to denote that solution, regardless of whether y has a multiplicative inverse or not. I am reasonably sure I have seen this convention used by other professional mathematicians.

revere wrote:This isn't quite right. You get the real projective line by adding only a

single"point at infinity" to the ordinary real line. If you addboth+∞ and -∞, considered as twodistinctelements, to the real line, you get the extended real number line , which is a different object.Jason wrote:This depends somewhat on the convention one adopts about the status of the single element of the trivial ring, { 0 }. Some texts (e.g. my ancient copy of MacLane and Birkhoff's

Algebra) consider this to be a ring with a multiplicative identity, in which case 0 is its own multiplicative inverse in that ring. Other texts require the multiplicative identity (if it exists) to be different from 0. Presumably such texts would consider the 0 of the trivial ring as having no multiplicative inverse. According to Wikipedia there are also texts which define the term "ring" in a way which excludes the trivial ring.Jason wrote:JK wrote:Indeed. I would disagree with Jason here. I don't think his condition for "x/y" to be meaningful is sufficiently general. If x and y are elements of a commutative ring (e.g. the integers) and the equation y × z = x has a unique solution for z in that ring, then it seems perfectly reasonable to me to use the expression "x/y" to denote that solution, regardless of whether y has a multiplicative inverse or not. I am reasonably sure I have seen this convention used by other professional mathematicians.

revere wrote:This isn't quite right. You get the real projective line by adding only a

single"point at infinity" to the ordinary real line. If you addboth+∞ and -∞, considered as twodistinctelements, to the real line, you get the extended real number line , which is a different object.Oops. It looks like the post which my previous one was intended to qualify did not end up being posted anyway. Here it is.

Jason wrote:This depends somewhat on the convention one adopts about the status of the single element of the trivial ring, { 0 }. Some texts (e.g. my ancient copy of MacLane and Birkhoff's

Algebra) consider this to be a ring with a multiplicative identity, in which case 0 is its own multiplicative inverse in that ring. Other texts require the multiplicative identity (if it exists) to be different from 0. Presumably such texts would consider the 0 of the trivial ring as having no multiplicative inverse. According to Wikipedia there are also texts which define the term "ring" in a way which excludes the trivial ring.Jason wrote:JK wrote:Indeed. I would disagree with Jason here. I don't think his condition for "x/y" to be meaningful is sufficiently general. If x and y are elements of a commutative ring (e.g. the integers) and the equation y × z = x has a unique solution for z in that ring, then it seems perfectly reasonable to me to use the expression "x/y" to denote that solution, regardless of whether y has a multiplicative inverse or not. I am reasonably sure I have seen this convention used by other professional mathematicians.

revere wrote:This isn't quite right. You get the real projective line by adding only a

single"point at infinity" to the ordinary real line. If you addboth+∞ and -∞, considered as twodistinctelements, to the real line, you get the extended real number line , which is a different object.Good explanation (minor typo where you say ab=c; you mean ac=b). To sum up, it is OK to say 0 divides 0 in a commutative ring (like the integers) but not that x divided by zero is meaningful in the field that embeds the integers.

Of course this would mean that according to the posts you and MarkCC put up, the integers are not numbers because they are "not on the real line." The integers that are spaced along the real line are real numbers, not the integers. They are the image of an injective mapping from the integers, the image being isomorphic to the integers but not identical to them (except by an abuse of terminology). On the other hand if they are "numbers," then you do have numbers where you can divide by zero, the numbers being the integral domain of the integers. There just aren't any units except 1 and -1, which makes the integers different than the rationals or real, but does it make them not numbers? For that matter, are the rationals "not numbers" either because they aren't on the real line?

Then there are my comments about plus and minus infinity which also aren't on the real line but an element in real projective space which is isomorphic to the numbers on the circumference of a circle. Are they not numbers? And the most troublesome example, of course, are the things we actually call numbers, complex numbers, most of which are also not on the real line and aren't a linear order, another of MarkCCs criteria for being a number.

I realize this is far afield of the intent of the original post. Sorry to be such a pain in the ass. But it is much more interesting than epidemiology.

'In particular, the binary operation division is only meaningful in an environment where we have multiplicative inverses. In the integers, for example, most elements do not have multiplicative inverses.'

I understand you are only explaining standard terminological conventions. It's fair enough for mathematicians to make use of standard definitions or else communicating about algebra would be even trickier than it already is.

But how important are such definitions in everyday conversation? Non-mathematicians are likely to become confused when you tell them that you are not allowed to teach school children that 30/6=5 without first teaching them about rational numbers and that, in fact, while it is perfectly acceptable to say that 6 divides 30 to write 30/6=5 is mere gibberish.

It's a tough business popularising science!

I think revere raises some interesting questions about what is a "number". But an interesting answer would not consist of an assertion about the list of criteria that mathematicians have handed down (for those who don't know, there is such no list).

Rather, it would be interesting to know how and WHY mathematicians have come to generalise and refine everyday concepts of number, making particular choices sometimes in different ways. It could be done from historical, logical, or practical (applications) points of view any of which could be starting point for popularising what mathematicians have learnt about this stuff.

from Revere:

" if we let x represent any other ring element we have that 0 = x0."

So x can be 1, or 7, or ... any number. And therefore by transitivity 1 = 7 = any number.

BAD result!

Therefore 0/0 MUST have no meaning.

Is this not a valid argument?

Karl: In a general ring, division isn't an operation because there is no guarantee that an element has an inverse (see Jason's post above for a good explanation of what that means).

revere-

Thanks for pointing out the typo.

I'm planning to a separate blog entry on the question of what a number actually is. You're right that I was a bit casual in my phrasing in my original post on this subject. My comment that infinity was not on the number line was meant simply as a response to Dr. Anderson. In the video, when he draws his number line, he labels two specific points as plus infinity and minus infinity. I was merely pointing out that it is incorrect to do so. I wasn't trying to lay out a definition of what is and is not a number.

And definitely don't apologize for harping on the details. Mathematicians get very excited when anyone shows an interest in their subject!

I wrote:On second thoughts I think JK and I have probably misinterpreted what Jason wrote. He appears to have chosen his words a little more carefully than I read them. He

doesn'tsay that "x/y" is gibberish if y doesn't have a multiplicative inverse, which is how I misread his above quoted statement.There is a standard construction in ring theory, called localization, by which a ring can be embedded in a larger one where some elements of the original ring without multiplicative inverses do have them in the larger one. When Jason wrote "If y is not the sort of thing that has a multiplicative inverse, ..." I suspect he had in mind the sort of element which

cannothave a multiplicative inverseeveninanylarger ring.My apologies for the misunderstanding.

Dude. you love math.

Why is this so complicated? Wouldn't any number divided by 0 be infinity? I'm not a math expert but wouldn't 0 go into any number an infinite amount of times?

Perhaps I'm missing something here... But hey, i'm a lamen. Kinda like ramen, but I dont have much of a noodle.

You say "And the reason 0 does not have a multiplicative inverse is precisely the one I mentioned in yesterday's post. For 0 to have a multiplicative inverse, we would have to be able to solve the equation 0x = 1."

Whereas from the actual paper, it states "Having nullity lie off the number line blocks the counter-proofs from real analysis that attempt to show that is undefined. In transreal arithmetic division is defined via the reciprocal, not via the multiplicative inverse. The reciprocal contains the multiplicative inverse as a proper subset, but also defines the reciprocal of zero. Hence, transreal arithmetic has an algebraic structure that contains a field as a proper subset, but which does define division by zero."

In other words, the multiplicative inverse isn't even needed. To show this, he expresses division as (a/b)^-1 = b/a, not a x a^-1 = 1. He goes on to state that 0 x 0^-1 = 0/0, therefore an multiplactive inverse doesn't exist, but the argument is that one isn't NEEDED in order to divide by zero. We all know that we can get the same answers with different methods, and different answers with different methods and they can all be valid. I'll leave it at that.

If Dr. Anderson can write 1/0 as infinity and -1/0 as minus infinity, then its not that bad to come up with 0/0 as 'nullity'. Guys don't waste your time and move on.

Thanks for this, its the first real explanation for division of 0 that makes sense.

I'm not a mathemetician at all (just took undergrad calc courses). This topic does touch upon one of my interests though. Please indulge me a little as I deviate from established mathematical formalism in which I haven't really been trained. I'll warn that I'm going to spout some pseudo math nonsense... think of it more as a concept to ponder then a mathematical proposition.

my understanding is there are "limits" such that:

x/y as y->0 = infinity

and

x/y as y->infinity = 0

It's my understanding that the use of limits are imposed because division by zero or multiplication by infinity blow the equation outside the finite domain and the finite terms are no longer recoverable through commutative or associative properties. So algebraic laws can no longer perform any useful work and sloppy algebra in this fashion can produce arbitrarily illogical statements.

First, I would assert that we generally do not understand the concept of zero as much as we tend to think we do as humans. We only ever encounter it as a finite clause. We restrict a set to a group of finite elements and then declare a zero property conditionally. (ie. there are Zero pink elephants in the room I am in right now).

"Nothingness" outside the constraints of a finite set is just as elusive to the human mind as the concept of Infinity.

OK. So here's my "math-like" comments that aren't meant to be mathematical propositions but that I would like comments on by mathematicians never-the-less:

0/0 = infinity/infinity = infinity * zero

There are no finite terms in that statement to be non-recoverable through interaction with the transfinite numbers. It's not generally a useful equality anyway, but I suspect that it is true. They don't really evaluate to anything. Well... I play with the idea that they have to evaluate to a finite term, but never any specific one.

I thought of this in a 400 level philo class on metaphysics. Which is why it looks not mathematically formal.

I reasoned that if physics is correct and all of reality is a field interaction of finite differentials, then the question of "where does reality come from" is analogous to "where do numbers come from". All finite numbers are derived from 1, + , and the characteristic of inversion or complements...

(I suspect that the concept of addition IS the concept fo 1)

1 is defined by it's relation to zero.

some-thing is NOT of no-thing

no-thing is NOT of some-thing

but

no-thing is also NOT every-thing

every-thing is also NOT no-thing

So... more pseudomath gibberish... I know it's painful to mathemeticians... my apologies... think of it as symbolism that uses the language of mathematics.

infinity * zero => f(1)

Infinite nothingness results in something. That was phenominal to me. Where does reality come from? Infinite Nothingness spawns the finite domain as it folds in upon itself.

The Brahman folds in upon itself and the largest concepts we can conceive of from within the cascading fractal is Brahma (infinity) and Siva (Zero) wrestling with one another and Visnu (1) maintaining the balance.

What's interesting symbolically is that Visnu's daughter was Maya (the illusion of reality), just like the daughter of the concept of 1 is all of the other finite numbers that derive themselves as functions of 1.

I think a lot of people are confused about infinity.

Inifinity is just a symbol mathematicians use to denote the following property on the extended real number line:

Lim f = INF if and only if

For all real numbers X, Lim f > X.

That's all (simplified, of course).

It's not actually a number, it's just a placeholder for an expression. And we see it has no meaning outside of this definition.

mark: Yes, sort of. Depends what you mean by "number." It is not a number on the real line, but there are other kinds of numbers: integers, rationals and complex

numbers, for example. So plus and minus infinity are a (single) number point on the real projective line. If you want to visualize that "number," it is at the top of the unit circle that sits with its bottom at the origin on the plane. All the other points of the real line are also "on" that circle, wrapped around it. Those points are not really "the same" points as on the real projective line, but they are topologically indistinguishable from them (ie., RP1 is homeomorphic to S1). This isn't different in my view from the integers or rationals "being on" the real line, so common usage allows this abuse of terminology most of the time.Anyway, that's my take, but I'm curious to know what a real mathematician would say about it (I just dabble).

On a much less sophisticated note:

I sometimes have to explain why you can't divide by zero to high school students who are not at all mathematically inclined.

I point out that one way of thinking about division is to understand it as recurring additions. So, 6/3 = 2 because if I add 3 to 0 twice I reach 6. Likewise, 10/2 = 5 because I can reach 10 by adding 2 to 0 five times.

So, on the simplest level of arithmetic, without recourse to advanced mathematics, students can understand the evaluation of a/b as the answer to the question "how many "b"s do I add to 0 in order to reach a?"

I then show them a problem like 3/0 and ask them to answer it using the above model. It quickly becomes intuitively obvious to them that the expression is meaningless, since no number of zeros will ever add up to 3.

Some of the brighter students notice that in the case of 0/0 any answer will work--which presents it's own problems.

These students I encourage to go on in math.

Ken: I like it.

http://www.sohbetci.com.tr

I think a lot of people are confused about infinity.

Inifinity is just a symbol mathematicians use to denote the following property on the extended real number line:

Lim f = INF if and only if

For all real numbers X, Lim f > X.

That's all (simplified, of course).

It's not actually a number, it's just a placeholder for an expression. And we see it has no meaning outside of this definition.