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.