This is going to be a long post, but if you want the truth behind the divisibility by zero confusion, this thread should illuminate for you exactly the apparent problem. The question as it stands is
I use the single quotes on the above operators called 'multiplication' and 'addition' over the ring R because they could be called anything else instead like 'jackkniving' or 'imballabloming'. These two operators are abstract (binary) operators, which only as a special case, when we consider our ring R to be the real numbers, come to be realized as the familiar multiplication and division that everyone learns in high school. The only thing important about our operators are the eight properties above
We're getting very close to our penultimate goal here, which is to craft a mathematical structure on the set of real numbers that will allow us to answer questions like what 'dividing by zero' means . A few very interesting and important things I want to point out here, is that a zero (element) depends on the ring in question. A different ring, a different zero (element). Later when we identify the set of real numbers as a ring, because it is a ring, we will have to designate some real number as the zero (element) and some other real number as the identity (element). In preview of that, I'll tell you now that when we take our ring to be the set of real numbers, we are going to let '0' be the zero element and '1' be the identity element.
You may have noticed so far no ideas taken from calculus. This is because we don't need a theory of the infinitesimally small (nor infinitely big) to work with the real number 0. Calculus deals with the 'closeness' of elements, allowing you to analyze limiting behaviors and rates of changes. 0 certainly plays a role in those kinds of studies, and hence helps you to better understand what 0 is and how it behaves, but those ideas have no influence on how we define 0.
So far the only operations we have to work with are multiplication and addition; to define whatever 'divisibility' might be (over rings) we have to get in a few more definitions. So we'll do that now, but not in the way you might have expected. Do you remember from high school math that to divide by a number x is the same thing as to multiply by 1/x. Well, this shows that 'dividing' is the same thing operationally as multiplication, we need not introduce a new 'divide by' operator. This subtle difference is crucial to understanding the divide by zero conundrum. Starting with R as a ring, here a few more definitions we need to form a sufficient axiomatic theory that will be our backdrop for analyzing the idea of 'dividing by zero' (and perhaps other ideas):
So a field is just ring with a few additional properties; mainly that every element has an associated multiplicative inverse 1/x. Specifically a field is just a more specialized type of ring, but our statements will be dealing with fields and not the more general idea of rings (some of which are not fields).
You may be wondering at this point why we are not defining a 'division' operator /, so that y/x is just equal to some number r. The reason is that we want as few operators as possible defined on rings; if we defined a divide operator we would have 3 operators in total together with addition and multiplication. The goal in mathematics is to get as far as possible with as little as possible. We absolutely could have defined 3 operators over rings; multiplication, addition, and division if we wanted to, there is nothing stopping us. But this axiomatic system would be less desirable because there would be a greater number of axioms (well definitions, if you make the distinction between definition and axiom). And since we can mimic division by instead multiplying by suitably chosen numbers, we don't really need a whole new definition. So it is best to adopt the theory that just includes the two operators of multiplication and addition.
These are all the definitions that we need. Now we are ready to prove some theorems that will hopefully bring us closer to understanding 0:
Theorem1:
The set of real numbers with the usual multiplication and addition forms a field.
Proof: This is really easy to prove, just show that multiplication and addition that you learned in high school satisfy the requirements of the 'multiplication' and 'addition' operators over a ring. An element of this set is called a real number.
The concept of a field is a much more general notion then just the set of real numbers (without considering topological aspects). It is very easy to show that the real number line is a field (albeit with additional topological structure) with the familiar operations of additional and multiplication. But to make this thread interesting, I'm going to talk about the 'divisibility by zero' on general fields, of which the real numbers are just a special case.
This is all the machinery we need to address the enigmatic division by zero issue. This last theorem answers all questions of divisibility by zero which comes merely as a natural consequence of our definitions (this is how mathematics works), but it might not be the answer you were hoping for:
Theorem2:
For a given field F, any element x in this field is divisible by zero if and only if x is zero.
Proof:This proof is broken into two parts since it is an if and only if statement:
1)The if part; if an element x is divisible by 0, the assertion is that the element x is 0. Or, said in other words, if an element is divisible by zero, it must be that that number is itself 0. So suppose that zero divides x. This means that for 0 there exists an element r in F such that 0 * r = x. So in other words we have that for some element r in F, 0 * r = 0 = x. Coincidentally then, this equation also immediately implies x = 0.
2)The only if part; if x = 0 then the assertion is that x is divisible by zero ie 0 happens to be divisible by 0 according to all our definitions. So suppose x = 0. Then we need to find an element r in F such that 0 * r = x when x = 0. However, conveniently enough, any and all elements f in r satisfy this equation 0 * r = 0.
Notice above in both parts the zero element could be the element r. So we will make the informal definition that: if a pair of numbers (x,y) is such that y is divisible x, then know there exists an r such that x * r = y, and we will call this number r the result.
Conclusion: This definition of divisibility is pretty standard http://en.wikipedia.org/wiki/Divisibility but all the definitions I used (for fields, rings, 'multiplication', etc.) may vary slightly depending on what textbook you consult and who you talk to. The point however is consistency. If theorem 2 didn't do it for you, my remarks here should.
In all this discussion, I never defined what is meant by 'dividing'. I had only used multiplication and addition. Divisibility is a relationship between two numbers (not a binary operation), and any pair of numbers may or may not have this relationship. Part of what we have shown then is that the pair (0,0) has the relationship status of being divisible. So the question reformulated in the terms of this axiomatic system is:
I will answer this question. And no, infinity is nowhere to be found here. You don't need to know calculus but your going to need a brain to get through this if you have never seen it before, but this stuff is way to beautiful to be dumbed down to accommodate the lazy. First allow me to present my credentials; I'm an honors mathematics graduate working as a programmer. I have a large bookshelf populated with math and science books ranging from differential topology to computational linguistics (although ive only read maybe 1/4 of them). The reason I'm making this post is because I like math, and I like talking about it. Also I'm really bored and in the midst of procrastinating. The definitions and theorems presented here are taken from this book: A Computational Introduction to Number Theory and Algebra [http://www.amazon.com/Computational-Introduction-Number-Theory-Algebra/dp/0521516447/ref=sr_1_1?ie=UTF8&qid=1328490906&sr=8-1]. I'm no authority on math, but the divide by zero thing is really simple to understand and I like to share ideas. For some this might be difficult to follow, I'm not going to go easy, but I promise that everything you need to understand is here and no prerequisites are needed beyond an open mind. First some preliminary definitions (more or less motivated by intuition) that we need to commit to in order to have a subject matter. As a reminder, I need to casually point out that the number your dividing is called the dividend and the the number your dividing by is called the divisor. So in 2/3, 2 is the dividend and 3 is the divisor. The first thing we need to do, is to define a ring. Later we will show that the set of real numbers is a ring, and so any properties of abstract rings are automatically properties of the set of real numbers:
If you don't want to read all the definitions and the few theorems, just read theorem 2 and the conclusion, that should give you enough
[li]A set is a collection of objects hereby denoted as {a,b,c,...} where a,b,c,... are the elements that belong to the set. Sets can be empty, have a finite number of elements, or an infinite number of elements. in the case of a set with an infinite number of elements, there may be countably infinitely many elements, or uncountably infinitely many elements. These definitions can be contested and elaborated, but such a discussion would only digress from the topic at hand.[/li]
[li]An operator is a function whose input (domain) and output (range) are the same set S. A binary operator is an operator that takes two elements from the set as its input (we can define operators to take multivariate arguments in general), and outputs an element which again is an element of the set S. The following examples are familiar operators on the set of integers: multiplication and addition.[/li]
[li]A ring R, is a set (nonempty) with two operators, which for convenience I will call 'multiplication' *, and 'addition' +, with the following properties: (we could denote addition by '%' instead, the symbol we use is arbitrary. I am only using + as the symbol for addition over rings because it is a familiar symbol to this readership)
Properties of 'Addition' : +
1)For any two elements x, y in R, x + y = y + x
2)For any three elements x, y, z in R, (x + y) + z = x + (y + z)
3)There is a unique element '0[sub]R[/sub]' in R such that for all x in R, x + 0[sub]R[/sub] = x
This is called the (additive) zero element. We call it the zero element because this unique element behaves like the familiar zero of the real numbers. This is the element we want to know if we can use as a divisor. The definition of 'dividing' elements has yet to come though.
4)For every x in R, there exists a unique y in R such that x + y = 0[sub]R[/sub].
We will denote the additive inverse of x as -x. You can think of this as a 'sign change'. This is not the same as stipulating a concept of 'negative elements' or 'negative numbers', but for matters of convenience, if you want to think of -x as a 'negative x' then by all means do so.
Properties of 'Multiplication' : *
5)For any x,y in R, x * y = y * x
6)For any x,y,z in R, (x * y) * z = x * (y * z)
7)For any x,y,a in R, a * (x + y) = a * x + a * y
8)There exists a unique element '1[sub]R[/sub]' in R such that for every x in R, 1[sub]R[/sub] * x = x * 1[sub]R[/sub]= x
This element is called the (multiplicative) identity element.[/li]
I use the single quotes on the above operators called 'multiplication' and 'addition' over the ring R because they could be called anything else instead like 'jackkniving' or 'imballabloming'. These two operators are abstract (binary) operators, which only as a special case, when we consider our ring R to be the real numbers, come to be realized as the familiar multiplication and division that everyone learns in high school. The only thing important about our operators are the eight properties above
We're getting very close to our penultimate goal here, which is to craft a mathematical structure on the set of real numbers that will allow us to answer questions like what 'dividing by zero' means . A few very interesting and important things I want to point out here, is that a zero (element) depends on the ring in question. A different ring, a different zero (element). Later when we identify the set of real numbers as a ring, because it is a ring, we will have to designate some real number as the zero (element) and some other real number as the identity (element). In preview of that, I'll tell you now that when we take our ring to be the set of real numbers, we are going to let '0' be the zero element and '1' be the identity element.
You may have noticed so far no ideas taken from calculus. This is because we don't need a theory of the infinitesimally small (nor infinitely big) to work with the real number 0. Calculus deals with the 'closeness' of elements, allowing you to analyze limiting behaviors and rates of changes. 0 certainly plays a role in those kinds of studies, and hence helps you to better understand what 0 is and how it behaves, but those ideas have no influence on how we define 0.
So far the only operations we have to work with are multiplication and addition; to define whatever 'divisibility' might be (over rings) we have to get in a few more definitions. So we'll do that now, but not in the way you might have expected. Do you remember from high school math that to divide by a number x is the same thing as to multiply by 1/x. Well, this shows that 'dividing' is the same thing operationally as multiplication, we need not introduce a new 'divide by' operator. This subtle difference is crucial to understanding the divide by zero conundrum. Starting with R as a ring, here a few more definitions we need to form a sufficient axiomatic theory that will be our backdrop for analyzing the idea of 'dividing by zero' (and perhaps other ideas):
[li]For any elements x,y in R, we say that x divides y or y is divisible by x if there exists an element r in R such that x * r = y. For shorthand we'll denote this situation by y/x. Note a very interesting fact that might not jump out at you right away is that with this definition, the status of divisibility of any two elements depends on the existence of some other element in R. Also, that we are in fact not dividing y by x as it may seem. We are only saying that when we have a pair of elements (x,y) of the ring R and you can find for this pair an element r such that x * r = y, then we are going to call this state of affairs 'x divides y'.[/li]
[li]An element x in R will be called a unit if x divides the identity element 1[sub]R[/sub], that is if there exists a unique r in R such that x * r = 1[sub]R[/sub].[/li]
[li]For each element x, the associated element r will be called the multiplicative inverse of x and can be denoted by x[sup]-1[/sup] or by 1/x.
Note that in reference to the above definition, I am tacitly suggesting that a multiplicative inverse exists only if 1 is divisible by x ie 1/x. For if x does not divide 1, then we could never write 1/x.[/li]
[li]A ring R with the property that every element has a multiplicative inverse is called a field. It might seem pedantic, but for proper logical bookkeeping we need a further stipulation that fields F have more than just one single element.[/li]
So a field is just ring with a few additional properties; mainly that every element has an associated multiplicative inverse 1/x. Specifically a field is just a more specialized type of ring, but our statements will be dealing with fields and not the more general idea of rings (some of which are not fields).
You may be wondering at this point why we are not defining a 'division' operator /, so that y/x is just equal to some number r. The reason is that we want as few operators as possible defined on rings; if we defined a divide operator we would have 3 operators in total together with addition and multiplication. The goal in mathematics is to get as far as possible with as little as possible. We absolutely could have defined 3 operators over rings; multiplication, addition, and division if we wanted to, there is nothing stopping us. But this axiomatic system would be less desirable because there would be a greater number of axioms (well definitions, if you make the distinction between definition and axiom). And since we can mimic division by instead multiplying by suitably chosen numbers, we don't really need a whole new definition. So it is best to adopt the theory that just includes the two operators of multiplication and addition.
These are all the definitions that we need. Now we are ready to prove some theorems that will hopefully bring us closer to understanding 0:
Theorem1:
The set of real numbers with the usual multiplication and addition forms a field.
Proof: This is really easy to prove, just show that multiplication and addition that you learned in high school satisfy the requirements of the 'multiplication' and 'addition' operators over a ring. An element of this set is called a real number.
The concept of a field is a much more general notion then just the set of real numbers (without considering topological aspects). It is very easy to show that the real number line is a field (albeit with additional topological structure) with the familiar operations of additional and multiplication. But to make this thread interesting, I'm going to talk about the 'divisibility by zero' on general fields, of which the real numbers are just a special case.
This is all the machinery we need to address the enigmatic division by zero issue. This last theorem answers all questions of divisibility by zero which comes merely as a natural consequence of our definitions (this is how mathematics works), but it might not be the answer you were hoping for:
Theorem2:
For a given field F, any element x in this field is divisible by zero if and only if x is zero.
Proof:This proof is broken into two parts since it is an if and only if statement:
1)The if part; if an element x is divisible by 0, the assertion is that the element x is 0. Or, said in other words, if an element is divisible by zero, it must be that that number is itself 0. So suppose that zero divides x. This means that for 0 there exists an element r in F such that 0 * r = x. So in other words we have that for some element r in F, 0 * r = 0 = x. Coincidentally then, this equation also immediately implies x = 0.
2)The only if part; if x = 0 then the assertion is that x is divisible by zero ie 0 happens to be divisible by 0 according to all our definitions. So suppose x = 0. Then we need to find an element r in F such that 0 * r = x when x = 0. However, conveniently enough, any and all elements f in r satisfy this equation 0 * r = 0.
Notice above in both parts the zero element could be the element r. So we will make the informal definition that: if a pair of numbers (x,y) is such that y is divisible x, then know there exists an r such that x * r = y, and we will call this number r the result.
Conclusion: This definition of divisibility is pretty standard http://en.wikipedia.org/wiki/Divisibility but all the definitions I used (for fields, rings, 'multiplication', etc.) may vary slightly depending on what textbook you consult and who you talk to. The point however is consistency. If theorem 2 didn't do it for you, my remarks here should.
Theorem 2 asserts two propositions. The first is that the zero element, 0, is divisible by 0, and the result of that is again 0. The second assertion is that if at any time we notice that some number is divisible by zero, that number then must itself be 0. In other words only 0 is divisible by 0.
In all this discussion, I never defined what is meant by 'dividing'. I had only used multiplication and addition. Divisibility is a relationship between two numbers (not a binary operation), and any pair of numbers may or may not have this relationship. Part of what we have shown then is that the pair (0,0) has the relationship status of being divisible. So the question reformulated in the terms of this axiomatic system is:
The answer is yes, and theorem 3 and its remarks should clarify everything. The one idea you should retain after all this is that it really comes down to a matter of definitions. You could rework all the above by defining a divide operator, and then just stipulating that no number can be divided by 0. That is what is meant if it is said that dividing by zero is undefined; it means literally just that. As for why people would not define division by zero, just think of what division means intuitively for non-zero numbers. When you then use the number 0, there's no intuitive meaningful interpretation for division by zero, so we either don't bother to define it, or make some other definitions like the ones above for logical bookkeeping purposes. Sorry if your disappointed, but if you want real deep mathematical truth your going to have to work for it. Please post your thoughts.
I still like my "proof" that 0.999_ actually = 1, but still an interesting read.
