My own uneducated dumbass way of looking at it is to look at division as a kind of subtraction. For example, 12/4 means “how many times can 4 be subracted from 12 to leave a remainder less than the absolute value of 4?”. The answer is 3. But if we ask the same question about 12/0, then even a infinite number of subtractions would fail to reduce 12 by any amount, meaning that the solution is a singularity. Therefore, generalizing that the same would be true for any number in place of 12, division by zero is undefined.
Exactly!
Very interesting, I’m going to have to look into this further.
The ancient greeks used to have very long and sometimes heated debates about whether zero was a number or if zero was the absence of any number. Too bad they aren’t around to see this thread. They would have enjoyed it.
Fields are one of the more straightforward constructs that you’re going to find as you get into higher and more abstract mathematics. A field is just a set F and two operations + and * that behave pretty much exactly like you’re used to. Specifically:
[ol]
[li]a + b is in F.[/li][li]a + b = b + a.[/li][li](a + b) + c = a + (b + c).[/li][li]There’s some 0 in F which satisfies a + 0 = a.[/li][li]For every a, there’s an element b which satisfies a + b = 0.[/li][li]a * b is in F.[/li][li]a * b = b * a.[/li][li]a * (b * c) = (a * b) * c.[/li][li]There’s some 1 in F which satisfies a * 1 = a.[/li][li]For every a other than 0, there’s an element b which satisfies a * b = 1.[/li][li]a * (b + c) = (a * b) + (b * c).[/li][/ol]
Most of the facts about arithmetic that you’re used to follow from these properties. The rationals and reals are both fields, but so is the set {odd, even} with odd + odd = even + even = even, odd + even = even + odd = odd, even * even = even * odd = odd * even = even, and odd * odd = odd.
Here’s a short proof that division by 0 is impossible in any field:
a0 = a0
a0 + 0 = a(0 + 0)
a0 + 0 = a0 + a0
0 = a0 (and by property 7 above, 0 = 0a)
So, if we define division by a to mean multiplication by b where b satisfies a * b = 1, division by zero is impossible because there is no solution to 0b = 1–note that this requires 0 and 1 to be unequal, but that does follow from the properties I listed above (although I don’t know the proof off the top of my head).
You have to think of pie. Which helps, because . . . well, we all like thinking about pie.
Divide a pie into two pieces. Easy as . . . well really easy.
Divide a pie into four pieces. Also easier, and still reasonable in terms of serving size.
So, divide a pie into seventy one pieces, all equal in size. Simple, although you need a protractor, and someone is gonna whine.
Divide a pie into 5.8 pieces. Ok, this one is tricky, cause you have to define what you mean by .8 pieces. But, if you will accept a piece .8 times the size of the other pieces, it’s still difficult but possible.
Divide a pie into pi pieces. Tricky, but, mathematically speaking still possible.
Now divide a pie into zero pieces. Now, no one is falling for the trick where you eat the whole damned pie, and tell us that there are zero pieces left. We know damned well you ate one piece, in fact the whole pie. We’re gonna kill you, this is pie we are talking about. No one ate any pie in the other examples, so you can’t eat any this time! No matter how you slice it, there is gonna be pie left. Some is more than zero, so dividing into zero pieces doesn’t work.
Tris
Actually, there is a system of math that does allow division by zero. I’ve forgotten the name - it came up in the last division-by-zero thread and I intend to hunt it down when I finish this post.
Basically it says x/0=unsigned infinity (unsigned because it can be either positive infinity or negative infinity - a paradox which I suspect led to division by zero being declared off limits and in bad taste).
There are some nice balances in this system. Since zero is neither positive nor negative, there is a certain esthetic satisfaction in its reciprocal - unsigned infinity - being both positive and negative. And the “anomaly” of multiplication by zero always yielding zero is balanced by the fact that division by zero always produces unsigned infinity.
Anyway; time to wake up the search hamster. . . .
It’s the real projective line.
Of note, it gets by by keeping 0/0 undefined, infinity * 0 undefined, infinity + infinity undefined, and such things. (As a result, it’s not a field, since neither multiplication nor addition are totally defined, infinity has no additive inverse, etc.). But it has its uses. A common guiding intuition behind its development, as I think was mentioned in the thread you’ve set the hamsters upon, was thinking of real numbers as the slopes of lines; unsigned infinity would then be the slope of a vertical line.
Ah, rats, Indistinguishable beat me to it (e provided the reference in the previous thread, too; if there are any divide-by-zero junkies out there, that thread is here).
That is very close to the mark. It is not quite that 0 does not exist as that even if you are given an effective (the technical term is recursive) description of a number, there is no infalliable test of whether or not it is 0. If you try to divide using recursively defined numbers, you cannot divide by a number unless it is decidably non-zero. Another aspect of this is that division is increasingly unstable as you get smaller and smaller divisors. It fluctuates to an unlimited degree with tiny changes in the divisor.
Let me also point out that if you stick to integers (whole numbers, the only numbers that allow unrestricted division are 1 and -1. You cannot solve the equation 2*x = 5, for example. If you stick to natural numbers, you cannot subtract any number save 0. The equation x + 7 = 5 has no solution.
Which, I imagine, is the same as saying “you cannot divide by a number unless it is non-zero”, as every non-zero recursive number should eventually be determinable to be non-zero (i.e., the set of non-zero recursive numbers is semidecidable), no?
(Of course, people aren’t generally viewing themselves as working in a universe restricted to the recursive numbers)
Try telling that to the IEEE. 
When would 0 and 1 ever be equal? Please point me to the thread where (I’m sure) you’ve answered this previously.
Unless I’ve misread something, it doesn’t follow from the properties you’ve listed; the trivial structure with just a single element (which is both 0 and 1 and adds and multiplies to itself) satisfies all the properties you’ve listed. Nothing wrong with demanding that 0 be unequal to 1, of course, but if you want to do so, you’ll have to add it in explicitly.
I’ll swear that I read somewhere that Z[sub]2[/sub] is the smallest field, but it looks like there are definitions of a field that allow 0 = 1. So I’ll add 0 != 1 to my axioms.
0 and 1 aren’t equal if you’re talking about the real numbers. But if you’re not careful, you can define a field in such a way that the additive identity and the multiplicative identity are equal. Since we usually denote the former 0 and the latter 1, it follows that 0 = 1 in that field. Again, it’s not in any number system that non-mathematicians care about.
Clearly, the only structure where 0 = 1 is the trivial one, with only a single element in it. (Proof: for all x, x = 1 * x = 0 * x = 0.) In my classes this was called the null ring, and fields were explicitly required to be non-null rings. Of course, this is only a question of definitions.
I’d wager that most mathematicians also don’t care about the null ring, since it’s a quite trivial and uninteresting structure.
As for this particular mathematician, I wouldn’t say I care about the trivial ring in itself (it’s not like there’s anything there to care about), but in terms of the category of rings at large, I’d certainly want the trivial ring to be in there, it being in some important ways more natural that way than otherwise.
Fields are a little different, since the partialness of reciprocation/division makes the theory of fields rather far from an equational theory to begin with, and so there’s less of an aesthetic push, in my mind, to avoid axioms like 0 != 1. I have no great problem with considering this to be part of the definition of a field, though I suppose my terminological moods can go either way.
Just a typo–this should of course be a*(b+c)=(ab)+(ac) (distributivity).
Anyway, the OP might also like to think about commutative rings more generally. A commutative ring with unity satisfies all of ultrafilter’s properties 1-11 except for #10; that is, not all nonzero elements necessarily have a multiplicative inverse.
But property #10 is the one that lets you divide by a nonzero number. So, while in the real number system (or any other field) you can freely divide by any number other than zero, in rings there may be nonzero values for which division is not fully defined.
A simple example is the ring of integers mod 10. This ring describes how the ones digit behaves under integer addition and multiplication; for example, 32=6 and 67=2 (since 6*7=42==2 mod 10). In this ring, division by 1, 3, 7, and 9 is well defined, but division by 2, 4, 5, 6, 8 (and of course 0) is not. You can check this by looking at the ones digits in a 10x10 multiplication table:
Multiplication table for ring of integers mod 10:
*|0 1 2 3 4 5 6 7 8 9
-+----------------------------
0|0 0 0 0 0 0 0 0 0 0
1|0 1 2 3 4 5 6 7 8 9
2|0 2 4 6 8 0 2 4 6 8
3|0 3 6 9 2 5 8 1 4 7
4|0 4 8 2 6 0 4 8 2 6
5|0 5 0 5 0 5 0 5 0 5
6|0 6 2 8 4 0 6 2 8 4
7|0 7 4 1 8 5 2 9 6 3
8|0 8 6 4 2 0 8 6 4 2
9|0 9 8 7 6 5 4 3 2 1
In the columns for multiplication by 1, 3, 7, and 9, each ones digit occurs exactly once, so you can determine that, for example, 6/7=8 (since 6 occurs only in row 8 of the 7 column). But in the other rows, some digits appear more than once, and others don’t appear at all. Should 8/4=2 or 8/4=7? The corresponding multiplicative equations, 42=8 and 4*7=8, both hold in this ring, so there’s no unique answer; we can narrow the possibilities down, but not completely. And what is 3/4? 3 never appears in column 4, so this quotient is not defined in the ring.
The problem is that this ring has zero divisors. In this ring, 2*5=0. Since division by zero is problematic, it’s not too surprising that division by its factors, 2 and 5, is also a problem. Fields have no zero divisors, so they don’t have this problem. But maybe the example will help you see that zero is not always unique in being a hard number to divide by.
Given that infinity is not a number, it seems like it hasn’t solved anything more than saying /0=Error
Given that -1 is not a number, it seems like it hasn’t solved anything more than saying 6-7=Error.
Infinity is not a real number, or a complex number, or an integer, or any of that, by the very definition of such systems. That doesn’t mean there cannot be some mathematical system with some element naturally labelled as “infinity”, with the system naturally thought of as one of “numbers”.
I guess I can also re-use my post from the last thread on the general meaninglessness of statements of the form “X isn’t a number”.
It seems to me that math, any type of math, is simply a sort of logic game. Given certain definitions (“axioms”), the results inevitably follow. I think what leads folks in these threads to talk past each other is that most of us are never taught more than the commonly-used mathematical “system”. Certainly I had never heard of Real Projective Line before the previous divide-by-zero thread; I just happened to figure out some of its “assumptions” from some of my own musings on certain mathematical “quirks”. I feel safe in saying that there is a lot of mathematical theory out there that just isn’t taught in general math classes.
I don’t mean to harp on the divide-by-zero thing, but one of the really interesting things in the other thread was the report of a calculator that would give the result “negative infinity” when dividing a negative number by zero, and “positive infinity” when dividing a positive number by zero (no report on what happens if you divide zero by zero), instead of the usual “error” result that most calculators give. I guess this illustrates my point that the results you get depend upon the assumptions you begin with.