This is in fact very easy when you think of the linguistics of the idea instead of mathematics. This is why I won't be using any equations, inequalities, or anything like that in my arguement. Existing maths don't allow dividing by 0, so of course any mathematical representation wouldn't work.

Here's the thing: Division of x/y is taking x and putting it into y parts. 8/2 is taking 8 (lets say apples) and putting them into 2 groups. We've all done this as kids:

@ @ @     
@ @ @    
@ @

Those are our apples. If we put those into two groups, we have this:

@ @      @ @
@ @      @ @

Each group has four apples. 8/2=4 (ok, maybe I lied about no equations).

Now, if we do 8/0, we specify no groups. If there are no groups of apples, of course you have no apples! If we were to say "parts" instead of "groups," you could look at the saying "I'll have no part in this."

If this person is having no part in this:

"this" / "no part" (just go along with me)

Then they are not involved with "this" at all. There is no "this" in the person's life. Saying that a person will have no part in something does not mean that this person will have an undefined part in said activity!

For the record, I truly hope that someone can prove me wrong beyond any doubt, because if I was right, it would be too hard to convince anyone.

Looking back on this half-decade old writeup (it is now September, 2006), I realize how ridiculous I was being.

In my original, rambling writeup, I claimed that if you take eight apples, put them in no groups, well, then you have no apples! Never did I touch on the feasibility of putting them in no groups. There is always going to be a group. In the case of my old writeup, that group is the group of apples that I am not considering.

So instead of no apples, I really have two groups. The empty group of apples that I want to think about (this is the only group I thought of originally), and the group of eight apples that I would like to conveniently ignore. Unfortunately, ignorance is not good math.

The trouble with this is that you're taking a mathematical idea ('divide by') and wrenching it out of context. It's been done before: we use the words "orthogonal" and "modulo", for example - both mathematical in origin - quite happily, outside of purely mathematical contexts, but that's because they've been given a meaning that is useful in the places where they're used. My argument will be that the new "linguistic" (contrast: mathematical) usage proposed has no such useful application, and is therefore nonsense.

Now, if you take your eight apples and 'divide' it into 'no groups', resulting in no apples, then where do the original eight apples go? I challenge you to do this with eight real apples, and make them all disappear by 'dividing' into 'no groups' (eating them isn't allowed - that's dividing them 'into' one person. :-)

Language (including mathematical language) just doesn't work like this - it's as though I were to say "I'll multiply the moon by a cow, and look - it gives me forty stars! Prove me wrong!" Unless you can provide some stable context (in this case - apples or moon - it needs a corresponding physical reality, because the mathematical one plainly doesn't apply) then you're just talking nonsense, and there's no reason to take you seriously.

Maths (complicated maths, for which there's no obvious physical analogue) gets away with it, because the system of rules creates a consistent and stable context which determines a semantics for the mathematical statements - which is why mathematicians can talk about such things as the square root of minus one, and still make sense. The sense is provided by the consistent universe of discourse (consisting of mathematical objects) to which the statements about sqrt(-1) refer (i.e. objects in the complex plane, clifford algebras, etc.)

If you make up new language without having such a stable context, there's no reason to expect anyone to understand or believe you, because there's no way to check the application of the new term, and there's no way to use your language to do anything useful. New language 'sticks' because it turns out to be useful in dealing with something, whether it's another part of the language (like in maths) or some regularity in our observations of the world (like in science) (or, going out on a limb, because it strikes a chord 'within', perhaps, as in poetry.)

With the best will in the world, I don't think your extension of the term 'divide by zero' accomplishes any of these!

Existing math does allow dividing by zero.

The branch of math starting with calculus is based almost entirely on the concepts of limits and derivatives.

The limit allows you to examine things that otherwise would have no answer, such as dividing by zero. The limit does not guarantee an answer, but allows you to consider if there is one, and find it when it exists.

The derivative at its most basic, determines the slope of a curve at a point, and in so doing makes use of limits to explicitly divide by zero in a specific, meaningful, controlled way. The point is considered to be a very short line segment with both end points on the curve. As the length of the segment approaches the limit of zero, the end points become coincident. The results of the division by zero involved in calculating the slope of the segment is defined on smooth continuous curves, and undefined at sharp corners and other discontinuities.

The catch here is that the use of the limit only gives meaning to division by zero in the context of a larger expression, and even then only when a variable is approaching zero, and only if the value of the expression is the same when approaching zero from both sides.

Dividing by the constant zero is not helped, and still has no meaning.

Derivatives in the sense that you are using them are a much more profound concept than division by zero: they are a way to resolve an indeterminate expression usefully.

When we take dx/dt (velocity), it is true that we are putting the limit as change in time approaches zero in the denominator, but there is a crucial distinction: change in position, in the numerator, is also limited to zero.

Zero divided by zero is a class wholly outside the scope of any non-zero constant over zero; the latter is undefined (or, in some applications in physics, infinity), whereas the former is indeterminate. Getting an indeterminate solution is not the end of the line; it just means that the guy with the pencil and paper did something clumsy and removed all the usefulness from an equation. For example, take the following solvable linear equation:

3x + 5 = 20

Now, I sit down with my pencil and decide that what I really want to do is multiply both sides of the equation by zero and then solve for x. Don't ask me for my logic; I'm intentionally being clumsy.

0x = 0
x = 0/0

So what's the value of X? According to my solution, it is indeterminate. That doesn't mean that X doesn't have a value; it simply means we haven't found it, or in some cases, that we cannot find it.

On the other hand, if you misused calculus and ended up with the following:


where n is any non-zero expression, the fraction would become a problem of division-by-zero and the mathematical system would explode like a wet turtle in a microwave oven.

Anyway, my point is that the fact that there is a zero (or a zero-limit) in the denominator alone does not mean that the concept of division-by-zero ought even enter our thoughts, or that the result is undefined.

And that's more than just clever semantics. A practical application can be found in physics: a photon has no mass, but it does have a measurable momentum. How is that possible?

Momentum is (elementarily) defined as mass times velocity:

P = m * v

However, this is where special relativity dives in and plays a trick on us: when something reaches the speed of light, its mass is increased infinityfold. So the effective momentum of the photon is:

P = (0 * infinity) * v

Of course, in this case, infinity can be reduced to any constant over zero, which changes the expression to:

P = (0/0) * v

Is that a problem? Not for the universe; photons have a definite and measurable (albeit slight) momentum.

So: indeterminate and undefined are completely different beasts.

Calculus only provides a mechanism for dividing by almost zero, to the extent that almost zero is so close to zero that it doesn't matter.

Dividing by zero gives meaningless answers, for example:
(x^2 means x squared where x is an arbitrary variable)

x = 1 {initial statement}
x^2 = x {multiply by x}
x^2 - 1 = x - 1 {subtract 1}
(x+1)*(x-1) = x - 1 {factorise left}
x + 1 = 1 {divide (x-1)}
1 + 1 = 1 {because x = 1 (see line 1)}
2 = 1

This contains a divide by zero because (x-1) = 0. This shows by contradiction that dividing by zero is not a valid operation.

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