h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

When first introduced to imaginary numbers based on i, I thought it would be a good idea to develop a mathematics based on the other insoluble that we come across early in math class - division by zero.

I invented the number z, which was 1/0, but it turned out to not work very well. What's 3(6z)? 18z. What's 2(6z)? 12z. What's 1(6z)? 6z. What's 0(6z)? It seems that the answer should be 6, but that's a discontinuity every bit as extreme as merely declaring that division by zero is nonsense.

Also, a look at the graph of y=1/x pretty cleanly leaves x=0 out of the picture anyway, and so I eventually decided that the "z" concept could safely be retired.

DM

Division by zero is an odd concept... what it amounts to is saying that no matter how far you go along the number line (in either direction), you will never go far enough to find a number that satisfies the question. This is why it is undefined rather than infinite. A definition simply wouldn't make sense, no matter what you set it to be. Mathematicians, when formally talking about infinities, refer to the cardinality of a set instead (the 'number' of entities in that set) using numbers like aleph_0 (the size of the set of integers and rationals).

That all said, there is a mathematical entity called the Riemann Sphere based upon wrapping the complex plane (all the complex numbers) into a sphere resting on the origin using polar projection so that the other pole to the one on the origin represents infinity.

There is a way of dealing with division by zero in maths. This is the concept of the LIMIT.

The idea is basically to do the division first, pretending that the number is finite, then to let it go towards zero. So, to take your example, if you wanted to divide 8*z by z, you first do 8*z/z=8, then let z go to zero. as we just have 8, the answer is eight.

If you wanted to do z^2/z, (z goes to 0) you would find that you have z^2/z = z which goes to zero.

This also allows you to do much more interesting things, like sin(0)=0 divided by 0:
it turns out the answer is 1! If you type it into your calculator, you will get an error message. But try this. sin(0.1)/0.1 turns out to be about 0.998. If you do sin(x)/x, with x getting smaller and smaller, you get closer and closer to 1, so you can say (in a way) that sin(0)/0 =1.

This procedure allows mathematicians to multiply and divide by numbers that are as close to zero or infinity as is needed, in a controlled way.

This is not proof, just speculation.

Consider the following: -

y = z / x ...where z is non-zero.

As x converges towards 0, y converges towards (+|-)infinity, therefore it would be apparent to me that any number divided by zero (with the possible exception of zero itself) results in positive or negative infinity - the sign of the result being determined by the sign of x.

z / 0 = +inf ...for positive z.
z / 0 = -inf ...for negative z.

The contradiction: -

y = 0 / x

As we all well know, 0 divided by any non-zero constant results in 0. But we also know that any constant divided by itself results in 1. So, is 0 / 0 = 0, or is 0 / 0 = 1?

Personally I believe 0 / 0 = 0, and here I'll explain why. Based on my first assumption (that x / 0 = (+|-)inf), we can state that...

1 / 0 = +inf
-1 / 0 = -inf

If we add these assumptions together in their fractional form: -

1 / 0 - 1 / 0 = inf - inf
1 / 0 - 1 / 0 = 0
( 1 - 1 ) / 0 = 0
0 / 0 = 0

Cheers.

Hmm... only applies if infinity is the same as a number.

Unfortunately division by zero leads not to a number, but to a concept. x/0 is undefined rather than infinite. Infinity is more a measure of set size than number size.

Please back that up with some form of evidence.

Take a look at my article in more depth - I explain a concept - and whether it's true or not, there is conceptual evidence there. You saying "no, that's just not true" will not help anyone prove or disprove a thoery.

The problem is inf - inf is also indeterminate. Also:
If
a=0/0, then 0a=0.
Any member of C of a satisfies the equation. But to put it into your context...
Where a=1,
(0)(1)=0
Where a=0
(0)(0)=0.
You can't pin down a to a certain value, which is why it's indeterminate.
And if you say
a=1/0, then 0a=1.
If you presume a is inf, we know that any number times zero is zero(*).
So:
0(inf)=1, and by (*) we've reached a contradiction. For any other value you reach a contradiction as well.

Ok, you're right. That makes perfect sense. So I take it that 1/inf is an infinitesimal, not zero? That would ensure that the statement (x/inf)*inf = x is true.

Yep, 1/inf is an infitesimal, not zero. But inf*x/inf=x is a little tricky. I belive the form inf/inf is undefined. I'm not sure. But in all sense it should be correct.

Am I The Only Person Who Finds The Actual Acceptance Of Multiplication And Division Too Extreme? The Resultant Planes Have Such Ridiculously Restricted Domains.

Multiplication Is Closed Over All The Real Numbers. So Is Addition. The Problem Is When You Start To Presume That Infinity Is An Actual Member Of The Real Number System, Which It Is Not! (see how annoying capping every first word is?)

Capping Every Word Keeps Me Sane Well It Used To...

Clarifying earlier post: inf * x/inf = inf.
If you want to know, ask and I'll clarify.
To paraphrase _The Mathematical Experience_, "finite arithmetic is simply not the same as infinite arithmetic"

Actually, could you explain that? I always had the impression that dividing by infinity depended on what kind of infinity you were using, or something like that. (Well, maybe not always - maybe just last year in Calc. Same thing.) Anyway, I find myself quite intrigues by this Infinite Math thing...

What would you like me to explain? Infinite arithmetic can be complicated: let's consider
3/inf=x. x is undefined, because, rearranging
inf*x=3, but inf multiplied by any number gives infinity.
To get a grasp with infinite arithmetic, you should search under a good search engine about "transfinite arithmetic" and "cantor".
I can explain the basics of transfinite arithmetic if you like.

A/B can be read A divided by B. Alternatively you can say A split into B number of parts (this is how it is often explained to children). So 4/2 makes sense because 4 can be broken into 2 parts and 'shared out'.
Now 1/0 means 1 split into 0 parts. What does this actually mean? At first it seems 1 split into 0 parts is 1 (i.e 1/0=1) because you haven't actually split anything so you must be left with the number you started with.
As I am sure you know 1/0=1 is false.
Assume the following exists: A machine that you could feed matter into and this machine can split the matter up precisely into equal ammounts according to the number you specify on a keypad. Imagine that it works for any number you type. What would pressing 0 do? It tells the machine to divide the matter into 0 parts or into a number of parts of size 0. Dividing into a number of parts of size 0 would cause the matter to no longer exist, breaking the laws of thermodynamics!
Of course, such a machine doesn't exist and can't exist as matter cannot be created or destroyed. By analogy division by 0 cannot exist.

Technically, dividing A/B means splitting A up into pieces of size B, right? Or is it B number of pieces? Or does it really matter?

But I guess, which analogy you use affects the way division by 0 works (or doesn't work).

The mean the same thing.

http://www.bbc.co.uk/dna/h2g2/alabaster/A675407

Written a whole article on dbyz and other sorts of undefined/indeterminate forms.

Division by zero, if my memory serves me, is already used in quantum mechanics (I think most bitterly in quantum electrodynamics--QED); physicists "cheat" equations by putting it through a process called "renormalification" in which they have an expression divided by zero and they know the solution they are looking for, so they merely ...take the solution they know is correct and call that the solution to the division. To me it seems somehow that imaginary numbers, maybe used how Sir Penrose has been trying to apply them, might well be the key to removing such logical inconsistencies from physics and mathematics all together.

I just thought I'd throw in my two cents and let somebody who is a little more knowledgeable in the subject either destroy or defend my statement, as I don't quite remember exactly (it's been a year or so since last I read about it.)

/SAS