Is This a Valid Way to Divide by Zero?

[Asterisk Propernoun]

When most people try to divide by 0, they either type it in to a calculator to find error, work it out on a sheet of paper to find repeating infinity (and I know writing infinity outside of calculus is against the rules, but it doesn't hurt to write it down just so you can see what will happen), or use calculus to get 'undefined'. Well, I believe that dividing by 0 is more of a philosophical process rather than a mathematical one. For instance, let's try using words to divide 10 by 2.

 

To have one of 2 equal pieces to 10 is to have: 5

 

That seems to work out nicely. Let's try this with 0.

 

To have one of 0 equal pieces to 10 is to have:

 

Well, that's an invalid statement in that you can't have one of zero. I'm thinking that, when you try to divide by a nonzero number, it's like filling a glass with milk; but when you try to divide by 0, it's like trying to fill a glass that isn't there. You can spill the milk all over the counter, your floor, and even flood your house with milk all you want, but you'll never get any closer to filling that nonexistant glass. I believe that this is why, when we try to divide by zero on a sheet of paper, we usually end up with the largest number we can think of.

 

Now, when most people set out to divide by zero, they normally look for a quantity. Now, imagining the empty glass as the essence of a quantity, I'm going to say that, when you divide by zero, your answer will be the absence of a quantity. This doesn't mean it's 0, because 0 is a quantity just like any other number. The absence of a quantity simply means you don't have a number, so your answer is literally nothing.

 

However, let's look at this differently. One can always say that they have 0 of something. For instance:

 

I have 0 ferraris parked in my driveway.

I currently have 0 dollars in my back pocket.

I have 0 solid gold keys that can unlock the doors to a solid gold house.

 

Keeping this in mind, can we replace the absence of a quantity with 0? Well, I can't think of any instances where one would need to divide by zero except for one, negative factorials.

 

5!=5*4*3*2*1=120

4!=5!/5=24

3!=4!/4=6

2!=3!/3=2

1!=2!/2=1

0!=1!/1=1

-1!=0!/0=0 (Let's just assume for the sake of the test.)

-2!=-1!/-1=0

-3!=-2!/-2=0

-4!=-3!/-3=0

-5!=-4!/-4=0

 

Now, when one uses factorials, they usually use it to see how many ways they can arrange n number of objects. For instance:

 

I can arrange two objects 2 ways: GH & HG, which equals 2!

I can arrange one object 1 way: G, which equals 1!

I can arrange the absence of objects 1 way, which equals 1!

 

Now, let's think about the negative numbers I've added to the pattern. If we assumed that you would get the absence of a quantity every time you divided by 0, and assumed that you could always replace that absence with 0, then all of the negative factorials would equal 0. I believe this is a valid answer. Consider holding negative five sticks of wood. You simply cannot do it, right? You can write negative signs on sticks as much as you want, but in reality, those are only positive sticks with negative signs written on them. Similarly if you're holding a handful of bills that says you owe money. You may think of it as negative currency, but they're just positive papers with numbers written on them. Since you cannot physically hold any number of negative objects, I believe it's reasonable to say that any negative factorial will equal 0. Since, because you cannot place negative objects on a table and shift them around, there will be no ways to arrange them.

 

If anyone can think of any other situations where dividing by 0 is necessary, then please tell me. It would be a good way to prove/disprove my idea.

 

Thanks.

[studiot]

First part

 

The idea that division by zero results in a blank (I prefer the word blank to your "nothing" since nothing can mean zero so blank is less confusing) is contrary to the closure requirement from set theory.

That is every operation of the type A/B must result in another member of the set

Division by zero cannot achieve this..

 

When folks start to contemplate dividing by zero they usually quickly move off set theoretical ideas of what a number is and limit their discussion to the counting property of numbers.

 

You have done the same here, and quickly shown that we do not need negative numbers for counting.

[Bignose]

In the real world, I really dislike the idea of redefining dividing by zero in any situation because 'undefined' has worked really, really well for us to date. Redefining division by zero to any value in any situation opens up the door to 'prove' that most any number can literally be equal to any other number.

 

And on a practical note, I like when the calculator or computer throws up an error because it tells us we did something wrong. If it didn't, and set division by zero to some value -- too many people rely too much on the computer and these possibly significant error would be missed. I mean, too many errors are missed as it is because the computer is implicitly trusted too much, so taking away at least some of possibilities of catching those errors seems like a really poor idea to me.

 

So, I'm going to stick with undefined.

 

Besides, the gamma function (the extension of factorial to the reals) has poles at each of the negative integers: http://en.wikipedia.org/wiki/Gamma_function The graph of which makes that obvious. It needs to remain undefined at those points, just like the tangent function is undefined at points, so that all its properties remain in tact.

[Asterisk Propernoun]

In the real world, I really dislike the idea of redefining dividing by zero in any situation because 'undefined' has worked really, really well for us to date. Redefining division by zero to any value in any situation opens up the door to 'prove' that most any number can literally be equal to any other number.

 

And on a practical note, I like when the calculator or computer throws up an error because it tells us we did something wrong. If it didn't, and set division by zero to some value -- too many people rely too much on the computer and these possibly significant error would be missed. I mean, too many errors are missed as it is because the computer is implicitly trusted too much, so taking away at least some of possibilities of catching those errors seems like a really poor idea to me.

 

So, I'm going to stick with undefined.

 

Besides, the gamma function (the extension of factorial to the reals) has poles at each of the negative integers: http://en.wikipedia.org/wiki/Gamma_function The graph of which makes that obvious. It needs to remain undefined at those points, just like the tangent function is undefined at points, so that all its properties remain in tact.

 

I'd imagine that the proof that any number can equal any other number through dividing by 0 would've been like saying 2=3 because 2*0=3*0. As for the rest of what you said, I see your point.

[ajb]

If anyone can think of any other situations where dividing by 0 is necessary, then please tell me. It would be a good way to prove/disprove my idea.

There are mathematical situation where you can divide by zero, but you will not be able to keep the axioms of a ring. The situation needs to be a bit more exotic than just the real numbers.