Understanding / by Zero

[conway]

I am looking for in depth reasons for why division by zero must remain undefined.

[fiveworlds]

Division by zero opens the portal to hell.

[Unity+]

I am looking for in depth reasons for why division by zero must remain undefined.

One of many reasons why division by 0 is undefined is due to the following reason:

 

Let a equal some number and the following be true:

 

a*0 = 0

 

Now, let's assume we had a variable b, some constant, and the following is true:

 

a/b = c

 

If we set b = 0, the problem would be that we could have multiple solutions for a, if we tried to solve for it.

 

a/0 = c.

 

a could be 1, 2, 3, or any number for that matter.

 

That is one of the reasons why it is undefined.

 

EDIT: I think this article explains it better https://en.wikipedia.org/wiki/Division_by_zero.

Edited June 26, 2015 by Unity+

[conway]

What is wrong with multiple solutions? If then b=0, then I can say anything / 0 is the thing itself. So while always different. The same rule is always followed to reach that sum.

The first part of the wiki article was mostly why / by 0 fails in regards to computing. Simply re write programming. The rest was mostly references to others work, (all very old work), and then an in depth discussion of how division is the inverse of multiplication, and "zero product property" exist so then nothing can be divided by zero. Well then why does the "zero product property" exist. Not that I am questioning this particular axiom, just suggesting it is "circular" proof.

 

 

Why on earth did I get a negative point for this post?

Edited June 26, 2015 by conway

[Unity+]

What is wrong with multiple solutions? If then b=0, then I can say anything / 0 is the thing itself. So while always different. The same rule is always followed to reach that sum.

The first part of the wiki article was mostly why / by 0 fails in regards to computing. Simply re write programming. The rest was mostly references to others work, (all very old work), and then an in depth discussion of how division is the inverse of multiplication, and "zero product property" exist so then nothing can be divided by zero. Well then why does the "zero product property" exist. Not that I am questioning this particular axiom, just suggesting it is "circular" proof.

 

 

Why on earth did I get a negative point for this post?

If there are multiple solutions in this context, then it is an inconsistency within mathematics. if n/0 equals 0, but n can be any value, then it would be an inconsistency.

[conway]

I am sorry I will clarify what I meant.

 

It could be that.....

 

n/0 = n .....but

 

0/n = 0 .......so no inconsistency would occur.

 

 

I really have no idea what I did wrong in my previous post to receive negatives. I realize saying this is at risk of receiving more.

Edited June 26, 2015 by conway

[Unity+]

I am sorry I will clarify what I meant.

 

It could be that.....

 

n/0 = n .....but

 

0/n = 0 .......so no inconsistency would occur.

 

 

I really have no idea what I did wrong in my previous post to receive negatives. I realize saying this is at risk or receiving more.

Well, let's test it out.

 

n/0 = n

 

n/0 * 1/n = n * 1/n

 

(n*1)/(0*n) = 1

 

Because of the order of operations(PEMDAS), the following would have to occur:

 

n/0 = 1

 

And the original:

 

n/0 = n

 

As you can see here, you would have an inconsistency because n could be any number and the result will always be 1. So, it could equal both n and 1. It could make sense for a parabolic function, but in this context it makes no sense at all.

 

EDIT: Scratch that, it wouldn't make sense for even a parabolic function because for each y value, the x would be a different value.

Edited June 26, 2015 by Unity+

[Endy0816]

The first part of the wiki article was mostly why / by 0 fails in regards to computing. Simply re write programming.

 

 

You can change how the computer handles division by zero. How a machine is able to spit out indeterminate and undefined in the first place.

 

Otherwise it will run out of memory trying to perform what you requested. Counting each time it subtracted zero from some number forever.

 

Well, let's test it out.

 

n/0 = n

 

n/0 * 1/n = n * 1/n

 

(n*1)/(0*n) = 1

 

Because of the order of operations(PEMDAS), the following would have to occur:

 

n/0 = 1

 

And the original:

 

n/0 = n

 

As you can see here, you would have an inconsistency because n could be any number and the result will always be 1. So, it could equal both n and 1. It could make sense for a parabolic function, but in this context it makes no sense at all.

 

EDIT: Scratch that, it wouldn't make sense for even a parabolic function because for each y value, the x would be a different value.

 

I think it was n/n that finally got me to stop when I was young.

 

Did get fairly far though with: n/0 = a. Don't try and define what it equals and you can push onwards. Helped with understanding singularity cases in general I think.

[Unity+]

 

 

You can change how the computer handles division by zero. How a machine is able to spit out indeterminate and undefined in the first place.

 

Otherwise it will run out of memory trying to perform what you requested. Counting each time it subtracted zero from some number forever.

 

 

I think it was n/n that finally got me to stop when I was young.

 

Did get fairly far though with: n/0 = a. Don't try and define what it equals and you can push onwards. Helped with understanding singularity cases in general I think.

But isn't the point of a variable to be a representation of a numerical value?

[ydoaPs]

I am looking for in depth reasons for why division by zero must remain undefined.

Get out a pen and some paper. Draw the function f(x)=1/x.

 

Don't just do the positive x's. Do the negative ones as well. Tell us what happens at x=0.

[John Cuthber]

I am looking for in depth reasons for why division by zero must remain undefined.

For the same reason that, if I tell you my pet is a poodle, you know it is a dog, but if I tell you it's a dog, you don't know whether it's a poodle, a pug an Alsation or what.

You are trying to undo a "many to one" function

Many numbers, when multiplied by zero give a result of zero.

[Strange]

What is wrong with multiple solutions?

 

In principle, nothing. For example, quadratic equations always have two solutions. However, x = n / 0 has infinitely many solutions. That is what "undefined" means.

 

Presumably, one could define new rules of arithmetic where n/0=17 or n/0=n. But I doubt it would be useful.

[Sato]

I disagree with the sentiment that the problem is [math]\frac{x}{0} = y[/math] having multiple solutions (in fact, infinitely many). There are many admissible formulae in our usual algebra that hold several solutions, such as is the case with quadratics; [math]x^{2} = n[/math] will have two for any non-zero [math]n[/math]. This property won't break anything.

 

The actual problem is that, if division by 0 is given the usual properties of division, then [math]\frac{0}{0}[/math] would equal 1. Still maintaining 0's canonical multiplicative properties, [math]x * 0 = 0[/math]. So, given some non-zero [math]a[/math] and [math]b[/math], [math]a * 0 = b * 0[/math]. Then, as both sides are the same value (= 0), their scaling by some factor [math]n[/math] will yield again the same value on both sides. If [math]n = 0[/math], then we have [math]\frac{a * 0}{0} = \frac{b * 0}{0}[/math], and by our usual rules of multiplication that is equivalent to [math]\frac{0}{0} * a = \frac{0}{0} * b[/math]. Given our admission of 0 to the usual rules/properties as described, we have [math]1 * a = 1 * b[/math], and by multiplicative identity [math]a = b[/math]. That is the problem, I think, or at least a much more salient and convincing one.

Edited June 26, 2015 by Sato

[ajb]

The problem, in one form or another, is that division by zero cannot truly be the inverse of multiplication by zero. As mentioned you do not have uniqueness, which is required for an inverse. Sato has shown another problem. you cannot keep all the axioms of the real numbers if you allow division by zero.

 

In short, you have to weaken the structures you are dealing with to allow division by zero.

[conway]

Unity+, Ajb, Sato

 

 

Your equations come down to * by 0 as well. I think like ajb and Sato pointed out, that the "problem" seems to consistently come back to this same issue. So then if this is the case why can we not change the "zero product property". What axioms ajb would break down and weaken. This is to say "assuming" you allow me to discard the "zero product" property I can rewrite all equations given by Sato with correctness.

 

 

YdoaPa

 

Could you spare me the time and tell me what happens.

 

Johncuthber

 

If I tell you I have a dog or a poodle.....in neither case does my dog disappear, or become undefined. Physicality allows for / by zero...ergo the equivalent of no operation being performed. Physicality also suggest multiplication by 0 is not necessarily 0. Which I think was the point of your analogy.

 

 

Strange

 

Then under these very assumptions, would the single fact alone , that it more accurately describes physicality, be useful enough, in and of it's self.

Edited June 26, 2015 by conway

[ajb]

One way of doing this is to look at wheels. I imagine there maybe other ways also...

 

In wheel theory you replace division with another operation '/' that is not inverse of multiplication in general; this allows you to make sense of /0. In doing so you lose 0x=0 and x-x =0 in general. This seems close to what you are saying.

 

Interestingly, you can always extend any commutative ring to a wheel. This means that you can extent the real numbers in this way. Again, this sounds close to what you are thinking of.

 

Google 'wheel theory' and see if that answers your questions.

Edited June 26, 2015 by ajb

[studiot]

Here is a simple explanation of extending fields.

 

http://www.sjsu.edu/faculty/watkins/infincalc.htm

[conway]

associative

 

distributive

 

multiplicative identity

 

multiplicative inverse

 

 

So then assuming only one axiom is added then technically all the above, including thoese for addition can be kept, and allow for (A*0=A) as well as (0*A=0), as well as (A/0=A)

 

For any A in S there exist Z₁, and Z₂, constituting A, so that any A in operation of multiplication is only representing Z₁ or Z₂, in any given equation. So that.

 

A = 2 = ( 2_{1 ,} 2₂ )

 

B = 4 = ( 4₁ ,4₂ )

 

such that ( A * B = 2₁ * 4₂ )

 

 

My apologies, one more axiom to add.

 

 

0 =

 

Z₁ = undefined

Z₂ = 1

Edited June 26, 2015 by conway

[John Cuthber]

 

Johncuthber

 

If I tell you I have a dog or a poodle.....in neither case does my dog disappear, or become undefined. Physicality allows for / by zero...ergo the equivalent of no operation being performed. Physicality also suggest multiplication by 0 is not necessarily 0. Which I think was the point of your analogy.

 

 

My point was this.

If I tell you I am thinking of a number, and if you multiply that number by two you get six, then you can "undo" the multiplication and deduce that the number I'm thinking of is three.

That's because there is only one number that gives 6 when you multiply it by 2.

 

However, if I tell you I'm thinking of a number and, when you multiply it by zero, you get zero, you have no idea what number I'm thinking of because there are many numbers that, when multiplied by zero, give zero (in fact all of them do).

An one to one function has an inverse, but a many to one function cannot have an inverse,because the inverse isn't a properly defined function.

 

Saying this "Physicality allows for / by zero..." is just typing meaningless words.

Multiplication by zero isn't the equivalent of "equivalent of no operation being performed." that's the equivalent of adding zero.

 

"Physicality also suggest multiplication by 0 is not necessarily 0."

Nonsense.

If I take no apples from each of ten trees, how many apples do I take?

the answer is zero, even if you don't like it.

[conway]

John

 

It may be I am misunderstanding you. This is to say there is another way of stating all examples you gave.

 

It is possible that you were thinking of zero, therefore I could have guessed.

 

If I hold 5 apples and divide them by zero I still hold 5 apples

 

If I hold 5 apples and multiply them by zero I still hold 5 apples.

 

Nothing in physicality disappears or has no value. It is only that value is undefined.

[pzkpfw]

...

If I hold 5 apples and divide them by zero I still hold 5 apples

...

By that you seem to be defining (x / 0) = x

 

OK, so use that: that would mean (x / (y - z)) = x, where y = z (because given y = z, then y - z = 0)

 

By standard algebra, (x / (y - z)) = x gives (y - z) = (x / x) = 1

 

So now you have both (y - z) = 1 and (y - z) = 0.

 

Which is an example of the contradictions you can get by trying to define a value for division by zero. By that definition, you provide an "answer" for some cases, but you mess up a bunch of other cases. Math needs to be consistent.

 

Question: do you have MS Excel or similar available? (I'm going to suggest a simple "test").

[Unity+]

John

 

It may be I am misunderstanding you. This is to say there is another way of stating all examples you gave.

 

It is possible that you were thinking of zero, therefore I could have guessed.

 

If I hold 5 apples and divide them by zero I still hold 5 apples

 

If I hold 5 apples and multiply them by zero I still hold 5 apples.

 

Nothing in physicality disappears or has no value. It is only that value is undefined.

The analogy is completely off.

 

Multiplication is seen as grouping subsets of a particular object that have n amount of that particular object.

 

Let's take your example. We will define a subset A that contains 5 objects called apple. Now, in this case, we want to know the amount of apples if we have B subsets of 5 apples.

 

There, the cardinality of the set of groups of apples will be B.

 

A * 5 = {{apple,apple,apple,apple,apple},{apple,apple,apple,apple,apple},{apple,apple,apple,apple,apple},{apple,apple,apple,apple,apple},{apple,apple,apple,apple,apple}}

 

In this case, we have a total of 25 objects called apple in the set A*B. Now, let's consider what would happen if we multiplied by zero:

 

A*0 = {}

 

We have nothing in the set.

 

Now imagine if we did not know the cardinality of A.

 

A * 0 = {}

 

In this case of multiplication, we know that there are zero subsets, therefore it is trivial. However, look as I do division with A*B.

 

A*B / 0 = {{apple,apple,apple...},{apple,apple,apple...},{apple,apple,apple...}....}

 

It is an undefinable set because you literally can't tell how many apples we started out with our how many subsets of apples we had in the whole set in the first place. Therefore, it is undefinable.(I think this is right).

[conway]

pskpfw, unity+

 

Again assuming the following axiom, I can solve for any equation presented by either of you in your previous post

 

 

For any A in S, there exist Z1, and Z2, constituting A, so then any A in operation of multiplication is only representing Z1 or Z2, in any given equation. Allowing that Z1 for zero = undefined, and Z2 for zero = 1. Allowing that Z1 for A = A, and Z2 for A = A

Edited June 27, 2015 by conway

[Unity+]

pskpfw, unity+

 

Again assuming the following axiom, I can solve for any equation presented by either of you in your previous post

 

 

For any A in S, there exist Z1, and Z2, constituting A, so then any A in operation of multiplication is only representing Z1 or Z2, in any given equation. Allowing that Z1 for zero = undefined, and Z2 for zero = 1. Allowing that Z1 for A = A1, and Z2 for A = A2

I guess we can assume any axiom in any case, but I thought we were talking about real math.

 

EDIT: By real math I am talking about the general axioms that are accepted. If you want to discuss newly made axioms, I think you should read up on axioms and what is considered a consistent axiom. Developing axioms that are inconsistent doesn't form formal mathematical structures. I guess you can try it, out of curiosity, but you might as well be standing in the middle of no where because that is where axioms that are inconsistent will most likely lead you. However, don't let my banter stop you. It might be interesting(or not).

Edited June 27, 2015 by Unity+

[conway]

This axiom is in addition to all axioms I posted in #18. Which was from an article linked in #17 by Studiot.

To be honest I think that my use of the word "undefined" was very poor. So that if I may restate the following

 

Z1 for zero = 0

 

Z2 for zero = 1