Division by zero,seems flawed

[Secret]

What i mean to say is that , in reality '0' does not exist ; because '0' means absence or non-existence .

Instead 0+ or 0- exists in reality . However small they may be but they have some value of their existence .

Dividing a number with 0+ or 0- is possible ; though the result will be infinity plus or minus . That is a different issue .

But dividing a number with 0 is just not possible . This will not make any sense also .

If it is considered that , l 0+ l = l 0- l ; then N/( l 0+ l ) = N/( l 0- l ) .

Here l x l means mod x and N is any number .

Even in such scenario, |0+|=|0-|=/=|0|=/=0
Thus N/|0+|=|N/0-| does not imply N/|0|

Thus N/0 is still undefined

Your application suggested above seemed to have something to do with surreal numbers, however I do not understand the specifics, thus I cannot comment any further on your suggestion without getting things wrong

http://en.wikipedia.org/wiki/Surreal_number

@Alphanumeric:
The hypothetical number Q you mentioned in the aside of your proof seemed to have something more pathological than just 0=1. I'll post the details of my investigation later when I have time

Note that I don't mean your proof is wrong, but that your proof is right and the "What if" scenario for the properties of the number Q is much nastier than what is presented in the post

 

[AlphaNumeric]

Of course, once you open that door even a tiny bit then all manner of pathological behaviour comes rushing in. Once you can 'prove' something contradictory then you can prove anything. It's known as the principle of explosion.

 

[Secret]

(This is the promised analysis that have been delayed from posting due to busy uni life)

Pathological properties introduced if n/0 is allowed​

It has been previously shown and proved that the reals (and complex numbers) does not fundamentally allow division by zero

For the purpose of the illustration of the pathological "can of worms" that division by zero can introduce, however, lets give the following definition:

Let q be the only element ∉ C such that
q x 0 = 1

Using the definition of multiplicative inverse and Def 1

For any number, a is a multiplicative inverse of b iff axb=bxa=1 and a is unique

q should be an inverse of 0
Therefore q[sup]-1[/sup]=0

Other definitions and results that would be used in the illustration:

A number a is an multiplicative identity iff axn=n and nxn[sup]-1[/sup]=a where n is any number

Property of 0 in the set C: 0xn=0 for any n∈C

Property of 0 in the set C: 0xn=0 for any n∈C

Commutative law for complex numbers: axb=bxa, a+b=b+a

Property 1: Multiplication of q is non associative
Assume we are given the following expression and were asked to evaluate it
qx0xn

If we consider
(qx0)xn
=1xn (Def 1)
=n (Def 3)

However if we consider this instead:
qx(0xn)
=qx0 (result 1)
=1 (Def 1)

Therefore multiplication involving q is non associative

we observed that
(qx0)xn=n but qx(0xn)=1, therefore we can apply a trick, inspired from the property of cross products

One property of the cross product: axb=-bxa

and have the following definition

Property of the multiplication of q:
(qx0)xn=nx(qx(oxn))

Property 2: Multiplication of q is non distributive
Consider
qx(0+0)
=qx(0x2)
=qx0 (Result 1)
=1 (Def 1)

Now consider
qx(0+0)
=qx0+qx0
=2
=/=qx0

Therefore multiplication of q is non distributive

Property 2 also highlighted the most disturbing property of division by zero (which compound with the observations in the the context of limits, result in it being undefined)

Property 3: "Spontaneous generation/Trivial Ring property"
Consider the following case again
qx0
By the property of the addition of real numbers and Result 1
qx0=qx(0+0+0+0+0+...)

Property 2 showed that
qx0=/=qx(0+0+0+0+0+...)
In addition
qx(0+0+0+0+0=...) = qx0+qx0+qx0+qx0+qx0+... = ???

Thus if the results of both sides are regarded as the same, then this implies the following disturbing fact
0=1=2=3=4=5=6=7=...
I.e. The entire complex numbers collapsed into the trivial ring (discussed earlier in another thread)

(This also explained why division by zero memes were often associated with black holes)

Conclusion: In order to define a sensible number system for division by zero, the following hurdles must be overcome:
1. Evaluate qx1, q+0, qxq, q+1
2. Fix property 3
3. Address and solve property 2 in detail
4. Proof or disprove that qxn=nxq (n∈C)

Again I must emphasize it is undefined anywhere in the reals, complex or possibly the quarternions (however I don't really have get into much detail with that yet)

 

[wellwisher]

Math is useful for many things but some of the basic procedures can lead to conceptual problems, when trying to find an analogy within reality. For example, 1/(1/2) =2. This violates the conservation of energy and implies perpetual motion. For example, if we start with one battery and divide by a half, aI will now have two batteries. The world's energy crisis is solved by math.

Division by zero also allows one to play the same conceptual trick in reality. If we divided the battery by zero we get infinite batteries. If we assume the conservation of energy, the dividing by zero would require infinite energy and therefore may not be possible within the physical reality of a finite universe.

 

[Secret]

Math is useful for many things but some of the basic procedures can lead to conceptual problems, when trying to find an analogy within reality. For example, 1/(1/2) =2. This violates the conservation of energy and implies perpetual motion. For example, if we start with one battery and divide by a half, aI will now have two batteries. The world's energy crisis is solved by math.

Division by zero also allows one to play the same conceptual trick in reality. If we divided the battery by zero we get infinite batteries. If we assume the conservation of energy, the dividing by zero would require infinite energy and therefore may not be possible within the physical reality of a finite universe.

@1st block
This explained why when doing maths on physical systems, one must be careful on the conditions they have, as some of the operations are simply forbidden by the physical conditions (i.e. there is no amount<0, at best you can only say the change in amount <0 (i.e. the amount decreses))

@2nd block
Technically you cannot say division by zero = infinity as if you apply such reasoning, you can say e.g. 2/0=1/0=3/0=infinity
Now assume if you can treat them as numbers then you will end up with infinityx0=1,2,3..... (because the definition of division/multiplicative inverse is a UNIQUE number a such that axb=1) which not only the rule of multiplicative inverse is violated, but also that 0xn=0 for any n in complex numbers is also violated. (Even when you say infinity is not a number, the multiplicative inverse rule is still violated)

Physically, as many of mentioned before near the beginning of this thread, divide some objects by zero means you are are not actually carrying out the division, thus the result is undefined (e.g. how can a basket of apples be distributed evenly among no people? there's not even a person to receive an apple! thus it makes no sense to say how many apples do each people get)

 

[hansda]

Even in such scenario, |0+|=|0-|=/=|0|=/=0
Thus N/|0+|=|N/0-| does not imply N/|0|

Thus N/0 is still undefined

Your application suggested above seemed to have something to do with surreal numbers, however I do not understand the specifics, thus I cannot comment any further on your suggestion without getting things wrong

http://en.wikipedia.org/wiki/Surreal_number

I think in this case , we should first understand 0 (zero) . Then only N/0 can be understood .

So , what is zero (0) ?

 

[Prof.Layman]

I think any new math that explains division by zero would have to show that 0/0=1. I read once a long time ago that division by zero is the biggest problem with theoretical physics today, so when my math teacher derived the tangent line to a circle something really caught my eye that I won't forget. They started out saying that h was the slope of the curve from two different points. Then they said that as the limit of h approuches zero that would give the tangent line, the two points on the curve become the same point. But, when this happens h is zero, and the h is canceled out from the top and bottom of the equation because before they said it was the limit. But then the tangent line is when h is zero, so then how did they just cancel out? Does writting the word limit in front of an equation change operation by zero? I didn't see how it affected division by zero other than just saying that it was okay in that instance. So then 0/0 would have to be able to be canceled out just like any other a/a, so then 0/0 would be one unless it just happened to be some kind of fluke where it just so happens in that instance. But if writing limit doesn't change rules of operations besides allowing canceling of zero's then why couldn't a/a just equal one for all reals? It seems to work in that instance but not in others. It would be like saying, well if I have no apples then I don't divide them one time...

 

[Secret]

I think in this case , we should first understand 0 (zero) . Then only N/0 can be understood .

So , what is zero (0) ?

I think I'll better google it cause I don't really know much about the specific properties of zero, other then it can be used to represent the magnitude of a number (e.g. 0.00001 vs 100000), an additive identity (i.e. 0+a=a) and that 0xn=0 for all complex numbers n

I think any new math that explains division by zero would have to show that 0/0=1. I read once a long time ago that division by zero is the biggest problem with theoretical physics today, so when my math teacher derived the tangent line to a circle something really caught my eye that I won't forget. They started out saying that h was the slope of the curve from two different points. Then they said that as the limit of h approuches zero that would give the tangent line, the two points on the curve become the same point. But, when this happens h is zero, and the h is canceled out from the top and bottom of the equation because before they said it was the limit. But then the tangent line is when h is zero, so then how did they just cancel out? Does writting the word limit in front of an equation change operation by zero? I didn't see how it affected division by zero other than just saying that it was okay in that instance. So then 0/0 would have to be able to be canceled out just like any other a/a, so then 0/0 would be one unless it just happened to be some kind of fluke where it just so happens in that instance. But if writing limit doesn't change rules of operations besides allowing canceling of zero's then why couldn't a/a just equal one for all reals? It seems to work in that instance but not in others. It would be like saying, well if I have no apples then I don't divide them one time...

When using limits

$$\lim_{h \to a} f(h)=L$$
e.g. $$\lim_{h \to 0} \frac hh=1$$

It means when h approaches arbitrary close to a/as close to a as you like, the value equals L. Note that h does not actually get to a, just as close as you want, therefore using the example above, h=/=0, thus you can cancel them out

f(a) has no relationship with $$\lim_{h \to a} f(h)$$, i.e. f(a) does not need to be = $$\lim_{h \to a} f(h)$$ (graphically it means as you approach a from both sides of the function f(x), you will eventually reach a certain value. However the actual value f(a) may be a dot above or below the point where the lines joined, therefore creating a gap), if they are equal, it means the function is continuous at a (i.e. there is no gap, jump or hole in f(x) at x=a)

0/0 cannot be =1 because 0xn=0, thus any complex number n can satisfy this equation. Division is an inverse of multiplication, thus it should undo what multiplication does. Therefore there should be only one possible number n that can satisfy nxm=1. We then say n is the multiplicative inverse of m. Clearly this is violated in 0/0 as 0xn=0 for any complex number n

 

[Prof.Layman]

But by definition isn't the tangent line the line that only intersects a curve at one point? How does it do this if the lim of h approuching 0 is then said to be when h is actually zero? Wouldn't this mean that the current definition of the tangent line would prove to be impossilbe and that it wouldn't be able to only intersect at a single point?

I don't see how zero times any number equalling zero proves anything. You could still put 0/0 in for n then say it is one and then get zero as a result. But if you had 1/0 as n then you would get 1 as a result, the zero's would cancel and give you 1 times 1. So then the statement 0xn=0 would not be true, but 1/0 does not satisfy any currently defined number and does not quallify for being a n since it would be undefined.

I think a system that involded a new way to deal with zero's effectivly would have to define n/0, and in this system we couldn't say that anything times zero is equal to zero. That could be part of the reason it doesn't work, because somehow it would have to explain how you could get the definition of tangent without using limits. In other words it would have to put out something that has been shown or verified to work everytime. If you didn't get the same result of the tangent line it would have to be wrong.

Scarry thing is that anytime you are dealing with unknowns they could be equal to zero at any time, and then you would just treat them as though they are any other number even though they could have been equal to zero at one time or another or the whole time you worked the equations...

 

[hansda]

I think I'll better google it cause I don't really know much about the specific properties of zero, other then it can be used to represent the magnitude of a number (e.g. 0.00001 vs 100000), an additive identity (i.e. 0+a=a) and that 0xn=0 for all complex numbers n

Following websites are interesting for 'division by zero' , 'zero' and 'division' .


http://en.wikipedia.org/wiki/Division_by_zero

http://en.wikipedia.org/wiki/0_(number)

http://en.wikipedia.org/wiki/Division_(mathematics)


Let us consider the operation of 'division' . What does division mean ? Division means dividing something into equal parts . So , N/m is less than N as long as m is greater than 1 .

When m is 1 , N is equal to N/m.

When m is less than 1 ; N/m is greater than N . Here actually , physically division becomes multiplication ; thats why N/m is greater than N . Here multiplication factor is 1/m , which is greater than 1 .

N/p is greater than N/m ; where p , m both are less than 1 and p is less than m . Going by this logic N/0 should be something greater than N/0+ . As N/0+ is infinity . So, N/0 also should be considered as infinity ; as something greater than infinity is infinity only .

 

[Prof.Layman]

Even if we did assume that n/0 was infinity, what better off would we be? We would have the same problem have having numbers that we could no longer deal with. So then it seems we would have to have multiplicitive rules for infinity in order to also have multiplicitive rules of zero.

I have come across one theory that can add/subtract infinities. It was called infinite sums. Basically you just write out the series of the reals involved, then you find a patern to those numbers. So if you said all whole numbers, then you would write out 1, 2, 3, 4, 5... Then you could write out all even numbers 2, 4, 6, 8, 10... Then you could add them down to get 3, 6, 9, 12, 15... Then by looking at the patern of the numbers you got you figure that line is every mulitple of 3. So then all whole numbers plus all even numbers would be all multiples of 3. For some reason this didn't become a valid mathmatically theory, but I am not sure exactly why. Some higher mathmaticas to me can seem even more arbitruary than this one. I think it may be because there was nothing to test it on, and the only thing I know of where it would be useful is string theory. But then, string theory can not be tested.

 

[hansda]

Even if we did assume that n/0 was infinity, what better off would we be? We would have the same problem have having numbers that we could no longer deal with. So then it seems we would have to have multiplicitive rules for infinity in order to also have multiplicitive rules of zero.

I have come across one theory that can add/subtract infinities. It was called infinite sums. Basically you just write out the series of the reals involved, then you find a patern to those numbers. So if you said all whole numbers, then you would write out 1, 2, 3, 4, 5... Then you could write out all even numbers 2, 4, 6, 8, 10... Then you could add them down to get 3, 6, 9, 12, 15... Then by looking at the patern of the numbers you got you figure that line is every mulitple of 3. So then all whole numbers plus all even numbers would be all multiples of 3. For some reason this didn't become a valid mathmatically theory, but I am not sure exactly why. Some higher mathmaticas to me can seem even more arbitruary than this one. I think it may be because there was nothing to test it on, and the only thing I know of where it would be useful is string theory. But then, string theory can not be tested.

Let us compare N/0 with N/0+ .

N/0+ = infinity . There is no doubt about it . So , we can write the following :

N = 0+ + 0+ + 0+ + ... upto infinity number of times 0+ .

But if we write, 0 + 0 + 0 + ... upto infinity number of times 0 ; the result will still remain 0 and the result will not add upto N , like the earlier case of N/0+.

So, we can write N in terms of 0 as , N = 0 * infinity + N . Here N remains as remainder , though N is greater than 0 .

Thus , N/0 is something greater than N/0+ .

 

[Prof.Layman]

Kind of reminds me of the 0.999... = 1 debates. But then came the question of can you denote the next number down from 1 as being 0.999...8. An infinite amount of numbers with a different value on the end of it. They all seemed to agree that it wasn't a valid form of notation, but they "prove" 0.999... = 1 by multiplying it by 10 and then subtracting all the 0.999... from it and then ruducing it to one.

It is kind of like what I did by adding different values of infinity. I was reading Guth's paper on inflation the other day and he says the reason why this is not valid mathmatics is because you could write all the values in any order you wanted to. If you write them in a different order than you get a different relation to those numbers. So then it would seem like the mathmatics that proves 0.999...=1 would work because we know that it is just going to be all nines or you could say that any number that changes the value of each digit would always change that value of the digits in the same order.

Mathmatics seems plagued with a problem with dealing with certain types of remainders, so I think you may be on to something. Say you divide 10 by 3. You get 3.333... Now multiply that number by 3 you get 9.999... You just did multiplication and division by the same number but you ended up getting a slightly different answer. So by saying that 9.999... = 10 you are simply correcting the muliplication/division errors brought about by the way remainders deal with numbers that do not divide evenly.

 

[arfa brane]

In modular arithmetic, the integers mod 4 has four elements: {0, 1, 2, 3}.

Under multiplication, we have: 0x0 = 0 (mod 4), 1x1 = 1 (mod 4), 2x2 = 0 (mod 4), 3x3 = 1 (mod 4).

So therefore: 0 ≡ 2 ≡ 0x2[sup]-1[/sup] (mod 4). So 2 is a 'zero divisor' in Z[sub]4[/sub].
But 0 doesn't have an inverse (i.e. 1/0 is not defined, so nor is 0[sup]-1[/sup]), and since 2 ≡ -2 (mod 4), it also says -2 ≡ 0x2[sup]-1[/sup] (mod 4).
Hence 2 is its own multiplicative and additive inverse in Z[sub]4[/sub].

 

[hansda]

0 and 0+ are ideal terms of mathematics. If we consider practical physics, 0+ can be considered as smallest unit for any variable. For example if we consider distance as variable, Planck's distance (LP) is the smallest distance and there is no smaller distance possible. So for distance, LP can be considered as 0+ . Here if we consider N as a finite distance , N/0+ = N/LP = some finite value(x) . So N = x*LP .

N also can be writen as N = 0*x + N or N = 0*infinity + N .

N/0 , if we write this as a*b + c ( the equation form of division ,where a=0; b=N/0 and remainder c=N )

Here we can see that N/0+ is a finite value but N/0 is infinity .

 

[hansda]

I think N/0 is incomplete division.

Let us consider the operation of division in the form of subtraction.

So, if we write N = a*b + c . Then (N-c) - b - b - b - ... (a times - b) = 0 .

But in the case of N/0 ; N - 0 - 0 - 0 - ... ( infinity times - 0 ) = N ( and not zero ) .

So, perhaps i think , N/0 is incomplete .

 

[Secret]

Despite the attempt in avoiding infinities and 1=2=3=4=... by simply left out the RHS of the non distributive property of D (which I still yet to find a solution to it), a new paradox had arose when attempt in division (despite applying the unique properties of the non associative multiplications in D)

Result derived from the division section
q^-m=0 , m>0 (sensible according to the definition)

1=q^(m-1) is not despite it might still make sense saying positive powers of q=1 and negative powers of q=0

BUT

Let m=2
you will end up with

q=1

Which contradict with the definition near the beginning (q=/=1)

I doubt this can be resolved

Given all these 3 weeks of failed attempt...

Is there a mathematical proof that for any conceivable number system, division by zero (including both n/0 and 0/0) MUST result in paradox???
If the above is false, then at least which axioms need to be broken for a number system to be divisible for all members within it
Is it impossible to have any arbitrary number system where every element has both an additive inverse and multiplicative inverse? If yes how to prove it?

So far in mathmatics the closest example is Riemann sphere, with 1/0=complex infinity. However 0/0 is still undefined even in such number system

There is also wheel algebra, but the division has two types (left and right) and is defined very differently from the usual one

View attachment 5684 (summary of my attempt in building D)

Other than that, it seems D obey nearly all the index laws (including that q^0=1)
The none associative property can be summarized using those three rules, which allow multiplication to be handled. Of those properties, (q.0).n=(q.n).0 is a surprise to me
I'm still at lost of dealing with the nasty non distributive property of D, thus I cannot define what is e.g. 1/(q+1)
My approach on the division is inspired by the division in complex numbers, which seemed to work quite well until q^-m=0 pops up
NB D is the following set (C,+,*,q) which I am trying to build
NB2 Note that distributive law breaks down only when said distribution will produce a q.0 pair (although this is precisely how division by zero collapse a number system into the trivial ring if distributive law is assumed to hold) and for the associative law of multiplication, breaks down partially only when (q.n).0 and q.(n.0) with n=/=q (anything within the brackets and the () as one object . term outside are commutative, thus can be rearrange in any order without affecting the result) and can be rationalized by (q.n).0=n.((q.(n.0))) for n=/=q

 

[Prof.Layman]

Of course, once you open that door even a tiny bit then all manner of pathological behaviour comes rushing in. Once you can 'prove' something contradictory then you can prove anything. It's known as the principle of explosion.

Someone could argue that this very statement in itself could fall into a catagory that is a result of the same underlying principle, but I won't, or have I already?

 

[Secret]

View attachment 5727

Sorry you are wrong. This is exactly what I did for 3 weeks in an attempt to construct a division by zero number system to no avail.

It seems in order to divide by zero, one must kill some of the field axioms which causes 0n=0 to result.
But then you end up something that is not a field, as the physics forum mentioned

View attachment 5728

Interestingly, (at least according to my 3 week analysis) the non associative property of a number system including division by zero with trying to preserve the field axioms is not that difficult to fix, with only q as the special case (Only 3 multiplication rules is required when manipulating 3 terms, with (q.0).n=(q.n).0 being the most surprising one which can make the multiplication less difficult to deal with than expected), until you start messing with division defined in a way similar to division in complex numbers, which end up with the killer q=1 contradiction. Still it is mush better than the non distributive properties of such hypothetical number system which I'm still at loss of fixing it

Someone could argue that this very statement in itself could fall into a catagory that is a result of the same underlying principle, but I won't, or have I already?

What principle? I don't think AN's statement is itself a result of the "Principle of Explosion", if that's what you mean

In modular arithmetic, the integers mod 4 has four elements: {0, 1, 2, 3}.

Under multiplication, we have: 0x0 = 0 (mod 4), 1x1 = 1 (mod 4), 2x2 = 0 (mod 4), 3x3 = 1 (mod 4).

So therefore: 0 ≡ 2 ≡ 0x2[sup]-1[/sup] (mod 4). So 2 is a 'zero divisor' in Z[sub]4[/sub].
But 0 doesn't have an inverse (i.e. 1/0 is not defined, so nor is 0[sup]-1[/sup]), and since 2 ≡ -2 (mod 4), it also says -2 ≡ 0x2[sup]-1[/sup] (mod 4).
Hence 2 is its own multiplicative and additive inverse in Z[sub]4[/sub].

Thanks! It intuitively explains what a "zero divisor" means, which confuse me since the first day I came across it some months ago

0 and 0+ are ideal terms of mathematics. If we consider practical physics, 0+ can be considered as smallest unit for any variable. For example if we consider distance as variable, Planck's distance (LP) is the smallest distance and there is no smaller distance possible. So for distance, LP can be considered as 0+ . Here if we consider N as a finite distance , N/0+ = N/LP = some finite value(x) . So N = x*LP .

N also can be writen as N = 0*x + N or N = 0*infinity + N .

N/0 , if we write this as a*b + c ( the equation form of division ,where a=0; b=N/0 and remainder c=N )

Here we can see that N/0+ is a finite value but N/0 is infinity .

I think N/0 is incomplete division.

Let us consider the operation of division in the form of subtraction.

So, if we write N = a*b + c . Then (N-c) - b - b - b - ... (a times - b) = 0 .

But in the case of N/0 ; N - 0 - 0 - 0 - ... ( infinity times - 0 ) = N ( and not zero ) .

So, perhaps i think , N/0 is incomplete .

0*infinity is undefined. I'll let the others to explain the details to you since I'm not good at tackling infinities, thus you cannot say such thing
Infinity is not a complex number. In the context of surreal and hyperreal numbers, arithmetic can be done on infinity as ordinal numbers, which I only briefly read about and don't really understand the details except the arithmetic is completely different from the more familiar complex numbers, and require careful treatment

You can still say N=0*x+N since for any x, 0x=0 (Even in your context that x=N/LP where N is an arbitrary complex number, x is still a complex number thus basically the equation reduces to N=N, which does not tell us anything useful)

I've also apply the second approach when I was still a 7th grader, but N/0=n (n is arbitrary) with reminder N does not do anything useful at all
After learning that division is multiplication by a multiplicative inverse, which requires the inverse to be unique, I understand that a/b cannot be any arbitrary n, thus the above approach does little help in the division by zero issue

P.S. Note that the two biochem.jpg are not identical, I'm just too lazy to use another name