Dividing by zero

[Rkyeun]

When you get to a 0/0 you have to stop and switch to a higher math. If there is argument about what 0/0 equals, then you are not using the same math as your opponent. If you're just using arithmetic, it's time to call it quits. The problem doesn't work, and you don't have an answer.

But in higher maths, you get to look at where those 0's came from, what they might have been, and what the graph approaches. And in those cases the problem will tell you what 0/0 is. It's undefined in most systems because you don't have enough data to go to the higher math and solve it, you don't know which two graphs crosses where or who's under what derivative. All you have to go on is 0/0 which doesn't work in lower math.

[houserichichi]

Just division by zero alone means you're not even working in a field (or even an integral domain), let alone the element 0/0. When we allow zero divisors we're back to working with ring elements - something the average Joe never comes across. But if we're doing that, we have to watch what we mean by multiplication in the first place. Is it real multiplication, matrix multiplication, function multiplication, or any of the myriad of others one could concieve.

As far as I understand, however, the argument for this thread was the reals (or possibly complex) which are both fields (and so by definition have no zero divisors). Anyways, moot point now I suppose.

[wendy]

There are certian circumstances where one can treat 1/0 as having a specific value. I use these in my geometry dabblings from time to time.

For particular values of infinity, one can treat it as additive-distinct, or multiplicative-distinct, or power-distinct (ie distinct on x to x+y, x*y and x^y respectively.

The geometric infinities are at the moment being some concern to me.

The horizon is the 1/0 type infinities, but we're picking up traces of even deeper ones, such as the antipodal approaches.

W

[houserichichi]

That was actually the point I was trying to get across in another thread in the relativity board - there are cases where points at infinity are defined, but this is not the case in the algebra of the reals.

[zero742]

I just want to thank everyone who has contributed their thought, ideas, and genius to this topic...I'm in calc II rite now, and after reading through this topic, I have a new inspiration for paying attention in class. Anyways, just wanted to say thanks and I look forward to reading and enriching myself and my life through these wonderful forums.

[Peligroso]

0/0=X

This means it could equal any number, but not every number at the same time.
the best way of thinking about this is differentiation. Basically trying to find out 0/0 of a variable function to find the reate of change and 0/0 is the limit and it has a value. U can see this from first principles.

[wendy]

Sometimes you can treat 0/0 as a number. Sometimes one can derive limits for it.

For example, the matrix of n*n, each element sqrt(n), is a matrix of determinate of 0. It has a definite inverse in the form of 2/0 N, where N is of the form

N i.i = 2 ; N i.(i+1) = N (i+1),i = N 1,n = N n,1 = -1

any other value has N i.j = 0.

You can easily show this.

One can just as readily calculate by allowing x-> 0 other matrix inverses.

One must always keep in mind, that in the world of recriprocation, 1/0 maps onto 0, and as long as one does not broach this, it is as every definite as 1/6 or 1/71.

[RQ]

Sometimes you can treat 0/0 as a number. Sometimes one can derive limits for it.

For example, the matrix of n*n, each element sqrt(n), is a matrix of determinate of 0. It has a definite inverse in the form of 2/0 N, where N is of the form

N i.i = 2 ; N i.(i+1) = N (i+1),i = N 1,n = N n,1 = -1

any other value has N i.j = 0.

You can easily show this.

One can just as readily calculate by allowing x-> 0 other matrix inverses.

One must always keep in mind, that in the world of recriprocation, 1/0 maps onto 0, and as long as one does not broach this, it is as every definite as 1/6 or 1/71.

The matrix assumes axioms about division by 0 and the rule is based upon these axioms. Those axioms might hold unproven but they can't be proven either. I'm a bit rusty on matrices. Might you refresh me a bit?

[wendy]

Dear RQ

Sometimes division by zero just realises to a large number k. Even showing that k = 2k, does not invalidate the process, since here might be important the mantissa.

In the recriprocal of matrices, the first term is "2/0 *". This is an algebraic constant of the form of 1/0 * some k. It's not so much that the matrix goes to infinity, but that it is meaningful to write 1/0 as if it has meaning.

The question is 'what is proof'. You can readily demonstrate the validity of these things, but is that "proof". I don't know. You can show, from AB=0, that kA.B = 0, where A, B are matrices, and k is an element of R.

On the other hand, what happens, when A and B are elsewhere derived, and that AB = cI/2, where I is the matrix identity, and c is a constant of curvature. In euclidean geometry, c=0, but this does not mean that we are at liberty to replace A with kA or B with kB.

In this way, it is then meaningful to write 1/B = 2A/c, and evaluate it accordingly as B' -> B then 1/B' -> B in a wholy coherent and consistant way.

It is therefore true that we can for some instances of 1/0, associate a workable meaning to.

[houserichichi]

Another very brief and simple answer is that general matrices do not form an integral domain (with matrix addition and multiplication) and so division by zero is allowed at times. As far as the algebra is concerned it's all a matter of satisfying particular axioms - and most matrices don't (though there are special cases). "m x n" matrices (with m not equal to n) form a ring with unity at best and thus are allowed to have zero divisors - that's straight out of the rules of algebra.

[PWrong]

I always thought that an axiom didn't require proof, that it was simply a rule that we choose to follow because it makes sense. So in theory, RQ could invent a new algebra with different axioms, as long as he made it consistent.

Houserichichi, speaking of matrices, what do you think of my idea about powers of matrices, like A^B, on the tetration thread? Could it work?

[houserichichi]

That's bang on, axioms are "rules" we assume true without proof and build off of them.

http://robotics.caltech.edu/~jwb/courses/ME115/handouts/algebra.pdf

That's a little breakdown of the axioms of a few algebraic structures. It may prove useful the next time someone mentions this topic in conversation (because we all know how often that happens in real life).

(I'll check out the matrix products now!!!)

[wendy]

I never had any problems dealing with 1/0. You just have to mind which 0 is being talked about.

It doesn't magically happen that all the properties of numbers go at once. Properties disappear gradually. Sometimes you can distingish 1/0 - 1/0 = 1, sometimes you can distinguish 1/0 from 2/0, and sometimes you can tell the index (ie 1/0 from 1/(0*0)).

Sometimes you can also pick up moduli and mantissa (ie modulus over logrithms), so you can tell 4/0 from 5/0, but not 4/0 from 8/0.

But whatever it is, it isn't "open". You can't pick up a 1/0 from one problem and use it in another, unless you build a bridge between the two, and haul it across the bridge.

But in the main, i had no problems with 1/0.

[houserichichi]

You must be very comfortable working in rings and groups then. :wink:

[wendy]

I have heard of rings and groups, but i don't use them. Couldn't tell one from the other, really. Geometry is the go here.

[houserichichi]

Ahh, so we start with different axioms entirely then - therein lies the reason to your madness (or perhaps to mine). :wink:

[RQ]

I always thought that an axiom didn't require proof, that it was simply a rule that we choose to follow because it makes sense. So in theory, RQ could invent a new algebra with different axioms, as long as he made it consistent.

Houserichichi, speaking of matrices, what do you think of my idea about powers of matrices, like A^B, on the tetration thread? Could it work?

Seeing that mocking me is your favorite thing here, you should know that the axiom 1/2=1 can never be disproven nor proven unless by a physical application in nature. There are many number systems and we use the one that makes sense in nature.

As for my previous arguments, I've found out that 0/0 does not equal 0, and it's a very simple proof with the assumption that 0/0=0.

It's on the fact that the two equation x*0=0 and x/0=0 are in fact the same two equations but from a different approach:

x*0=5

x*0/0=5/0

Since 5/0=0 then

x*0=0

thus

5=0

I had a different way of disproving it + another one but I forgot. Hopefully the matter that division by 0 is undefined has been concluded.

[houserichichi]

Pwrong is right though, you literally CAN make your own system up and get division by zero to work, but you couldn't call it an "algebra" since those are actual things that follow basic rules. There are structures called rings where you have two binary operations (called the addition and the multiplication) where it's very acceptable to have a division by zero. In fact, only when you hit something as complex as an integral domain is division by zero not prohibited at all.

[RQ]

As for my previous arguments, I've found out that 0/0 does not equal 0, and it's a very simple proof with the assumption that 0/0=0.

It's on the fact that the two equation x*0=0 and x/0=0 are in fact the same two equations but from a different approach:

x*0=5

x*0/0=5/0

Since 5/0=0 then

x*0=0

thus

5=0

I had a different way of disproving it + another one but I forgot. Hopefully the matter that division by 0 is undefined has been concluded.

Never mind, this is false. This is like saying that 1=2 because 1*0=2*0

[Krickette]

My math teacher told us the 2=1 thing...He just said "don't divide by 0 or the world will explode" or something to the relative equivalent thereof...

[Rkyeun]

Dividing by zero doesn't make the world explode.
...it makes it implode.

See also: Black holes.
Black holes are 'funnel' shaped, meaning it's a hollow sphere that gets longer and thinner as it extrudes kataward. That area of hollowness grows increasingly small as the distance "from the center" (henceforth called X)approaches 0, but it never actually closes. And in a graph of Y=-1/X^2 we can see a fair approximation of a cross-section of a black hole. Note the lack of actual center, which is why astrophysics has chosen the mathematical concept of "singularity" to name a black hole.

In summation, don't divide by zero. You land in a black hole, where classical physics begins to break down because you can prove just about any absurd 5=0 thing you want.

This moment of silliness brought to you by insomnia, the letter X, the number 0, and Diet Cherry Vanilla Dr. Pepper with Lime.

[houserichichi]

Diet Cherry Vanilla Dr. Pepper with Lime

SWEEEEEEEET

[wendy]

One can divide by certian kinds of zero, in a meaningful way. These zeros are the inverses of the geometric radii. Note that we still have 0k = k, but something like a symmetry group like {3,6} has an implied radius, and one can calculate the diameters of figures derived from {3,6} in a consistant way.

The diameter of x3x6o (truncated {3,6}, a tiling of hexagons), is 3/0, while the tiling o3o6x (hexlat = {6,3}) is sqrt(3)/0. When one places the surfaces concentric, against a common symmetry, the two are different size (one has cells three times as big).

[houserichichi]

Sorry, I'm not even remotely familiar with your notation. What exactly do x3x6o and o3o6x mean? I looked them both up via google and came up empty handed - what is the common name I should be searching under?

When you say that the zero is the inverse of the geometric radius what exactly do you mean? The radius measured about which point, that is? I looked it up in your polygloss but wasn't able to find a reference on first glance.

[wendy]

The x3x6o notation is my inline version of the dynkin-symbol, x is @, o is o, and numerals are branches, thus x3x6o = @--3--@--6--o

x3x6o is the truncated {3,6}, which has hexagons (vertex-figures) + hexagons (truncated triangles). o3o6x = x6o3o is the hexlat, a tiling of regular hexagons {6,3}.

I distinguish between the different kinds of infinity: one of these is the radius of the geometric horizon. Since {3,6} is horizon-centred, it has this radius.

W

[Eric B]

Hi Wendy.
I discovered this board, and I am going to leave the Polycell e-mail list because I prefer board discussions to e-mail discussions.

Anyway, this problem illustrates what I was saying regarding apeirotopes (poytopes having an infinite number of sides). For those not aware of that discussion; I proposed renaming "hyperspheres" as "apeirotopes", a term currently used for flat d-1 dimensional surfaces that are the limit of a polytope as the number of facets approaches infinity. In this theory, an infinite straight ine can be seen as identical to a circle. How? Because the line is formed by taking a polygon, fixing the length of sides, and then increasing the number of sides to infinity. But on the other hand, if we instead fix the radius, and then increase the number of sides to infinity, the length of the sides shrinks to zero. Whet we end up with is the entire set of points (infinite) the elngth of the radius around the center point, and this is the definition of a circle. The straight line is really an "infinite magnification" of those infinitesimal points! At the end of each radius, is an identical point, which can hypotheticallybe "magnified" to in infinite line, divided into the original fixed length "sides, now lying at 180° to each other.

This is a paradox; which stems from messing around with infinities and zeroes (which is actually a type of infinity: infinitely small!)
So for the problem of 0/0 or 0⁰ leading to "1=2"; in a sense, that is actually true! When you are at infinity, then basically, the entire line of finite numbers collapses down to a single point, with all numbers equal! For ∞-n=1, and ∞-x=2; n=x=∞, so ∞-∞=1 and ∞-∞=2; hence 1=2. They become equal, because from the perspective of infinity, both 1 and 2 are both infinitely small (like 0), and thus equal!

This is just a paradox that stems from the nature of infinity.

[wendy]

One can quite easily demonstrate that 0 ^0 = 1, because this does not invoke division. One shows that n ^m = 1 [*n]_m, meaning you repeat the enclosed square brackets m times, eg

2**3 = 1 *2 *2 *2

Since zero is a legitimate count, the meaning of 0^0 is clearly 1 followed by 0 statements of *0, ie 1.

Making 0^0 anything else, would make any product interterminate, since

p = p * q^0 (count of 0 q in the product) then put q = 0, [legitimate], then put 0**0 = 5, then p = 5p.

Because you can derive a value of 0**0 without invoking a division, and this value is the same as any other product of 0 numbers, ie prod{}, then the value of 0^0 is exactly 1.

On the other hand, one can not assume that having 0^0 decided makes all dividion by zero work.

In the case of the geometric radius, one has the sequence 1/R, which chases a linear progression with R, and so it is meaningful to deal with 1/0 as a very large R. This is why we can get proper values with matrix division.

W