Sep 6, 2011

Divide By Zero

This blog will now implode.

Mathematically it is said that dividing by zero is a logical impossibility. When you try to do the calculation on a calculator, it usually gives you an error or NaN (Not a Number) which is a result of the IEEE floating point standard. Furthermore, in the unlikely event you decide to divide by zero on paper or manually, you always end up with logical impossibility such as the idea that 1 = 0 or that any number can logically equal any other number.

In classical mathematics this isn’t possible. If we say that 1 equals 0 then what meaning does any number actually have if they are all equal to each other? Clearly two of something cannot equal one of something. However, I’d like to attempt a thought experiment.

In classical mathematics we say that 1 does not equal 2 because we treat them as unconnected entities with no common underlying basis. Let us demonstrate with two apples -

O O

O

So we have two apples and then underneath we have one apple (represented by the crude use of a capital O)

So how does one apple O equal two apples O O or vice versa?

I like to think of this in terms of quantum proofs instead of classical proofs. Just as in classical physics, things work very differently than on the quantum level where the same rules do not apply, we see a bit of clarity in a division by zero in this case when looked at through classical mathematics versus what I will call hyper-mathematics.

The underlying issue with zero is that in a function like [ y = 1/x ] as x approaches 0, y approaches negative infinity. This is a problem in and of itself because we really don’t like to think about the nature of infinity either. So in my mind, I believe the understanding comes specifically from the special set of numbers such as defining +∞ and –∞ respectively and also the duality of the opposing nature of +0 and –0

Is there such thing as a positive zero and a negative zero? Well, I suppose it would depend on whether we’re talking about the nature of zero to begin with. Zero is a representation of ∞ nothing or in this case –∞ , so I’d say that 0 whether it is +0 or –0 is inconsequential because it’s still 0 and 0 is interchangeable with -∞, while ∞ is an infinite positive of something. I suppose then we are talking about probabilities of something or nothing in infinite amounts and that the probability is actually both something and nothing at the same time, and thus leads us to declarations of 1 = 2, but moreover that anything is actually anything else and nothing at all in ∞ abundance.

The only thing I know about that exists and doesn’t simultaneously is in the realm of quantum physics where we have decoherence or the collapsing of dual wave into a single upon conscious observation. In this manner we say there are quantum computers where normal 1 and 0 become both simultaneous in a qubit configuration. Here is an inkling of something equaling something else at the same time in a manner that says it is both, and the determining factor is simply conscious observation. I’d say, then, that it’s not entirely impossible for 1 = 2 in the same manner that in a qubit 1 = 0 in a quantum state. But what does this really mean in the end?

For practicality reasons, just like we use classical physics (Newtonian and Einstein) for the large stuff, when we get down to the building blocks themselves, that no longer applies and we switch to quantum physics where 1 = 0 in some sort of dual existence. I’d say that classical mathematics is for counting apples, while hyper-math is for counting infinities and probabilities within each construct.

But, you may say that infinity is simply that… there cannot be more than one infinity, can there? Again, I suppose this depends entirely on our understanding of infinity and reality itself. For instance, we can say that reality is infinite because it extends forever, but then we’re talking about a dimensional plane of existence which is infinite but only a level, wherein other levels are separate infinities and may not intersect our own layer of reality which is itself an infinite plane. So we get an idea of having multiple infinities in the greater construct of reality, which in itself comprises infinity when taken together.

I suppose then that ∞ + ∞ = ∞(2) which can still be represented as merely ∞

The devil is in the context.

If we wanted to demonstrate the notion of multiple ∞ we simple grab four mirrors. Face two mirrors toward each other and do the same with the other two side by side with the first set. What we have, then is a demonstration of ∞(2) where two spaces are separately infinite. This is a basic demonstration just to get the gist of the concept.

Let’s take this a step further and look at holography.

A holographic plate contains a three dimensional information construct of the item that was effectively imaged, but if we cut the holographic plate in half, we do not have half of the image, but two complete sets of the information, and no matter how much we divide the holographic plate we will always end up with a complete informational scene, as if holography captures the information in an ∞ state and no matter how much we divide ∞(x) we end up with many infinities all intact. In effect, we aren’t dividing the information, but only our available viewport of that information. This is a lot like understanding the difference between ∞ and ∞(x) in context.

Information about any given point in the scene is recorded across the entire medium, rather than one fixed area as in a photograph. Cutting a hologram in half results in two complete representations of the original scene, each merely having a more limited viewport. Think of viewing a street outside your house through a 4ft x 4ft window, and then through a 2ft x 2ft window; You can see nearly all the same things through the 2ft window, but you can see more at once through the 4ft window.

This is where the application of dividing by zero and the many natures of ∞ make sense.

Zero, in this case is the complimentary opposite of an infinite. Where ∞ is taken as a positive something in infinite capacity while zero is an infinite nothing in capacity.

When we say that 1=X where X is every possible number to ∞ and even infinite nothing (zero) we can represent this with holography where the plate can be divided ∞ and still retain the information of the whole in each division. But I suspect this goes deeper because then we start to ask what happens when we divide all the way down to atomic, subatomic, or quantum level?

I suspect then that we begin to understand the nature of ∞ and zero a lot better than we do right now, and the overall implications for what we call an understanding of reality and multiple realities.

It would mean, then, that the very stuff everything is made of contains an infinite amount of information. It is a probabilistic infinity, and that brings us back to 1=2. Everything equals everything else and is based on fundamental configurations to attain a specific probability at a higher level, but even at a higher level all things contain the infinite information to comprise infinite anything else including itself. While we can divide the apple infinitely, and we get fractional apples in classical mathematics, in hyper-math we end up with multiple infinite instances of informational constructs, just like the holographic plate – and the slices of said apple represent the limited viewport effect of the whole.

In the end, the question remains: Is it mathematically possible to divide by zero in a logical manner? The answer, I suspect, depends entirely on whether you want to deal with the nature of reality and subdivisions of infinity in the process. For all practical applications, the answer is no for the same reasons we don’t teach quantum mechanics to our middle school children. However, for a more inclusive answer, we can say that yes it’s not only possible but gives us an incredibly deeper understanding of everything by doing so.

After all, wouldn’t the division of ∞ into ∞(x) really mean the same thing as dividing a holographic plate? In both instances, we end up with all the information and just a smaller viewport to each individually. I think that sums up our understanding of reality at the moment. We see a smaller viewport of the full ∞ represented as ∞(1) or if we take a multiverse theory into account ∞(x) where our viewport on reality is simply narrowed. Our notion of “real” numbers are actually the abstract concept in the grand scheme of things, because numbers represent fractional viewports of the infinite information as a whole.

Do we need to calculate that on our calculators? Probably not, because we’re dealing with abstract concepts that fly in the face of traditional mathematics. Will dividing by zero create some sort of rip in reality?

I don’t think so… this blog is still here, and so are you. I’d say then it’s pretty safe.

But don’t attempt it on a calculator… because it will likely break its tiny mind.