Division by Zero - Impossible?
68We are sometimes bothered by the question: 'what happens when you divide by zero?' What really stumps us, though, is not what the result is, it is how one goes about performing the action. How can one slice an apple in zero pieces, for example. Our minds can't get past that conundrum, so often the question is abandoned as absurd.
However, an answer may yet be found.
Mathematics are often used to form a model that describes reality. For example, much of theoretical physics exists in a math-only form. Math is used to develop complex theories long before they can be tested experimentally, and even to make such experiments possible.
Yet, sometimes, the process needs to work the opposite way.
Great advances in mathematics have been made by incorporating the principles of science into the discipline. Cartesian curves, Geometry, and Algebra all arose out of translating elements of reality into mathematical concepts. The entire field of calculus was developed so that existing theories of motion could be described in mathematical terms.
Division by zero is one of the last strongholds of the much-despised term "undefined". Yet, for this too, we might find a reality-based model that can help us apply a definition.
Consider a massive star. When that star reaches the end of its lifespan, it undergoes a gravitational collapse, in keeping with the dictates of Einstein's Relativity and the theories derived from it. If it possesses a mass within certain limits, it collapses beyond all measurable boundaries, beyond even the electromagnetic limits of its sub-atomic particles. It becomes a Black Hole.
By definition, a Black Hole is a massive body of infinite density. Now, bearing in mind that the collapsing star does not automatically become more massive just because it has collapsed in this certain way, how can it be that it has an infinite density?
Well, density is basically mass divided by volume. As the star becomes a Black Hole, it shrinks into a singularity, a point of zero volume. Thus, at that point, there is an infinite density, even though there is only a finite quantity of mass.
Ergo, take any mass, divide it into zero volume, produce a Black Hole. Or, in mathematical terms, M/V=D where V is 0 yields M/0=Infinity. Even if Black Holes don't actually exist, the math is still valid.
Division by zero is possible, if only in extreme circumstances, and it yields a definite, predictable, defined result: infinity.
Not only do we now know that division by zero is possible, but we have an answer too.
*Post-Script
Since originally publishing this piece, I've come across a number of "proofs" as to why a result of infinity cannot work, most of which involve some variant of showing how it can be used to make 1=2. It should be noted that every one of these so-called proofs is based on a fallacy. It is easy to miss; Einstein and Newton made the same mistake. The fallacy involves substituting an infinately large number for infinity. The problem is that any finite number, no matter how large, is still finite, and is therefore not the same as infinity at all.
One can use a similar 'error' in math to prove that any number equals any other number. For example, presume that x = 0.9999... Now, take 10x (9.999...) and subtract 1x: 9x = 9. Reduce to x: x = 1. So, by the most basic algebra, 1 = 0.999...
In short, it is not possible to calculate ∞/∞ without additional parameters (which would change ∞ to a set number); the result will always be both infinately large and infinitely small at the same time, but never 1, 2, or any other finite number.
Furthermore, the term 2∞ is meaningless. Think of it like multiplication by zero: 2 * 0 = 0. It is the same with infinity: 2∞ = ∞.
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Comments
Here's a few sources:
imagine.gsfc.nasa.gov/docs/science/know_l2/black_holes.html
archive.ncsa.uiuc.edu/Cyberia/NumRel/BlackHoleAnat.html
Other references include Stephen Hawking, Richard Feynman, and pretty much every Physics Prof in the world.
No, having infinite density would not alter the body's gravitational slope in the slightest - gravity is based on mass, and the mass remains unchanged. It merely takes up less space.
And, yes, you can't slice anything into zero pieces. You can't slice anything into -1 pieces either, but that number works just fine in division. Just because you can't apply a certain number to a specific situation doesn't make the number or a formula that it's involved in automatically invalid. Show me how to cut an apple negative one times, and we'll discuss it further. Else, by your reasoning, the number -1 can't even exist, let alone function in division.
This isn't a matter of 'cutting'; but you are right in that our languague confuses us. We tend to think in terms of analogy, and the old 'slicing the apple' is a remnant of First Grade math.
Division by zero is neither a mathematical flaw or a linguistic garbage dump. It is our own mental blocks that keep us from reaching understanding. Once past these, the problem becomes quite simple.
Here's another example: take a standard x:y graph. Using the "rise over run" method, draw a line with an infinite slope. See? You can't ever quite reach infinity until the denominator (x) falls to zero.
If you try to think about this in terms of physics, you run into the exact same problem you did with mathematics.
It is impossible to have infinite density with a finite amount of energy. This is not possible in any volum, even zero. However, when you compress any amount of mass into zero volume, you get infinite density. So that's a blaring contradiction.
Lets say you compress a finite amount of energy into zero volume, you get infinite density. And yet the only possible way to get infinite density in any volume -even zero- is to have infinite energy.
So you compress finite energy into zero volume and get infinite energy? That right there is thecreation of energy which, according to all known branches of theoretical physics, is impossible.
What you also failed to mention is that your model of a singularity existing in zero volume is only one of many models, and is the only one among them in which it is possible to contain mass in zero volume.
The model you describe is the singularity as described by General Relativity.
But as I said, there are other models, and the two most popular are the singularity as described by Quantum Mechanics, and the singularity as described by String Theory. And in both Quantum Mechanics and String Theory, it is impossible to contain energy in zero volume.
Now out of all the possible models of a singularity ever thought of, only one of those models actually allows for energy to be contained in zero volume. And the really crazy thing is, the one that does allow for such a thing comes from the most antiquated of these theories.
Now obviously, out of all possible models of a singularity both known and not yet thought up, only one of them can be right. All others will be wrong.
I agree with much of what you said, but, to be fair, QT predates GR by several years. Heck, it beat SR by two years: Plank published in 1903. What's more, after a century of work, QT is still full of holes big enough to fit a Proton through, whereas GR has passed every single test.
And don't even get me started on String Theory... oh, boy. It seems they'll call any whackbrained idea 'science' these days, even the untestable ones.
Finally, infinite energy only enters the picture if you're dealing with QT equations - which are flawed. There is no rational reason why the total energy would change at all, regardless of the change in volume - Thermodynamics is pretty clear on that.
String Theory may be untestable, but my point is still valid. And you’re right, the precursors to modern Quantum Mechanics did predate General Relativity.This is because, unlike what you said, General Relativity is actually just as flawed as Quantum Mechanics. General Relativity has passed all its tests but one, it cannot be used to predict events on a quantum scale, Quantum Mechanics can. But Quantum Mechanics’s flaw is exactly the same as General Relativity’s. It can’t be used to predict events on a larger scale.And within its world of events on the quantum scale, Quantum Mechanics not only has solid mathematical foundations, but can be tested, and has passed its tests. It only breaks down on the larger scale, just as General Relativity breaks down on the quantum scale.So both of these theories are flawed in the same way, they can predict events on one scale but not another. But if one tries to use both at once, both collapse into nonsense and contradiction. The perfect example is that of General Relativity allowing energy to exist in zero volume, while Quantum Mechanics does not.So either, both theories are wrong, or both theories are incomplete. Actually that’s what String Theory was, an attempt -however desperate it may have been- to reconcile the flaws in each theory which come about by use of the other.Each theory produces fundamental flaws in the other. The perfect example is the point of division by zero, which appears to be simultaneously possible and impossible -which is impossible-.
Sorry about that last post's structure. I tried pasting it from notepad, and apparently that deleted the line breaks and the spaces after the periods.
I know the feeling - I have a few posts floating around with all kinds of crazy fonts due to pasting...
I also appreciate what you're saying about scales. It seems to me that 'incomplete' is the better term in both cases. There's so much that's identical between the Quantum and Universal levels, it seems there has to be a link, yet both theories seem to hit a wall.
My thought - not even a hypothesis - is that there's some major disconnect; something we just aren't seeing. I suspect a big part of what's missing is that our understanding of the relationship between energy and matter is largely assumptive, and probably way off base. But I just can't put my finger on the problem...
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BartholomewKlick says:
6 months ago
Can you cite the source that tells you that a black hole has infinite density?
If a black hole had infinite density, wouldn't it also have an infinite gravitational curve? Thus, wouldn't everything have been sucked into it long ago?
The way I've always seen it, division by zero is impossible because you can't slice an object into peices and have the result "there are no peices of the object." Division by zero isn't a conundrum because of some flaw in math, it is a bit of nonsense yielded sayable through the power of language, yet just as innane as, say, "The brown dog submarine plunged romeo five."