Zero divided by zero?

What is the value of 0/0? At first glance you might think that this actually has the value of 0, considering that if zero is divided by anything it is still zero! However, this isn't actually really true as you will soon find out.

Consider the basic algebraic principles: they state that if one number or expression is equal to another, then you may apply basic arithmetic operations to one side as long as you mirror them on the other. For example, the expression 7=14/2 is mathematically true. Multiplying both sides by 2 will give you 7x2=14 which of course is also true according to our number system. More generally, if a=b then ac=bc as well.

We can also apply this principle to the expression 0/0. Say, we are assuming that we do not know the value of 0/0, so we will set the expression equal to the letter X. So now X=0/0. Applying the principle that multiplying zero by both sides will result in an equal expression, we do so, giving us Xx0=0. But wait a minute; the value of any number multiplied by 0 is equal to 0, so can't X take on an infiinte set of values, not just 0? Strange as it may seem this is mathematically true, however we can't simply leave the definition of 0/0 as "all real or imaginary numbers", since this goes against the rules for forming legitimate funtions.

If we define the function f(X)=X/0, we should be able to set Y=X/0. However, this means that again Yx0=X, which as we have seen means Y is equal to any possible number. This cannot happen however, since the function isn't valid. It is not a many-to-one or one-to-one function. It is in fact a one-to-many function meaning that in our system of mathematics it is undefined due to the fact that for the value X=0, we get more than 1 equally likely result. So, in actual fact 0/0 is not 0, it is undefined.

There are a few strange things to note however, about multiplying and dividing by 0. The first of these things is that if Y=X/0, then if X takes on any value other than 0, the resulting value for Y is infinity.

So for example, if X=3, then Y=3/0, which of course is infinity. This becomes stranger however when we take Y as infinity in terms of the algebra.

if ∞=3/0

∞x0=3

and

0=3/∞

Now, that last statement makes sense: if 3 is divided infinitely, then it will approach 0. This at least seems feasible in terms of convergence to a limit. However, what about the statment ∞x0=3. Haven't we already assumed that any number multiplied by 0 is equal to 0? Infinity however multiplied by 0 seems to be equal to an infinite number of differnent numbers, as with the value of 0/0. So does this not mean that since both expressions are equal to the same vast range of different values, we can set ∞x0=X, where X comes from the expression Xx0=0, so X=0/0 and has a range of different values, just as ∞x0 has. However, since both expressions may take on a range of different values, then at the same time as ∞x0 is equal to 3 in one case, X may be equal to any number, for example 4, in the other. We then come out with the strange result 3=4. Now, this proves at least that many to one functions cannot exist at the same time as our normal number system, since strange results occur. The only solution to the problem is to set 0=∞. Now everything begins to seem more rational. By saying this we can now say that ∞=3/0 is not valid in the first place, since any number is 1 divided by itself, and that value only. So the only acceptable value for ∞=X/0 is X=0 (or X=∞ since we are now assuming 0=∞). This now becomes a much simpler 1 to 1 function, which is acceptable in mathematics.

Our other problem of Xx0=0 now becomes one which is much easier to solve. If 0=∞ then Xx∞=0 and the only value that X can now take is 0. This also means however that strangely ∞^2=0.

In summary, the strangeness of the whole dividing by zero thing, seems to have a logical solution: that 0 and infinity are one and the same. If you have read Godel, Escher, Bach by Douglas Hofstadter, you will be familiar with the term strange loop - and this is one. A strange loop is essentially a figure which comes back to its original value or state, as you get further and further away from the original: a strange sort of paradox. It is a weird concept, but one which perhaps could be used to explain why the universe seems to curve in a strange way I have realised: since it has been said that the universe's conept of space is undefinable in physical terms since if you travel infinitely far away from the place you are at then after an infinite distance you will come back to position 0 (or back to where you started). But of course it would be impossible for us to ever reach that infinite distance, since we would need to be able to travel at an infinite speed, and one thing is for certain: the speed of light is the limit.

Drawing Hands by M.C.Escher - a drawing which embodies strange loops

Relative Matters

While reading about the amazing NASA New Horizons Space Probe (which is due to arrive at Pluto in less than a year!) I came across a webpage tracking the movement of the probe. On the webpage (http://pluto.jhuapl.edu/mission/whereis_nh.php), the software NASA uses to produce nice visuals of where the probe is and other information about it includes a special statistic: heliocentric velocity.
Now, initially I didn't know what this was, but looking at the description given:

Heliocentric Velocity. The current position graphic also notes the spacecraft's heliocentric velocity - its speed with respect to the Sun - in kilometers per second. One kilometer per second is equivalent to 0.62 miles per second, or 2,237 miles per hour.

This might seem like a pretty normal statement, but think about it carefully. If there has to be a central reference point for this probe's velocity to be measured from, then that must mean that all velocities are relative (which according to the theory of relativity, they are). This must also mean however, that the sun has a velocity, and that velocity is relative to other large stellar objects, or even things as small as atoms perhaps, which have a definite position in our universe. This makes sense, in earthly terms, where we measure our velocity relative to what can be regarded for all purposes as the stationary ground.
But, say that there existed just one object in the universe, with no other atoms around to help measure its velocity. Would that object ever have velocity? Actually, wouldn't that mean that even if a force was exerted on the object (somehow without another object causing it) that the object would seemingly simply be as good as stationary, since there would be no reference points to measure any acceleration in its movement from. Do not be fooled by the fact that if you were standing next to the object, that you would see it moving away from you, since this assumes that you would also be in this completely empty universe, meaning that there are now two objects which can be measured relative to each other (you and the object).
But of course, when I said that a force was caused, and implied that this happened without another object causing it, then this would be impossible. In fact, the only way for that object to have a force effected on it, would for it to split in two (in an unexplained and spontaneous explosion) and so resulting in two parts of the object having pushed away from each-other with equal force. Of course, we could not actually measure how large that force was in terms of our standard units since that would mean that we were there as well, and would be measuring the forces relative to the current system of our universe, which is wrong since the single object (now dual-object) universe is a different universe with different physical rules of relativity - our universe contains a complex set of objects relative to each other not comparable to that universe.
The way our universe is comparable to that single-to-dual object universe is that they seem to both have started in the same way. Perhaps our universe started as a very similar object. Of course, since that object was the only thing that existed, there was nothing to compare it to and so it existed relative to itself both in size and motion: meaning in fact that it had infinitely small size and motion (or infinitely large depending on how you want to think about it). This concept seems similar to the concept of a singularity - the thing that our universe is thought to have evolved from. Perhaps physical laws and rules emerged at the beginning of our universe in a similar way to the object splitting in that alternative universe, and did so for no apparent reason (this concurs with the spontaneous expansion idea of the Big Bang Theory). Of course, maybe I can now attempt to explain the complicated notion of 'non-existence' outside the universe: perhaps this simply implies that outside the boundaries of relative relationships between matter in our universe, nothing exists since there is simply nothing to measure it relative to, since that space is immeasurable relative to objects in our universe and we cannot simply get out a ruler and say how large objects outside the 'boundaries of the universe' are or how fast they are travelling.
This relativity might also explain (in a convoluted manner) the deveptively-simple, ever-so-weird concept of gravity. The explanation might be that gravity is simply a result of the relative equilibrium of all the particles of the universe having been disturbed by that unknown force which caused the Big Bang to begin. Ever since then, there has been an imbalance in that equilibrium, however since the universe is so complex and matter-full, the particles were unable to regain their original structure (the structure they held when they were a singularity) and have simply been trying to "locally compensate". What I mean by this is that instead of the whole network of the particles in the universe simply returning to their original relative positions, they have formed 'mini equilibria' in certain areas of the universe, such as on that lovable place we like to call earth, or even in our galaxy as a whole. Were every single particle in the universe to suddenly become aligned in such a way that they might be able to come together again to reform overall equilibria, that might be more favourable, and perhaps it has happened before (a la the theory of repeating universes: where universes continually form and collapse upon themselves to form cycles of singularities). Perhaps, this will happen again in the future? Nobody knows whether it will however, and if it will, when it will happen? Or is time, in this context, strangely relative as well? I leave that question to you.

Credit to:
http://pluto.jhuapl.edu/mission/whereis_nh.php
http://www.nasa.gov/mission_pages/newhorizons/main/#.VBSbcMJdVA0