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

Do you believe in fate?

We oftentimes hear about the suggestion that there is such a thing as "fate", "destiny", a "plan" for humankind. These words are tossed around in everyday speech in a myriad of different contexts, whether it be the often religious notion that God has a plan for humanity and that things were set out for us from the very start or perhaps the astrological notion that the stars dictate the way different sorts of people will act each day due to various 'correlations'. Personally, I don't put much faith in astrology, but others do and recently I have come to see reason for them to do so. Why, you may ask, would any extra-terrestrial happening dictate what I might feel like eating for breakfast tomorrow. Well, perhaps these things don't affect each other directly. It is the notion of cause and effect and there is often a misnomer about what people mean when they say two different events are related to each other. In the case of the positions of the planets and the stars, what is to say that certain movements of these objects in space could actually be related to the actions of is on earth, not because they themselves caused them, but because there is a meta-cause of both these things. It is the same problem as the idea of the number of drownings going up at the same time as ice cream sales. This does not mean that one causes the other, but that a higher-order cause was involved in bringing about both, namely the fact that both these statistics peak in the summer because more people go to the seaside when the weather is hotter and swim as well as being more inclined to buy a nice, cold ice-cream cone.

Consequently you might question what this higher-order cause might be, and it is a very good question to ask. Perhaps it is a very complicated cause to do with time. Time is one of the only things that links everything in the universe with everything else, among other theories of humans such as the idea of the three spacial dimensions. Perhaps, the pseudoscientific notion of biorhythms comes into play and means that each month the stars will be in certain positions and at the same time, purely on the basis of the regular rhythm of life, you will also be full of a certain hormone which makes you more likely to feel a certain way. In this sense looking at the stars to determine our fate is completely redundant, we may as well make less work for ourselves by analysing what time it is and our past history to determine a more accurate and less confusing view of why we might feel a certain way, rather than simply because 'the moon has passed close to the line of the equator'.

Everything in the world, and in the universe as a whole for that matter, obeys a regular rhythmic pattern dictated by the forces and objects around it. These ideas stem from the most basic physical principles we regard as 'most likely true based upon our current knowledge', and were discovered by scientists like Newton and Einstein among many many others. This is why we can predict the position of the earth at a certain time, or the position of another planet relative to the earth at that time. We would see it as strange to think that tomorrow Mars would suddenly, without reason, be moving much faster around the sun than it was today. We would have to see evidence for this, otherwise we would have to probably rethink the most basic, axiomatic principles governing the physical universe.

In fact this idea of these firstmost principles is very interesting. It is strange nowadays to use the word 'theory' to describe such things as gravity and atoms for example, since these 'theories' have become so well accepted, and seem so true that they are now almost always referred to as 'facts of life'. But of course, they are facts, but only in the realms of our own understanding. We have built other theories upon them and other theories came before them that they had to in some way agree with in order to be seen as credible. But the limit of our ability to test theories we make to determine whether they are true or not comes when we hit the barrier of what we are able to observe by any means about the universe around us. If we could in some way 'observe' time in some way then I am sure we would have a very different system of physics at the present moment. However much you think about it physics is at its roots a human explanation for something we do not fully understand and possibly never will. Just because something seems to fit, however well, with our understanding, in actuality it is never good enough to be regarded as real 'fact' since somewhere along the line, often at the very beginning we had to make an assumption, or a 'given statement' such as "all things fall down on earth". What is to say that they do? All that we have to observe is five basic senses at the end of it (not even in agreement with the number of recommended senses for something to be objective and not simply down to mere coincidence), and to be honest those alone will never be good enough to say for certain that all things fall down, or in fact that the objects and systems we interact with in everyday life are really what we see them as. At the end of it they are only viewed from our biased points of view. So where does the real truth lie? Well, in actual fact until the real truth - if there is one - is discovered for certain we have to believe every theory to be true in its own right, however crazy it might sound, since at the end of it everything is still only subjective to how credible we as a population think it to be, and that never equals certainty. For now however, I suppose our theories about life, mathematics, the universe and everything are all in actual fact true, since the concept of truth is, if you think about it, as much of a human invention as anything else. It belongs to the concept of logic and lies within the same thought system as Newton's laws of motion, Einstein's theory of relativity and the laws of thermodynamics and so all these things are true in that sense.

Coming back to my original idea about fate, there is a flaw in cause and effect - that is expressed as the very weird and wonderful notion of the human brain and consciousness. If everything, as most people believe it does, follows the currently accepted physical laws, then true randomness is in theory impossible, since everything must follow a rigid and structured system. By these premises that must mean that human thought is simply a direct result of the atoms and chemicals in our brains interacting with electrical signals in a complicated yet theoretically predictable manner meaning that fate has always been decided for us. This would also mean that if we were able to computer-generate a model of a human brain at a specific moment in time with every detail included, and programmed it using all the physical laws then this should mean the brain would work almost exactly as that of the person who's 'brain snapshot' was taken (this is a similar notion to modelling the atmosphere to predict the weather), give or take a few differences resulting from the absence of the external factors the person was subjected to: stimuli. But these differences are of course very important since in the long run they might add up to make for completely different paths of events. Of course, these stimuli would theoretically come from other predictable physical events and objects causing them which in theory could also be programmed into the computer. But of course these stimuli were in theory caused by other stimuli and so on and so forth until eventually you realise that to obtain an accurate model of what that person's brain would be doing at a certain time you would need to program the entire universe into the computer. Now, here comes the big problem - what about the computer itself, it of course is an external stimulus and part of the universe, so how do we program it into the equation? Well, as you may have guessed you would now need an infinite number of computer simulations each inside of one another to do this, much as you get with two mirrors placed in front of each other forming an infinite number of images within one another, which of course is impossible. But perhaps there is some truth in this: perhaps we ourselves are just one of an infinite number of sub and meta universes each inside each other in a never-ending chain. However, as we have seen with this theoretical computer simulation of the universe, there is perhaps a beginning. What could this beginning be? God perhaps? Well, again we are jumping to conclusions - my theory could possibly be false from it's very origins and therefore further reasoning would be futile, but it could just as well be true for all anyone knows. In a sense it is actually true by definition in the strange way I described before - true just as every other theory is. It just goes to show that seeing isn't actually believing, but the other way round. If you believe something, then it true by your own definition since the universe is a very, very relative place. What might be true for you may not be true for others. Because of my computer argument I can see how fate might simultaneously exist and not exist. To predict it all you would need would be a brain of infinite capacity - God as some might describe it. You might say this is impossible, but of course so was motion according to Zeno of Elea in his Achilles and the Tortoise paradox. What is impossible for someone may not be in another situation or train of thought for reasons that people are still trying to understand. That, I believe, is one of the beauties of life and consciousness.

Credit to Wikipedia for this image of an extract from the page on axioms... Just something to think about, is this truly true?

Credit to Godel Escher Bach among other things for inspiring me to write this article.

The Sad Truth About Students

Only programmers will understand. If you would like to learn how to program in C++, thenewboston has a YouTube channel with lots of beginners tutorials. You'll be hooked once you start.

Reading - A short poem

The written word – oh how it tumbles and its myriad characters rise

As if Black, Letters Like Flies rise up and Follow

The lines written on the page of sorrow

Where countless fall, die, torment, flee

As if aflame from Reality

And we, we lucky few who survive the trouble, toil and spew

That spills over into every word, chapter, verse and rhyme,

We finally sit down comfortably, sublime.

To enjoy what is left of our shattered universe,

For what we see is but the written verse.

Computers don't always do what they're told to

Thinking about computers, we know that one of these days they're going to stop obeying Moore's law, at least for silicon microprocessors. Moore's law was a prediction made by a man called Gordon Moore in a paper published in 1965. It predicted that the number of transistors on processors would double every year. Surprisingly this law stayed true for a time long after the ten year period Moore initially suggested. However, unfortunately all good things must come to an end and Moore's law was no exception. Gradually the doubling period has lengthened to 18 months and in 2010 it became 3 years. There is of course a reason for this and these are the limitations of manufacturing silicon computer processors with too many chips. Firstly is the fact that, when we have too many tiny transistors in too small a space, they will generate immense amounts of heat, hot enough to burn themselves up, which we obviously don't want. The second limitation is what I'm interested in however, and that is that there is of course a limit on how small we can make transistors, since the universe is made up of atoms which are of a limited size!

This might not have been something Moore had thought of, or even a stage he thought it was possible to get to, for which this might become a problem, but the reality is that this problem is one which is actually very close to becoming a major issue for the future of computing.

But why is it an increasing problem to have transistors made up from smaller and smaller numbers or atoms. Surely we should be able to go down to one atom transistors theoretically, since in reality this is all that is needed to either let electricity flow, or not to let electricity flow.

But in reality our limits are set to five atoms. Why you may ask? Well, it is due to the unpredictability of electrons when we look at them closely. Something which a part of quantum theory.

Quantum theory, Schrödinger's cat, electron cloud - they all make up this weird and wonderful modern model of atoms. The main principle behind this theory is that, contrary to what you might think, electrons cannot only be thought of as particles, but a more accurate description is 'clouds' of space where the negative charge might be at any point in time, and has a certain probability of existing within. Yes, probability and physics. Not something I would've thought might go hand in hand, but apparently, electrons can be anywhere. It doesn't just stop there however. At this small a level, these electrons can demonstrate having equal probabilities of existing everywhere within this cloud, and since we have no method of determining the exact position of electrons at this small a level, we simply have to RELY on this theory of chance - making it really difficult to build computers with transistors this small. But of course we can use this to our advantage in fact, and this can be seen in the modern quantum counters, but that's a story for another day.

What I'm interested in is the fact that this underlying principle of atoms becoming unpredictable at a nanoscopic level is actually present throughout the world of probability and statistics. It's actually not a remote and unexplained scientific theory, but something we experience on a daily basis. Let me explain.

Think of the age old model of probability - flipping a coin to get either heads or tails. There is an equally likely probability of getting each outcome, as many of us know from the hours well spent working out probability problems in our maths workbooks at school. And as we know, often probability is actually correct. Plotting a graph of the cumulative outcomes of an infinite number of trials would eventually give us an eerily accurate 50% outcome for both events. However, as we flip less and less coins the probability will probably become less accurate, up to the point where you have only flipped two coins and can get only four outcomes: heads then tails, tails then heads, heads then heads, and finally tails then tails. Looking at these outcomes they are all equally likely. But only two of them fit the overall model of probability, with a 50:50 outcome for heads and tails. This means there is actually only a 50% likelihood of getting an accurate probability, making it very difficult to predict the number of heads and tails. However, it doesn't stop there, since things take a turn for the worse when you get into the realms of flipping just one coin. There are only two outcomes - heads or tails - and neither fits the conditions needed for our 50% probability! It's physically impossible! The only way to get half heads and half tails would be for both to happen at exactly the same time. Sounding familiar?

Now we see that the behaviour of electrons at smaller and smaller levels is weirdly similar. At a large level, the properties of substances are quite predictable - give or take some slight universal randomness - just as the number of heads and tails are after flipping a coin umpteen times. However, just as the probability becomes less and less 'probable' as we get less and less trials for heads and tails, with less and less atoms, it becomes less and less likely for us to be able to say where the electrons are and predict how the material will behave as a result. Finally we get into the realms of just a few atoms and the probability gets very weird and out of hand, just as it did when we had just two flips of the coin. Imagine this, but on the scale of electrons having an infinite number of equally likely possible locations they can take, rather than just two outcomes as with the coin. Since there are infinite outcomes and probabilities for the locations of electrons, the probability is very hard to predict even on a large scale. We can simply say a certain number of electrons at least will PROBABLY exist in certain 'clouds' as a result, using an equation called Schrödinger's equation, which is basically an equation like "number of trials divided by two" for the number of heads in a certain number of trials of a coin, except applied to an infinite number of locations for electrons. Finally we reach the one atom scenario, where predicting where the electrons are going to be becomes an impossible chore, without saying that the electrons are at each and every point at once. Perhaps this apparent physical impossibility explains the existence of alternate universes with infinite numbers of possible events. Perhaps it might one day prove the existence of God. Who knows? Well, maybe the scientists in another universe, but none in this one - most probably not at least.

                              

The never ending end and the means justified

One question we may ask ourselves often is, 'is what I'm doing really worth doing?'. Think about the last time you might've thought this way, perhaps it was at school or at work and you were wondering whether your job or revision methods were the best way forward.

Well, I have a little piece of advice for you. STOP listening to yourself for a while and just go with it. As much as we might hate to think otherwise, life is unpredictable and unexplainable both physically and philosophically. We can never really know whether what we class as 'consciousness' is really our own beings making moral decisions, or whether we are mere spectators in the grander scheme of things, witnessing this madness through the senses and thoughts of another.

My point is however, not to try to convince you that you are not a real person, it is to tell you that since you might be you should try to achieve the most out of your life. How do I do this you may ask? How do I live the 'ideal' life?

In truth I was actually tricking you slightly, since I would like to say, and I know this for certain, that there is no more of an ideal life than the one you are living as you read this article. The very experience of your unique life pathways in itself is a wonderful thing and the emotions and experiences, good or bad, that you encounter along the way are also unique and should therefore be treasured. It's all part of being human - a richness that no other species will probably ever experience. Put this down to what you will but the statement remains.

Now, there is one thing you can do to improve your life. It may seem counter-intuitive but I think that it is the truth. That is to stop worrying about changing your life drastically and do things step by step. The most important thing is to recognise that you are never going to be the same one day as the next. You are, however you might feel, a different person every new year, every new day, every new moment. Scientifically we might attribute this to neuroplasticity and the ability of the brain to change its structure as we experience new things.

If you are still sceptical, just imagine the day you were born. Are you the same as you were then? No, of course not, but you didn't feel these changes coming on. In fact you are actually made up of a completely fresh set of atoms and particles to those that came out of your mother's womb.

Now think back to when you were just a child, aware, but still not mature. Mature. You know you became it as you grew older, but was there one day when you just woke up and found yourself to be mature? Of course not, maturity was a gradual process due to you understanding more and more about the world. And it hasn't stopped for you yet, no of course it hasn't. You are still developing second by second and there is nothing you can do to stop yourself from changing.

But you do of course have a say in the way that you change. You can change for better or for worse, depending on how you choose to think, act and say the things you do. Each small action will lead to a small change, a fact that you could use to your advantage in accomplishing your life goals.

If you want to stop smoking for example, gradually start doing more and more things to contribute towards this ultimate goal. Eventually you will come to a point where you have overcome your addiction and this is the same for everything. All things are on a continuous spectrum and nothing can be fixed in stone, just as the universe in itself is not really that fixed.

If you just keep in mind how much you want something, those clouds of dreams and aspiration will condense into a real form and you WILL accomplish your goal.

The end never comes before the means and the means will not always seem like they are working, because the changes come so gradually. Rome was not built in a day and neither will you be.