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You can find in this page: Random numbers, Infinite sums, Fractals, Schroedinger's cat, Russell's paradox paradox on probability and parallel axiom.
Consider an infinite sequence of numbers, say 3,5,7,1,6,345,674,2,42,34..... is this sequence random? Or let me rephrase the question: does it exhibit patterns of any sort? I suppose it is reasonable to call a pattern-less sequence random, therefore the discussion on random-ness now becomes a discussion on patterns.
Can we create a truely random sequence? Perhaps we can, but it won't be as easy as we think. Yes, we can write a program that tells the computer to create "random" numbers. But are those numbers really random? Computers are dumb and it can do no more than what the program orders it to. The so-called random number generator that computer programs use takes a parameter from outside (usually from the clock), and feeds it into some complex formulae. Therefore strictly speaking those numbers are not random. Or try picking a random even number between 0 and 10. Chances are the very first number you think of is 6. People have a tendency to pick the middle values when being asked to do so within a given range. Human is pretty rubbish, in general at creating random numbers and sequences.
Now consider the infinite sequence 1,1,1,1,1...... clearly it has a pattern and we can easily predict the next value, and the value after. One way to describe the sequence is: "1st value=1, 2nd value=1, 3rd value=1, 4th value=1..(and goes on forever)". An alternative way to do so will be: "every value in the sequence is 1". The difference between the first and the second description is that the length of the first is infinite while the second is finite (and very short). It's the second method of describing a sequence that I'm interested in . How about the sequence 1,0,2,1,0,2,1,0,2....? I can say "for any natural number n, let the 3n-th value=1, 3n+1-th=0 and 3n+2-th=2". This description is longer than the one for the previous sequence, but nonetheless is still finite.
It is quite evident that the complexity of a sequence is related to the length of its description. One definition of random sequences would be: an infinte sequence is random if its shortest description is infinitely long. This definition can be easily extended for finite random sequences (sequences with finite number of values). Imagine converting the description above from english to some programming language. Then the program for the first sequence will be something like "let n-th value=1, repeat for next n". The program consists of 2 steps. With this in mind I can say a sequence of x values is random if the shortest program describing the sequence requires x or more steps. The cool thing about this definition is that it also provides a way to measure "random-ness" of a sequence.
There are probably more ways to define random-ness of a sequence. But this is the one I've heard of and I find it quite elegent. Bye for now
I've shown previously that the infinite sum 1/4 + 3/16 + 9/64..... tends to one. The unusual thing about this is that number of terms added is infinite, and yet the sum itself is finite. This is a geometric series. Instead of using the diagrams to estimate the sum, there is a general formula which most secondary school students should know pretty well. Yep, it's a/(1-b), where a is the first term and b is the constant multiple for each term. Note that b has to be less than 1 otherwise absurd results will arise. (ie a/(1-b) becomes negative while the sum should be infinity.)
Now consider this, 0+0+0+0..... = 0 and 1+1+1+1+1 .... tends to infinity. Yeah I know they're pretty obvious...so obvious that no one ever cares about these two sums. But what makes the first sum tends to a finite value and the second to infinity? How about 1+1+1+1+1+0+0+0+0...(with all other terms 0)?? Clearly this particular sum tends to a finite value (5). One conclusion we can make is that in order for the sum to converge to a finite value, the size of the terms have to get smaller (and tend to zero) as you go further and further.
Okay, so does the sum 1 + 1/2 + 1/3 + 1/4 +1/5..... converge to a finite value? Clearly as you go further and further down the list, the n-th term is in the form 1/n which tends to zero as n tends to infinity. The truth is, the sum does not converge, or to put it in another way, it tends to infinity. Why? I'll first show a few more terms of the sum....
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9....... = 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9....
Now 1/3 > 1/4 and so 1/3 + 1/4 > 2/4 = 1/2. And 1/5, 1/6, 1/7 > 1/8 and so the second bracket > 4/8 = 1/2.
Therefore the sum is greater than 1 + 1/2 + 1/2 + 1/2....which tends to infinity. Hence the sum doesn't converge.
The fact that the size of terms gets smaller as you go further down the sum is not sufficient to prove a sum converges (although clearly it is a requriement). So what else do I need?
There are a few ways to determine whether a sum converges or not, which I'll not mention here (for now anyway). But I can tell you that 1^p + (1/2)^p + (1/3)^p..... does converge as long as p > 1. Yes, even p = 1.00000000000001 will make it converge.
Maths is beautiful. No kidding. (1)
Most people associate maths with numbers and funny symbols, and mathematicians with boring nerds with thick glasses. Well they're sort of right, but look at these:
Click the thumbnails to view them in full size.
Nice pics, aren't they? So who is the talented artist behind these works? Computer and maths. Really. These pics are known as fractals. Fractals are funny things, in that within each fractal there exist patterns which repeat themselves as you continue magnifying a particular part of it. And in theory you can carry on with this magnifying process forever, and you can still find these patterns (Of course you can't magnify forever with the pics in this site because of the pixels..). Try and see it for yourself with the full size pics. Or have you been to a gym? Sometimes you can find an image of yourself on a mirror, and within it another smaller image... each image just gets smaller and smaller but it is still you. (These reflections are NOT fractals, but they should give you a rough idea of what fractals are.)
So how do computers and maths create these fractals? They look more like works of art than equations... In fact they are graphical representations of mathematical models. Mathematicians construct the models, enter the data into some programs.... and voila. The left pic is part of what is known as the Mandelbrot Set.
We normally say we live in a 3-dimensional world. Those who know some physics will point out that we actually live in a 4-dimensional space-time continuum. So is it necessary that dimension must be in the form of an integer? No.... fractals are known to have dimensions like 1.442, or 2.135.... weird, isn't it? So do fractals only appear on the displays of computers and are they too difficult to be constructed by normal people like us? No and No! To be continued... To find more fractal images, click here to Big Al's Fractals .
Maths is beautiful. No kidding. (2)
How long is the coast line of Hong Kong island? Okay, I suspect no one knows (including myself). But if you really what to know, how will you go about in measuring it? Measure the perimeter of the island from a map and then calculating the true length? Yeah, that's what most people will do. But it's not accurate, is it? Afterall, a map can only show the obvious features of the island. What appears to be a straight line segment on the map is probably not straight in real life, and we therefore can assume the length you obtain from measuring from a map is an underestimate of the true length. So, if I want a more accurate measurement, I'll have to measure the island physically. To ensure a good accuracy, I use a metre ruler for measurement. I'd expect my result to be higher than the estimate I made earlier. Surely that's accurate enough, right? But using a metre ruler means I'm making an assumption that the bit of coastline which lies between the ends of the ruler is straight. And that's still some way from 100% accuracy. In order to improve the accuracy, I'll have to choose a shorter ruler. And yet every time I use a smaller ruler, my estimate of the length increases. Therefore, by the time we use the shortest ruler we can possibly think of (one which is short enough to measure individual atoms), the estimate will explode into some astronomical value (in fact the value is infinity). What this means is that in nature you can find many things that have details "within" details (and this goes on forever), as long as you look hard enough. Fractals are not as artificial as those pictures (in the post below) suggest afterall. Heard of Pascal's triangle? In contrary to popular belief (Europeans anyway), Pascal's triangle was invented by the Chinese centuries before Pascal "rediscovered" it. To be continued...
Maths is beautiful. No Kidding. (3)
To finish off the topic on fractals, I'll make an attempt to show that the probability of a term inside the Pascal triangle being odd tends to zero as the triangle gets infinitely large. The reason I use the word "show" rather than "proof" is because technically speaking it isn't (even close to) a proof. I remember reading a convincing argument for it years ago. Unfortunately I do not have the text with me and so I can only show what I can remember.
Now, to those who do not know about Pascal triangle, it is basically numbers arranged in a triangle whose size can be expanded infinitely. There are a few rules that determine the value at each particular spot inside the triangle. 1. Each row has one more terms than the row above. 2. The first and the last terms of each row are one. 3. Any term between the first and the last of the row is the sum of the two terms above it. This is what a Pascal triangle looks like:
1
121
1331
14641
The next row will be 1,5,10,10,5,1... and so on....compute the next few rows yourself as my arguments will be more obvious with a larger triangle. Colour the odd terms in white and the even terms in blue.
This is what you should get for a 3-rows triangle:
Yes,
you may say the pic on the left doesn't really resemble to the coloured triangle
you just made. This is why I want you to compute the next few rows. Once you
have done so, and coloured it, this is what you should get:
Now this
one should look more like the one you made. The problem with the 3-rows triangle
is that there is only one even term , and hence it'd be difficult to imagine
the blue bit as a triangle.
If you carry on expanding the triangle and colouring it, you will soon realise that the 3-rows coloured triangle is the building block for the larger triangles. This shows that the coloured Pascal triangle is in a way a fractal. You may say the fractals we see preserve patterns no matter how much we magnify them, and in the case of Pascal triangle, it stops when the size is 3 rows. To solve this problem all you need to do is to expand the triangle to infinity.
Now the probability of a term being odd is represented by the white area. Suppose we have a infinitely large coloured Pascal triangle in front of us. By considering the 3-row triangle we can say that the white takes up 3/4 of the total area. However, when we are dealing with the infinitely large triangle, the "white" area is no longer pure white. Instead, 1/4 of it is blue (ie 3/4 is white).( If you don't understand what I mean, the second triangle should illustrate my point pretty clearly.) I can repeat this argument for ever, since the triangle is infinitely large. And the white area is 3/4 x 3/4 x 3/4......which tends to zero.
This is strange, given that each row (except the first) has at least two odd terms.... the key here is that the triangle used in the argument above is extremely large.
An interesting point to make is that this implies the blue more or less covers the entire triangle. Or to put it in another way, 1/4 + (3/4 x 1/4) + (3/4 x 3/4 x 1/4).....= 1/4 + 3/16 + 9/64..... tends to one (a finite value) eventhough there are infinitely many terms in the summation.
Speaking of cat, how can I not mention the Schroedinger's cat?? It is one of the most famous cat around (at least for physicists anyway). So what is so special about it? Well, to be honest the cat doesn't exist in real life. Rather, it's part of a thought experiment created by...surprise surprise... Schroedinger.
In this experiment, a cat is locked inside a box. This is no normal box as no one can observe what happens inside. Nope, not even infrared/ heat sensors, or x-rays will reveal anything. There is plenty of air, water and food for the cat so it won't die of thirst or hunger or lack of oxygen. In other words the cat can live normally and happily inside the box....er... no, not quite actually.
There is also a concealed bottle, full of lethal gas. The cat will die instantly if it makes any contacts with the gas. Next to the bottle there is a hammer which is connected to a Geiger counter. (A Geiger counter is a tool measuring radiation.) And next to the counter is a sample of radioactive material. Now, if the sample decays, the counter will trigger the hammer to smash the bottle, and the cat will die.
Question: What will happen to the cat in...say 5 hours time? Dead or alive?
Answer: Both dead and alive.
Reason: We cannot make any observation to conclude whether the cat is dead or not. All we know is that the state of the cat depends solely on whether the sample has decayed or not. Now, according to physics, until we observe the state of the sample, the sample is in a "superposition of states". That is to say, the sample both has, and has not decayed. Therefore, we can conclude that the cat is both alive and dead.
But a cat CANNOT be both alive and dead!!! If I open the box, the cat is either alive or dead, but not both!!! Physicists will tell you that as soon as you open the box, you're in effect observing the state of the radioactive sample. And when you make an observation, these superposition states collapses and becomes "has either decayed or not". That is why you'll not see a cat both alive and dead.
Absurd? Ridiculous? Weird? Bizarre? Apparently Schroedinger was using this thought experiment (a paradox really) to illustrate the absurdity of particle and quantum physics. These days people tend to use this paradox to show the unusual and cool side of physics....
What is a set? Without getting into mathematical jargons, a set is basically a collection of "members". All even numbers form a set. So do all the bottles of beer inside your fridge. If there isn't any beer bottles inside the fridge, you call it an empty set. Simple, isn't it?
Now consider this. Can a set be a member of itself? Some sets can, and others cannot. For example, the set of bottles is not a member of itself. That is because the "set of bottles" is a set, not a bottle. On the other hand, the set of non-bottles is a member of itself, since as stated earlier, a set is not a bottle. If you're still not sure, then just accept for now that there are sets which are not members of themselves.
Now comes the paradox... Let's call the set of all sets which are not members of themselves S (There is no reason why you have to call it S, but most people do) . Is S a member of itself? Clearly either it is or it isn't. Suppose S is a member of itself, then by the definition of S it must not be a member of itself. Now suppose S is not a member of itself, then again by the definition of S, it is a member of itself. Weird, isn't it?
In general, we accept that, if we make a wrong assumption and we make logical deductions from that assumption, then we will reach a contradiction. For example, let A=1, B=3 and I claim that A is larger than B. If A is larger than B then A-B must be positive but 1-3=-2. Therefore my claim must be wrong. In the case of S, we already know that S either is a member of itself, or it isn't. However, either way leads to a contradiction. So what's wrong?
There is nothing wrong with the definition of S, since S does exist. What is wrong is the definition of set. The paradox is known as the Russell's paradox and it's pretty famous because it exposed the flaws within the definition of set. So the collection of beer bottles is not a set afterall then? It is still a set, and most of the sets we can think of are still sets. The S in Russell's paradox just happens to be a special case.
It's so obvious it cannot be wrong!
I'm sure everyone knows what parallel lines are. Consider this statement: There is a infinitely long parallel line AB and a point C, which doesn't lie on the line AB. There exists one, and only one straight line which 1. passes through C 2. is parallel to AB. The statement shouldn't be hard to picture. Is this statement true?
For 99% of the population the answer is a definite yes. I mean, it's so obvious once you think about it and most of us cannot think of a case when the statement is false. Let me tell you some more of this statement. It is known as the parallel axiom. Along with other axioms, they provide the foundation for Euclidean geometry (the one we all learn when we were at secondary school). What are axioms? We answer questions using reasons, and we can ask for reasons for the reasons we first gave. It follows that we can repeat that process many times, but there will be a point when we face with a number of statements which appear to be latently obvious, and yet we cannot prove. We decide to believe these statements as true and call them axioms.
From what we observe in life, Euclidean geometry appears to be correct in all situations. So can we say that the parallel axiom is correct? Surely if the axiom is wrong, then Euclidean geometry, which is derived from the axiom, must also be wrong. Therefore, the converse (the opposite) must be correct.
Not so! Those who think the previous paragraph is correct have fallen into what I regard as a common trap in logic. If you do, don't worry about it. Everyone, from politicians to the drunken man on the street to myself falls into it every now and then.
When mathematicians decided to reject the parallel axiom, a whole new set of geometries became possible. Are these geometries wrong? Nope. Some of them are so similar to Euclidean geometry that it's hard to notice any real differences between them. Whereas others become extremely useful in different branches of science, such as relativity.
You're confused? You don't know whether to believe the axiom or not? Do weird stuff of this sort only exist in maths? Ask yourself any simple questions and keep reasoning and see what happens.
You want another example? Here is one. Suppose A > B and B > C, is it necessarily true that A > C. Don't think of A, B, C as numbers, think of them as objects like fruits, and the relation > as "prefer to". eg. I prefer apples to oranges. Some economists can debate on this for ages...
I have another paradox to entertain you. To a certain extent this one is both similar and very different to Russell's paradox, which I mentioned quite a while ago. This paradox is different in the sense that it deals with probability, rather than the naive set theory. However, just like Russell's paradox, it exposes flaws within our early understandings of matters that we almost took for granted.
Suppose there is a fair coin in your pocket and you decide to toss it, and then to record the result. The reason I use the phrase "fair coin" is to ensure the probability of having head facing upwards is exactly 1/2. That is not very exciting, so you repeat this over and over again and record the results, using H to represent head and T for tail. Therefore your results may look like this: HTTHHTTHH..... Now suppose you do this infinitely many times. What are the chances of having heads, and heads only?
Not very high. That's right, and in fact the exact answer is (1/2)^n, where n is the number of times you toss the coin. Clearly as n increases to a very large value, the answer gets closer and closer to zero. I am therefore safe to say the portability of getting all heads after infinitely many tosses is zero. In everyday language, you will NEVER get all heads. That's pretty convincing, right?
How about the probability for THHHHH.... where every time you toss, the result, with the exception of the very first one, is head? Is it higher than HHHHHH....?The answer is the probability of happening is the same in both cases ie zero. Reason? THHHHH... is just as unique as HHHHH... and therefore the two must have the same probability. In fact, every possible sequence is unique and so they too have the probability zero. With this in mind I can now come up with a conclusion: Nothing EVER happens when you toss a coin infinitely many times, since the chance for any sequence to come up is zero.
What do you think?
