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Posted 19 November 2013 - 05:16 PM
Posted 19 November 2013 - 05:23 PM
Wolfram Alpha to the rescue. Given that the circumference is on the order of 10^{62} units, I suppose you could say 62 digits.
Posted 20 November 2013 - 02:22 AM
Don't represent it using digits then. The ancient greek mathematicians got by pretty well without having a proper number system to assist them, instead, performing math using geometry. If you represent Pi using geometry, it's about the most ordered image possible
Posted 20 November 2013 - 07:14 AM
It's not that nice in geometry - try constructing a square with the same area as a given circle using a compass and unmarked straight edge ;)
http://en.wikipedia.org/wiki/Squaring_the_circle
EDIT: You can't do it with origami either! Although you can use origami to double the cube and to trisect an angle, both of which are not constructible using ruler and straightedge.
EDIT2: Although if you have magic scissors which can cut the circle up into non-measurable sets you can cut up a circle and reassemble it into a square (assuming the axiom of choice) - http://en.wikipedia.org/wiki/Tarski%27s_circle-squaring_problem there is a solution which only requires 10^{50} pieces! Which is much harder than cutting up a ball using a magic knife and then reassembling it into 2 identical copies (good if you do it with a ball of gold! Infinite money), since that can be done with just 5 pieces! http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Edited by Paradigm Shifter, 20 November 2013 - 07:27 AM.
Posted 25 November 2013 - 06:23 AM
I like thing that are evenly spaced, or symmetrical or balanced. Irrational and prime numbers bother me just because you can't make them line up evenly with things.
Like so many parts of mathematics, irrational numbers seem a little unwelcome at first but there are plenty of beautiful things to discover about them. I suggest you learn a bit more, but understand that it is up to you to decide what kind of arithmetic you will use to solve any particular problem - for example, most of computer science only really requires you to use the integers. Like fractions, complex numbers and even negative numbers, you can think of irrational numbers as an optional extension.
So what's beautiful about irrational numbers? They aren't just a special exception for awkward numbers like e and pi - there are actually more irrational numbers than rational numbers (look up Cantor's diagonal argument). Yet between any two irrational numbers you can always find a rational, and vice-versa, between any two rational numbers you can always find an irrational.
When you think about it it they come up in real life as well, just you aren't necessarily aware of them. Wouldn't it be strange, for example, if you measured the distance between two atoms and it turned out to be exactly 1 metre? Or an exact fraction like 1.4 metres? Or even a long fraction like 1.453162343055682m? It seems much more likely that the number would be an infinite series of random-looking digits, which would make the distance an irrational number.
Pi is actually an extraordinarly organized irrational number. Although you cannot express it as an exact fraction, you can express it exactly as the limit of an infinite series such as:
pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
I think I'll stop there. I encourage you to find out more, and I do hope I've make this better for you not worse...
Posted 25 November 2013 - 07:49 AM
When you think about it it they come up in real life as well, just you aren't necessarily aware of them. Wouldn't it be strange, for example, if you measured the distance between two atoms and it turned out to be exactly 1 metre? Or an exact fraction like 1.4 metres? Or even a long fraction like 1.453162343055682m? It seems much more likely that the number would be an infinite series of random-looking digits, which would make the distance an irrational number.
That assumes a naive model for the universe in which we live. The distance between two atoms is not a number that can be determined with arbitrary precision, so it doesn't make a lot of sense to ask whether it is rational or irrational. I am not a physicist, but my understanding is that this is not just a limitation of our instruments, but a feature of nature.
The way I think of the world these days, everything that matters is discrete. Real numbers are a convenient approximation in situations where the numbers are large enough. This is often the case in physics; but there is nothing "real" about real numbers.
Posted 25 November 2013 - 07:59 AM
Like so many parts of mathematics, irrational numbers seem a little unwelcome at first but there are plenty of beautiful things to discover about them.
Though, everything with a grain of salt. Equally, like many parts of mathematics, irrational numbers have the property that they aren't very practical. Mathematics work exceptional well for mathematical problems, and they can be fun to think about, but they're not necessarily very suited for reality.
Mathematics allow you to calculate the likelihood that in a group of n people, two will have the same birthday. Which, of course, works fine for purely mathematical problems such as hash functions. On the other hand, it's complete bollocks for actually calculating the likelihood of two people having the same birthday in any group of real random people, since the base assumptions are wrong. Real birthdays are, in contrast to the mathematical thought experiment, very unevenly spaced. Not only due to seasons, but also due to holidays.
Mathematics tell you that you can't divide by zero. An electric engineer will tell you that it's very easily possible. Division by zero is the moment when there's that little electric arc, and current goes to infinity, just before your board goes up in flames.
Mathematics tell you that the likelihood of a goat being behind Monty's doors changes when he opens another one, and you're twice as likely to win if you change mind. Unless Monty's assistant is swapping prizes behind the scene (which of course we cannot know for sure), the likelihood doesn't change. Again, while the reasoning is correct, the assumptions and conclusions are wrong (the mathematican coming up with the proof has probably never seen the show).
Mathematicians come up with thought games like prisoner's dilemma where you can either defect or cooperate (= confess), but ignore the fact the only realistic outcome to asking would be that each prisoner will blame the other one alone, but moreover the police doesn't ask. The police are lying, they'll tell each suspect that his partner "has already confessed" and that the only way of getting a somewhat lesser sentence is to also confess. If that doesn't make you confess, they'll not let you sleep for 72 hours. You'll confess anything they want anyway only so you get to sleep. Again, the mathematician's base assumptions are unrealistic, and it's a fun thought experiment, but that's all it is.
Mathematics tell you that all numbers on a roulette wheel are equally likely, and that all lottery tickets are equally likely to win. Yet when you look at reality, some numbers do occur significantly more often than others. Mathematicians speak of "fallacy" in this context, but in reality the fallacy is with them. They assume all lotto balls are identical in size and weight and start from the same position, etc. when in reality they don't. They assume that the roulette ball is equally likely to drop into the field you've betted on as to drop into a field close to the eddy current brake. Which of course isn't the case, that's what casinos make money with.
Mathematics tell you that pi is neither 22/7 nor 355/113. However, for all practical means, it is. You yet have to draw a circle on the ground and demonstrate a practical difference with your tape measure. Wise people like Aristotle or Liu Hui already knew this 2,000 years ago.
Mathematics allow you to calculate the length of a circle the size of the known universe when in reality you neither really know the size of the known universe nor whether or not it expands (you only know one possible interpretation of what you see), nor do you know whether the universe is even approximately of circular/spherical shape, nor would you be able to tell whether the result is correct.
Posted 25 November 2013 - 08:00 AM
When you think about it it they come up in real life as well, just you aren't necessarily aware of them. Wouldn't it be strange, for example, if you measured the distance between two atoms and it turned out to be exactly 1 metre? Or an exact fraction like 1.4 metres? Or even a long fraction like 1.453162343055682m? It seems much more likely that the number would be an infinite series of random-looking digits, which would make the distance an irrational number.
That assumes a naive model for the universe in which we live. The distance between two atoms is not a number that can be determined with arbitrary precision, so it doesn't make a lot of sense to ask whether it is rational or irrational. I am not a physicist, but my understanding is that this is not just a limitation of our instruments, but a feature of nature.
The way I think of the world these days, everything that matters is discrete. Real numbers are a convenient approximation in situations where the numbers are large enough. This is often the case in physics; but there is nothing "real" about real numbers.
Yeah, fair point, I was assuming a simple Newtonian / Euclidian universe - just trying to build up some kind of intuition about what irrational numbers are.
Posted 25 November 2013 - 08:11 AM
Mathematics tell you that the likelihood of a goat being behind Monty's doors changes when he opens another one, and you're twice as likely to win if you change mind. Unless Monty's assistant is swapping prizes behind the scene (which of course we cannot know for sure), the likelihood doesn't change. Again, while the reasoning is correct, the assumptions and conclusions are wrong (the mathematican coming up with the proof has probably never seen the show).
I agree with what you said about mathematics being based on assumptions, but I think you've got this one confused. The Monty Hall problem isn't about absolute probabilities (which are simply 1 for the door with the prize behind it, 0 for the other two doors) it's about conditional probabilities given the information available to you. Initial that is nothing so they begin at 1/3, 1/3, 1/3, but when the presenter opens one of the doors you did not choose this affects your probabilities in an asymmetric way. If you aren't convinced, draw up all nine possibilities (3 positions of the prize and 3 possible guesses) and see how often you do better by changing!
Edited by Geoffrey, 25 November 2013 - 08:12 AM.
Posted 25 November 2013 - 08:21 AM
Do the Monty Hall problem with 1,000,000 doors instead. Pick a door at random. Then the host opens up 999,998 doors and they all have goats behind them. Do you change your pick? If yes, how many doors do you think you need to have to make it a better choice to change? Correct answer: 3, as in the original problem.
Paul Erdos also disagreed with the Monty Hall problem, until he was shown a computer simulation ;) And Erdos used probabilistic arguments to prove results in combinatorics!
Posted 25 November 2013 - 09:20 AM
I really don't get why people get stuck on the Monty Hall problem. When you learn more information about a situation, the probabilities do change.
But we don't generally encounter evolving situations like this much in education, so most people aren't really equipped to accept the counterintuitive result (I know I wasn't, the first time I saw it). This is exactly why it's such a good problem, because it teaches young mathematicians that their intuition can be wrong. If you got it right the first time, congratulations, but maybe one day something else will come along to keep your ego in check.
Posted 25 November 2013 - 09:37 AM
I can give you another probability problem that I like even better than the Monty Hall problem. Two people play Russian roulette the following way: A neutral referee goes into a room and secretly tosses a fair coin, either puts a bullet in a revolver or doesn't --depending on the result of the coin toss--, spins the drum, comes out of the room and hands the gun to one of the participants. Then the participants will take turns pointing the revolver at their own head and pulling the trigger. The revolver's drum has 6 chambers for bullets.
Imagine you are participating in such a game, your opponent goes first, pulls the trigger and no bullet comes out, then you go, then the opponent again, then you, then the opponent once more. Still no bullet. Now you are at the sixth position, so if there is a bullet in the revolver, you are going to get it. What is the probability that you will get your brains blown out when you pull the trigger?
Posted 25 November 2013 - 09:44 AM
Zero of course. I point the gun at the opponent's head, pull the trigger, and if there is no bang start running very fast.
Posted 25 November 2013 - 10:11 AM
Do the Monty Hall problem with 1,000,000 doors instead.
Yes, yes... Vos Savant's well-known explanation. Very entertaining, very smart. Except it's based on a wrong assumption. Monty isn't forced to, and indeed doesn't always open a door with a goat. And that's the point.
I watched that show a few times when I was around 10 years old, and sure enough, they'd sometimes (admittedly rarely) bring out the prize on the first door. If the host doesn't like you and you choose the goat, they just give you the goat. They'd occasionally have only one "zonk" (that's what the "goat", which is a fluff animal, is called here), too. In that case, you'd have a car in one vault, a home trainer or some equivalent value in another, and a "zonk".
You thus have a 1/3 (or 2/3, if there's a second prize) chance of "surviving" round 1. But of course you always "survive" since your vault isn't opened. Insofar the 1/3 chance is entirely irrelevant. You could just as well not make any kind of choice and only decide in round two.
The second round is an independent event. Which leaves you with two vaults, or a 50/50 chance.
The only "gained information" that you have between rounds 1 and 2 is that there are now 2 doors instead of 3. So the chance has "gone up" to 1/2 from 1/3, but it's none higher for either door. And it sure isn't magically 2/3 for one door because you didn't decide to change your mind.
That left aside, you don't know if the host is cheating and assistants move prizes around after you made your final choice. There's no way of knowing.
Edited by samoth, 25 November 2013 - 10:16 AM.
Posted 25 November 2013 - 12:42 PM
Of course, a good description of the Monty Hall problem makes the assumptions explicit. The applicability to the show that inspired the problem is something I don't particularly care about.
Will anyone give my problem a try? You can write a simulation, if you can't think of the math...
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