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Author | Topic: A question of numbers (one for the maths fans) | |||||||||||||||||||||||
fallacycop Member (Idle past 5550 days) Posts: 692 From: Fortaleza-CE Brazil Joined: |
All that means is the pie can never be fully resolved in a 10 based number system. That doesn't mean that can't be resolved. Yes it does. Ever heard of irrational numbers?
For practicle purposes, say like in construction, 3.141 is good enough to build even the tallest building.
practical purposes are completely besides the point. we are talking about pure math here.
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fallacycop Member (Idle past 5550 days) Posts: 692 From: Fortaleza-CE Brazil Joined: |
If it went on forever, then you could take an infinite line and make any size circle out of it. But if you travel down an infinate line, you never go over it twice. that would depend on how you define your infinite line and on how fast you are traveling. there are ways of making a one to one correspondence from every point in an line to every point (but one) in a circle of any size.
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lfen Member (Idle past 4707 days) Posts: 2189 From: Oregon Joined: |
All that means is the pie can never be fully resolved in a 10 based number system. That doesn't mean that can't be resolved. RR, I had heard that Pythagoras was so upset with irrational numbers that he pledge his students to secrecy but I'm not sure about this story:
D) IRRATIONAL NUMBER: This story would have been a pleasant one, had not one of Pythagoras’ disciples, Hipposus, in his infinite curiosity while exploring various combinations of the three sides of right angled triangles,stumbled on the simplest of them all. He found that for a right angled triangle of unit length for two of its shorter sides the hypotenuse turned out to be a number he could not handle at all-2. 2 is obviously not an integer. Not it can be expressed as a ratio of two integers- a rational number, as we call them today. Then what is it? Does such a number exist? For a person 2500 back, this is a daunting question indeed. The existence of this number cannot be denied as it flows from his master’s famous theorem. It is said that since this unpardonable discovery could not be laid to rest with the newfound system of logic of the Brotherhood, Pythagoras handled it the only he could. To his eternal shame, Pythagoras ordered that Hipposus be drowned. Thus to my knowledge, Mathematics got its first martyr, long before any other physical sciences could claim such a distinction.http://www.geocities.com/mathimoh/irrational.html Yes, the incommensurateness of the rationals and irrationals is a puzzling even counter intuitive finding. Let say you have a line length of 1 and another line length of pi. It seems like you could find a small enough length that would evenly measure both of them. Pi would have three something times as many of those tiny lengths than would the line of 1, but no way. It can't be done. There is no length however tiny that can be used to evenly measure the two lines. Here is a simple proof by contradiction (I believe that is what it's called. It's been quite a while since I've done math)
Let us suppose each side is an inch long; then how long is the hypotenuse? Let us suppose its length is m/n inches. Then m/n=2. If m and n have a common factor, divide it out, then either m or n must be odd. Now m=2n, therefore m is even, therefore m is even, therefore n is odd. Suppose m=2p. Then 4p=2n, therefore n=2p and therefore n is even, contra hyp. Therefore no fraction m/n will measure the hypotenuse. The above proof is substantially that in Euclid, Book X.” (Bertrand Russell, History of Western Philosophy) http://www.thebigview.com/greeks/pythagoras.html They are called irrational numbers because they can't be expressed as the RATIO of two integers: .3333 is 1/3 but pi is not a repeating decimal. The decimal is only getting closer and closer to pi. It will never equal pi. I still find this a strange thing about everyday life. Incommensurability is a mystery to me. It is one of many hints that reality is not what I think it is. lfen
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Modulous Member Posts: 7801 From: Manchester, UK Joined: |
We can only approximate it. I use pi all the time, and 3.14 gets me close enough. Yet it is a real ratio which real calculations can be performed on, without the need to approximate. It seems like you are rejecting a great swathe of mathematics. eΠi+1=0 springs to mind. We cannot say this is true by your definitions. Likewise we know that Π = c⁄d, but now you are proposing that (Πd)⁄c ≠ 1 It's fine if you want to reject any number you cannot write down - but you need to appreciate just how that might affect the rest of maths. Incidentally, if it suits you to think in terms of approximation, then you know that if you made any approximation of 0.9999.... it becomes 1. If you don't like the decimal representation of an infinite geometric series, you can think of it in terms of fractions - it's just more difficult to write and less likely that people will understand the number being represented. It is strange that every decimal number has different ways of being represented - even though we accept it as common for fractions.
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lfen Member (Idle past 4707 days) Posts: 2189 From: Oregon Joined: |
Rat,
Take a look at this wikipedia article:Zeno's paradoxes - Wikipedia My reasoning is, that if a number never ends in an equation, then you can't logically get to the next number. Taking an finite symbol to "represent" infinity is taking a huge risk, and a leap of faith. It works in the formulas, but is it really correct? You are having the same problem Zeno had. He proved that Achilles could never catch up to a tortoise because in order to catchup he would have to first go half way. Once there he would have to go half of the next distance and then half of that and he would always be some fraction of the distance behind the tortoise. This brilliant (but ultimately fallacious proof) demonstrates something we know not to be the case. But if I recall correctly the mathematics to handle this were developed much later by Newton and Leibniz. .999... is shorthand for an infinit sum .9+.09+.009+ ... and calculus has proved that the value of the sum is 1. It's good to think about these things but naive assumptions can be false. There are some good books on mathematics written in a popular style. Why not read one? lfen ABE: Here is another link to a overall math view of this:http://www.andrews.edu/~calkins/math/webtexts/numb13.htm Edited by lfen, : see ABE above
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cavediver Member (Idle past 3673 days) Posts: 4129 From: UK Joined: |
eΠi+1=0 springs to mind eiΠ+1=0 please Though could this be the first mention of Euler's Identity at EvC?
Modulus towards the beginning of the thread writes: rR has a point in that he stated it can't be done. I have the same strength of point in that I said it can. That's as advanced as the debate is right now Hmmm, and 100 posts in and I don't think anyone has advanced beyond this yet
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Modulous Member Posts: 7801 From: Manchester, UK Joined: |
Though could this be the first mention of Euler's Identity at EvC? I hope not, but I can't be sure. It would seem a great angle for the fine-tuners to try and come from. Unfortunately, it's not like its an easy thing to search for!
Hmmm, and 100 posts in and I don't think anyone has advanced beyond this yet Come one, we've moved at least a little bit! If you fancy trying to mathematically prove something to somebody in this medium, go for it. I'm sure it'll be as entertaining as usual. I considered starting calculus 101 in light of rRs infinity issues.
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cavediver Member (Idle past 3673 days) Posts: 4129 From: UK Joined: |
Come one, we've moved at least a little bit! Yes, but the direction matters
If you fancy trying to mathematically prove something to somebody in this medium, go for it It's not the typesetting, it's teaching formal mathematics to an engineer pearls before swine, teaching pigs to sing, etc
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RickJB Member (Idle past 5020 days) Posts: 917 From: London, UK Joined: |
riverrat writes: Infinity is a subjective concept. If infinity exists, then so does God.0.999... = God What the... What are you smoking, Riverrat?
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cavediver Member (Idle past 3673 days) Posts: 4129 From: UK Joined: |
What the... What are you smoking, Riverrat? Give him some slack, it's a damn site shorter proof than Godel's
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kuresu Member (Idle past 2543 days) Posts: 2544 From: boulder, colorado Joined: |
Zeno's problem is that he assumes that man runs like the graph of a square root, or maybe more precisely, runs like the value e
All a man's knowledge comes from his experiences
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sidelined Member (Idle past 5938 days) Posts: 3435 From: Edmonton Alberta Canada Joined: |
RiVeRrat
sidelined writes: We need not concern ourselves with the infinity aspect of the values to the right of the decimal since all we need to know is that they are equal to one another riVeRrat writes: Says who? Since you can't get the end of the number, then you never really know if it is equal or not. It is not an absolute. Let us go through this again We are subtracting the infinite quantity 0.999... from 9.999... ok? Since the quantity that is being subtracted is to the right of the decimal point then we have the following operation. .999... minus .999... These 2 quantities, though infinite, are equal since the same number repeats itself and there is a one to one correspondence among all the member of the sequence. Therefore, it follows logically, since subtracting ANY given quantity from that same quantity equals zero. Another way to look at it is this. We know that.9 is .1 in difference from the value of 1 .99 is .01 in difference from the value of 1 .999 is .001 in difference from the value of 1 .9999 is .0001 in difference from the value of 1 Now to illustrate the concept let us make a table of values
.999... + 0 = 1 is correct because,since it continues off into infinty,it cannot ever show a difference value that can be added to it in order to arrive at one. No difference value is the same as zero difference value. Now if I had graphing capabilty or if I were better versed in calculus and the concept of limit then I caould show you how the quantity shrinks to zero in the approach to the infinite value. Edited by AdminJar, : formatting
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kongstad Member (Idle past 2899 days) Posts: 175 From: Copenhagen, Denmark Joined: |
riVeRrat writes: You can't subtract .9999999.... logically. That assumes infinity has an end. The number 0.99999... is defined as an infinite series. If you accept that the number exists, you must accept the concept of infinity. You are of course free to refuse this, but this would have fundemental effects on a lot of other things. When youaccept the notation 0.9999... you accept that the concept is that the 9's are repeated an infinite number of times. Now if you accept this or 0.9999... and 9.9999... then you must accept that9.9999...-0.9999... =9 is meaningful Instead of placing nine an infinite numbr of times, you subtract an infinite number of nines. One way to prove that 0.9999... equals 1 is to use infinite series. 0.9999... is defined as the sum 9/10 + 9/100 + 9/1000 + ... Does this sum converge on a number? The difference from 1 is 0.1 in the first instance, then 0.01, then 0.001 etc, the difference i 1/(10^n) in the n'th instance. Now if 1 does not equal 0.9999... then 1-0.9999... must be some number different than 0. Assume that 1 does not equal 0.9999... , then there must be difference, lets call it d, between them, that is1-0.9999...=d, where d>0 Now we know that the difference between 1 and 0.9999... is 1/(10^n) in after n instances. But we can make n arbitrarely large, since the sum that makes u 0.9999... is an infinete series, but then there will be a number N where 1/(10^N) is smaller than d! But that is contrary to our claim that d>0, so we can conlude that d=0, that is 0.9999...=1! This all hinges on the fact that 0.9999... is a sum of infinite many addends, so if you accept the notation 0.9999..., yuo must accept the conclusion that it equals 1.
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riVeRraT Member (Idle past 445 days) Posts: 5788 From: NY USA Joined: |
There may very well be a point to what you are saying. But if there is, you have not made it yet. where is the beef of your logic? we say that 1.0 is a way of representing a number, and that 0.999... is another way of representing that same number. Just as 1/3 is a way of representing a number a and 0.333... is another way of representing the same number. you seem to have objections to the former but not to the latter. why is that? I object both, but accept both. Can you get .999... by multiplying 2 numbers?
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kuresu Member (Idle past 2543 days) Posts: 2544 From: boulder, colorado Joined: |
On my Ti-86, a number with twelve decimal places is assumed to go on for inifinity. So .999999999999 stretches on for infinity, as should pi (but my calculator stops at eleven for some reason).
Anywho, .999999999999 / 2 = .5. So, .5 *2 = 1 or .999999999999 again, my calculator considers 12 decimal places to strech for infinity, so I have twelve for the decimals here (except .5, of course) All a man's knowledge comes from his experiences
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