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Post by Progenitor A on Sept 12, 2010 17:27:31 GMT 1
Also in the Spectator:
10/3 = 3.333333333333333333333333333333333333........
An infinty of three's! Ahhhhhg........! INFINITY!
But 10/3 = 3+1/3
God !
Where has infinity gone!
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Post by abacus9900 on Sept 12, 2010 17:46:34 GMT 1
Also in the Spectator: 10/3 = 3.333333333333333333333333333333333333........ An infinty of three's! Ahhhhhg........! INFINITY! But 10/3 = 3+1/3 God ! Where has infinity gone! That's just the way the decimal system converts to ordinary fractions, naymissus. I suppose you could say it shows the incompatibility of one system with another, but as it works in practice it does not really matter. In theory, the decimal form of the fraction could go on forever but what would be the point? It's just an idea, in the final analysis, that can be applied in the real world.
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Post by alanseago on Sept 12, 2010 17:56:53 GMT 1
Goes along with the value of pi when I was at school. I remember insisting that there must be precise figure.
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Post by abacus9900 on Sept 12, 2010 18:07:40 GMT 1
Goes along with the value of pi when I was at school. I remember insisting that there must be precise figure. Yes, and people are still trying to break the current record, but why? It's meaningless!
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Post by lazarus on Sept 12, 2010 18:33:52 GMT 1
Goes along with the value of pi when I was at school. I remember insisting that there must be precise figure. Yes, and people are still trying to break the current record, but why? It's meaningless! It doesn't actually loop, or hasn't been found to repeat yet, so it may well have answer. I do remember thinking in maths at school that we could put men on the moon but can't work out the exact diameter of the hatch on the rocket.
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Post by marchesarosa on Sept 12, 2010 20:33:31 GMT 1
"can't work out the exact diameter of the hatch on the rocket".
Why ever not ? Don't they have a tape measure?
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Post by alanseago on Sept 12, 2010 20:41:58 GMT 1
You cannot calculate with a tape measure. Its a man thing.
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Post by speakertoanimals on Sept 13, 2010 17:21:00 GMT 1
(1/3) isn't really like pi, in that although both involve infinite deciaml expansions, (1/3) is a recurring decimal, whereas pi isn't.
It's the difference between rational numbers, which can be written as p/q where p and q are integers, and irrational numbers, which can't.
Rational numbers either have a finite or a recurring decimal expansion (or binary, you can chose another basis if you wish), whereas numbers like pi can't have a finite or recurring expansion, however hard you try.
To those that may doubt these statements or think that pi MAY end if we just keep trying, look at what the greeks discovered (but got a bit upset about).
It is fairly straightforward to show that square root of 2 cannot be written as p/q, that the decimal expansion of square root of 2 carries on for ever without repeating -- irrational numbers exist, even without looking for complicated stuff like pi.
It can also be shown that pi is also irrational.
Hence these computations of umpteen digits of pi are not trying to 'prove' what is already known, but instead are good tests for various computational techniques. Indeed, modern formulae for computing pi, which are rather weird and wonderful, contain so many digits that their only use is for testing new supercomputers. What a better advert for your new machine, than saying how many digits of pi it can compute setting a new world record, or how fast it can reach the old one?
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Post by baalmaiden on Sept 14, 2010 18:54:34 GMT 1
What boggles the mind is that there are many infinities. There is an infinity of even numbers, likewise odd numbers and when you get to fractions, well.....
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Post by speakertoanimals on Sept 14, 2010 19:18:46 GMT 1
Actually, when it comes to fractions (well rational numbers p/q where p and q have no common factors and are integers), there are as many of these as there are integers.
Why? Imagine points (p,q) in the xy plane, each point is a rational number. We can imagine a spiral path, which comes out from the origin, and eventually passes through each point once and only once. This then gives us a list of the rational numbers, hence there are as many of them as there are integers.
The CONTINUUM, that is another beastie, and there are a bigger infinity of numbers between 1 and 1.001, than the whole infinity of integers. Cantor was a clever chappie!
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Post by Progenitor A on Sept 14, 2010 19:48:51 GMT 1
What boggles the mind is that there are many infinities. There is an infinity of even numbers, likewise odd numbers and when you get to fractions, well..... Not only that , but there are some infinties that are bigger than other, e.g. there are more even numbers in the even number infinite set than there are prime numbers in the prime number infinite set! Infinity is an embarrassment out of which mthematicians struggle to escape.
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Post by abacus9900 on Sept 15, 2010 8:51:55 GMT 1
Of course it is because it is absurd!
How can one infinity be more than another? Infinity is not quantifiable so it is a meaningless and untestable concept. This is a good illustration of how sometimes mathematicians indulge in skewed thinking.
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Post by eamonnshute on Sept 15, 2010 9:22:45 GMT 1
How can one infinity be more than another? Infinity is not quantifiable so it is a meaningless and untestable concept. Think about finite sets. How can you decide if one is bigger than the other? Easy - just pair them up and see if anything is left over. If the sets are {a,b,c} and {d,e} then we can pair a with d, b with e, and c is left over, so the first set is bigger. We do not need to quantify the sets to compare them! We can do the same with infinite sets. If we compare the set of integers with the set of numbers between 0 and 1 by pairing the members of the two sets then we find that there are numbers left over in the second set, which means that the second set is bigger. This is how you do it. First write down the integers and pair each one with a number between 0 and 1, for example: 1 -> 0. 32856435568....... 2 -> 0.8 7846674676...... 3 -> 0.67 013257545...... etc. Now suppose we make a number which is different from every one of the right-hand numbers. We can do this by taking the first digit of the first number, the second digit of the second number, and so ad infinitum: 0.370... Now change each digit, eg 0.481.... You can see that this number is different from the first number in the list, and the second, and all of them. So we have proved that there are more numbers in the list than we can count! So there are more numbers between 0 and 1 than there are integers.
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Post by Progenitor A on Sept 15, 2010 9:34:33 GMT 1
How can one infinity be more than another? Infinity is not quantifiable so it is a meaningless and untestable concept. Think about finite sets. How can you decide if one is bigger than the other? Easy - just pair them up and see if anything is left over. If the sets are {a,b,c} and {d,e} then we can pair a with d, b with e, and c is left over, so the first set is bigger. We do not need to quantify the sets to compare them! We can do the same with infinite sets. If we compare the set of integers with the set of numbers between 0 and 1 by pairing the members of the two sets then we find that there are numbers left over in the second set, which means that the second set is bigger. This is how you do it. First write down the integers and pair each one with a number between 0 and 1, for example: 1 -> 0. 32856435568....... 2 -> 0.8 7846674676...... 3 -> 0.67 013257545...... etc. Now suppose we make a number which is different from every one of the right-hand numbers. We can do this by taking the first digit of the first number, the second digit of the second number, and so ad infinitum: 0.370... Now change each digit, eg 0.481.... You can see that this number is different from the first number in the list, and the second, and all of them. So we have proved that there are more numbers in the list than we can count! So there are more numbers between 0 and 1 than there are integers. So infinity is countable? And there is meaning in term such as 'infinite gravity' as long as we use the right numbers?
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Post by eamonnshute on Sept 15, 2010 9:42:22 GMT 1
So infinity is countable? And there is meaning in term such as 'infinite gravity' as long as we use the right numbers? The infinity of integers is obviously countable, but there is at least one other infinity which are not countable, as I have demonstrated. As for infinite gravity, when physics gets infinities it sets the alarm bells ringing - it means that there is probably something wrong with the theory. So infinite gravity probably doesn't exist in Nature. And thinking of infinity as just another number is wrong, it has a rather different meaning.
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