Why are large calculations so hard to compute?

I don't know too much about computers, but I started to wonder one day why modern computers can't calculate something like the value of pi to as many decimal places as we want. Isn't the computer only doing one calculation at a time (the next step in the division process) and then storing the number it gets, moving on to the next one? I mean, if each number takes up one byte of information, and you had a 30 gigabyte HD you should be able to calculate pi to that high a precision too right?

My thinking must be flawed in here somewhere, because if it was this easy people would have done it already. If someone could point me in the right direction on this one it'd be great. Thanks!

 

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Ah. I was under the impression that you could calculate it as you would any other number. Say 5/3. You would divide 3 into 5 (answer being 1) then carry over the remaining 2 and divide 3 into that again and so on. Wouldn't it just move to the next number after it has stored the answer from the previous?

But still, if computers are able to run a couple hundred thousand calculations a second (or more?) why does it take so long to divide pi?

 

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It really depends how accure you want pi to be. If you just want a few millions digits, then it takes only a matter of seconds divide pi. And I won't call that is long.

 

Intuitively, from the perspective of information content, Pi contains an infinite amount of information. To compute its value totally would require either an infinite amount of memory (hopefully with the values already present) or an infinite amount of time.

An interesting consequence is that the sequence of Pi's digit contains all possible encodings of all possible information that ever could exist.

Similarly, a computer given infinite time can enumerate all possible encodings of all possible information that ever could exist.



This should be disturbing. Enjoy.

 

pi

Conceive a sphere constructed with the earth at it center, and imagine it surface to pass through Sirius, which is 8.8 light years distant from the earth... Then imagine this enormous sphere to be so packed with microbes that in every cubic millimeter millions and millions of these diminutive animalcula are present. Now conceive these microbes to be unpacked and so distributed singly along a straight line that every two microbes are as far distant from each other as Sirius is from us... Conceive the long line thus fixed by all the microbes at the diameter of a circle, and imagine its circumference to be calculated by multiplying it diameter by Pi to 100 decimal places. Then, in the case of a circle of this enormous magnatude even, the circumference so calculated would not vary from the real circumference by a millionth part of a millimeter.

This example will suffice to show that the calculation of Pi to 100 or 500 decimal places is wholly useless.

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IIRC, there is no algorithm that can compute the n-th digit of pi without computing the n-1 preceding digits. So, while an algorithm exists, you require an unbounded amount of information to compute an arbitrary digit of pi.

 

Pi is an infinite sequence of random numbers.

 

Interesting, James. If you have a reference to this algorithm, I'd appreciate it. Obviously if this is true, then what I've been saying is indeed incorrect.

Thanks

 

Algorithm for finding nth digit of pi can be found at http://www.mathsoft.com/asolve/plouffe/plouffe.html .

 

If you had a long enough string of random digits most of the time you could probably find patterns in it anyway. There are lots of different possible patterns, so even a totally random string of digits is likely to have some redundancy. At least that's what I'd think intuitively (it would just take a lot of computation time and most compression algorithms focus on text because that's the most wasteful).

I wouldn't consider the digits of pi random either, but if you chopped of the first 100 digits and gave the string to someone I bet they would never guess it was pi [or rather part of it ...]. I think random is relative to your amount of knowledge (Try to imagine a physical mechanism that's random). That's why random strings and strings with high information content are essentially indistinguishable.

 

Just because the sequence is always the same doesn't mean it's not random. You can make a random sequence of numbers, and use it in as many instances as you like and it's still a random number.

 

Alpha, what is your definition of random?

Even if the sequence internally has no obvious pattern, I would say it's relationship to the radius and circumference of a circle qualify it as giving the sequence an ordered pattern.

 

Is there a repeating pattern after so many decimals in pi? Ie. ...24242424 or something like that? Or is it a whole swack of different numbers? I just don't see how any value can be a chain of infinite numbers, if there is no pattern in them at all.

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I heard somewhere that quadrallionth digit of Pi is zero and after that it repeats itself??...??

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bye!

 

Zion, possibly the entity was stating a useful engineering approximation similar to using 3.14 at the “value” for Pi. Perhaps the entity meant that the quadrallionth digit of Pi is zero and the next sequence of a quadrallion quadrallion digits of Pi happen to be zero. After that the entity didn’t care…being an engineer and not a mathematician. Hehe.

Is Pi erotic…err…that is, ergodic?