#
Does pi Have an End?

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I've known of pi for many years now. I can recite it to about ten digits, or something like that. I know some people can do it to hundreds of digits and probably even thousands which is amazing. What I'm wondering is; will it ever reach an end?

I've seen pages which have listed it to millions of digits and it just seems to go forever. I've heard it really is infinite but...is it entirely infinite? Literally, I mean. If it is, is there a mathematical formula that suggests why this is?

I have a feeling I know the response will be that it is infinite but I'd love to hear input anyway. It's such a strange number.

A fellow named Lindemann proved that Pi is infinite. There is no “nutshell” description of this proof that does not use the Calculus, so putting it here would be difficult.

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and

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Doesn't end. And it doesn't even repeat in any kind of pattern.

1/3 is "infinite", but at least it has a repeatable pattern (.3333333333.....).

They've been striving to get to the last digit since antiquity. They reached 1 million digits in 1971. Today Pi has been taken to trillions of digits on supercomputers.

kokopelli

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If you are interested, here's the square root of 4 to a million decimal places:

http://www.gutenberg.org/files/3651/3651.txt

From the link:

**Quote:**

4. Actually, slightly more than 1 million digits are given here. These

digits were computed by Norman De Forest using a custom utility and a

command with a lot of dollar signs in the command line. They were

computed during his copious spare time on a standard IBM PC over the

course of about 6 minutes and 40 seconds. We do NOT guarantee the

accuracy of these digits. Although these digits have been checked once

we encourage others to check them as well.

I would like to think it does, just to piss off the yammering purists.

That is why I like Phi, instead. We know already how its never ending story turns out.

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π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999… *and so on!*

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I read this book recently and it discusses the proof for this theorum in great detail, it's an excellent read.

It talks about a lot of the nerdy mathematical formulas used in The Simpsons television show, they actually have a lot of educated writers on staff. Most of them have PhD from MIT or Harvard and put a lot of thought into the mathematical jokes in the show. You'll find your answer here.

kokopelli

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He is correct.

Consider countable vs uncountable.

A set is countable if you can pair each member of the set with a unique integer. If you have a set of three items, say {red, green, yellow}, you can say red is the first member of the set, green is the second member of the set, and yellow is the third member. In fact, any finite set is countable.

Many infinite sets are countable as well. For example, the integers since each integer can be paired uniquely with itself.

A set for which you cannot pair every member with a unique integer is called uncountable.

Remember that rational numbers are numbers that can be expressed by one integer divided by another non-zero integer. For example, 1/2, 3/4, 99/5132413241235123412, ... . There are countably many rationals. Irrational numbers are numbers that cannot be written as the ratio of two integers. Pi and e are examples of this.

Another way of looking at it is that rational numbers are numbers that can be expressed in decimal in which the digits repeat after some point.

It is easy to find two integers that represent any such number where the digits repeat. First, determine how many digits repeat. For example, 5/27=0.185185185... with the three digits 185 repeating over and over. Let x=0.185185185... . Multiply the number by 10 to the power of the number of digits that repeat. For x=0.185185185, multiply by 10^3 or 1000. Thus, 1000x = 185.185185185... . Subtract the first from the second 1000x-x=185.185185... - 0.185185... . You get 999x = 185. Thus, 0.185185185... = 185/999. When you divide top and bottom by the common denominator, 37, you get 5/27.

On the other hand, the perceived size of something doesn't necessarily mean that one is larger than the others.

Consider two concentric circles of various sizes. For example, draw one circle of radius 10 and one of radius 1 with both having the same center. There are exactly as many points in the outer circle as in the inner circle and vice versa.

To see this, pick any unique point on the outer circle and draw a line between that point and the center of the circle. It will cross the inner circle at one precise point. Thus, for every point on the outer circle, there is a unique point corresponding to that point on the inner circle. Thus, the inner circle contains at least as many points as the outer circle.

Similarly, choose a point on the inner circle and draw a line starting at the center of the circle and through that point on the inner circle and extending out to the outer circle. Again, for every point on the inner circle, there is a unique point on the outer circle associated with that point. Thus, the outer circle contains at least as many points as the inner circle.

And therefore, the number of points in each circle are identical.

If pi did have an end, it could be expressed as a fraction: every terminating decimal can be expressed as a fraction, as is clear from the following reasoning:

Consider 0.1 - this can be written 1/10; consider 0.01 - this can be written 1/100; consider 0.001 - this can be written 1/1000 and so on ad infinitum.

Clearly, any terminating decimal can be written as a sum of such decimals and, therefore, as a sum of fractions which is itself a fraction.

It can be proved fairly easily, albeit with a little calculus, that pi cannot be written as a fraction, therefore it is implicitly non terminating.

Next time around God will do better.

Sometime in the future God will toss this universe back into the vat, melt it down, and recast a whole new universe.

And in this future universe the value of Pi will be exactly 3.

in the next universe the circumference of all circles will be exactly three times their diameter. And that will make life for beings in this future universe much easier than it is for us.