[Jimbee (OP)]

Why does it seem so hard to calculate the value Pi? Pi is defined as the ratio of a circle's circumference to its diameter. And it is non-terminating and non-repeating. But in calculating it, the further out they go, the more difficult the task seems. To-date, the furthest it's been calculated is to 105 trillion places this year.

Is it because the further out you go the larger (or actually smaller) the digit that far out is? For example, for 3.141592653... the three would be 3 divided by 100,000,000.

Because some time back, I got this idea. .01001000100001... is an example of a non-terminating and non-repeating number. But the pattern is obvious. So could it be there is a more simple way of calculating Pi based on some pattern that we haven't discovered yet?

[alancalverd]

I understand that there is a proof of no possible pattern. Unless of course you are a rotifer or corona virus in which case decimal numbers are irrational as everything that matters is really a rational multiple of  π

[paul cotter]

Pi, like e, the base of Naperian logs, is a transcendental number with no pattern as far as we can tell.

[Bored chemist]

It's generally believed (but not proven) that pi is a "normal" number in the technical sense that there are equal numbers of 1s 2s 3s etc in the decimal expansion of it.
That rules out numbers of the sort you constructed which has a lot more 0s than 1s and none of any other digit.
Other patterns might be  possible.
Good luck finding one.

[evan_au]

Pi can be represented by a very simple pattern:
π/4 = 1 - 1/3 + 1/5 - 1/7 + etc

This series is very slow to converge, so it's not of any practical use to calculate π

But it does illustrate some important aspects:
- 1/1000001 is zero in the first 6 decimal places
- So any pattern that you might detect in the first 6 decimal places will be destroyed by adding in this factor which does not affect the first 6 decimal places.
- The series continues forever, so it will destroy any other pattern you detect in the decimal expansion

It is necessary that the digits of π (or any transcendental number) do not form a repeating pattern
- But it is not sufficient

If you want to represent π, you could use the infinite series, or adopt the standard mathematical shorthand, and just write "π".

https://www.forbes.com/sites/kevinknudson/2016/04/18/everything-you-wanted-to-know-about-pi-part-4-infinite-series/

[Eternal Student]

Hi.

      I quite like everything that's been said already.   However, it's just faster and more natural to start with some minor arguments before going on with anything else.

     

I understand that there is a proof of no possible pattern.

      Possibly - but a lot of texts on the subject would just consider "a pattern" as a simple repeating pattern of digits in the decimal expansion.    There certainly isn't one of those.      @Jimbee 's question is harder because they allow more complicated things to still be considered as "some pattern".
    This leads to the comments made by @evan_au   --->   There certainly are some rules or algorithms (e.g. those based on summing an infinite series) that you could use to generate as many decimal digits for Pi as you require.   The most generalised notion of there being some pattern or rule for the digits would just be that a suitable algorithm to generate the digits does exist.

   @evan_au ,  you make this point well but then switch back to discussing "patterns" as if they could only mean a simple pattern of repeating digits.   

- 1/1000001 is zero in the first 6 decimal places
- So any pattern that you might detect in the first 6 decimal places will be destroyed by adding in this factor which does not affect the first 6 decimal places.

     An algorithm exists to determine an arbitrary number of digits.   So "some rule" exists to determine the next N digits given the first N digits,  even if all I do with the first N digits is just count them so that I know I will need the N+1 through to 2N  digits calculated.

- - - - - - - - - -

I think it's time to return to the first line of the OP and go along with the spirit of @alancalverd 's first reply.

Why does it seem so hard to calculate the value Pi?

       It's only because the language and development of mathematics has brought your attention to it and given some cause to think that we may want to express these sorts of ratios in a certain way.
       As suggested by @alancalverd , we could use a system where Pi seemed perfectly natural and could even be considered as just one unit and written as  1.    However, that would make some other numbers seem strange or "irrational", so we'd just be moving the problem somewhere else.
       The only thing we can't do is make ALL numbers seem perfectly rational.   The ancient Greek mathematicians and philosophers were very worried about this, so you're in good company if you feel that all numbers should be rational or well behaved.   Indeed it turns out that, regardless of what you take as your number 1, when you develop this into a complete number system there will always end up being vastly more numbers that are "irrational", transcendental or badly behaved.
        This is worth making clear:   There are vastly more irratonal numbers than rational ones.   It is a miracle that we can do so much while confining ourselves to just some numbers that seem natural - but we can.     If you dropped a pin into the Real number line and picked a number "at random" the chance of it being a well behaved rational number is zero (or "infinitessimally small" but it would need another whole forum post to discuss whether an infinitessimal is precisely equal to zero).   The truth is that most real numbers will be un-natural slippery little scallywags.   Almost everything you see - some length, some weight (etc.) will be an irrational number of units in some system that anybody else may choose to measure those things in   (e.g.  metres, inches or "hands" etc.).  So these things would always require an infinite and non-repeating set of digits to write it in a decimal form for that other person.
     The good news is that, regardless of what quantity you start from and consider as one unit and write as just 1,  an identical structure emerges and behaves as the number system we are familiar with and were taught about in school.   Thus, we have no difficulty in recognising that when we add 1 sheep to a field that had 2 sheep in it, we have a field with 3 sheep in it.    Similarly, if we add 1 cow to a field of 2 cows, we have 3 cows in the field.   This holds perfectly well even though     1 sheep     is clearly not the same as     1 cow.    It doesn't matter, the mathematical structure is the same and will  behave the same whether your unit is the cow or the sheep.   However, it does matter if we need to make an exchange, attempt to make some comparison or find a ratio between these two systems.   If we went to market and wished to barter and swap our cows for someone else's sheep then the chances are that 1 cow is actually worth an irrational number of sheep.   I'm going to guess a cow would provide a farmer with 3 times as much meat as a sheep but it's a unlikely to be exactly 3 times.   As discussed earlier (dropping a pin into the real number line) the cow-to-sheep ratio is never going to be a perfectly rational number.  So, in principle, we could never make a fair exchange of some sheep for any number of cows - beacuse no natural multiple of cows would ever be exactly equal to some other natural multiple of sheep.   Problems with the ratio of two quantities being badly behaved (irrational) are to be found everywhere and we do encounter them ALL the time.   Human beings are just very good at making approximations and getting a working solution.   In the history of man, we have made an exchange of some sheep for some cows.   The absolute impossibility of any fair exchange existing just went un-noticed.     

Best Wishes.

[alancalverd]

@Jimbee 's question is harder because they allow more complicated things to still be considered as "some pattern".

The essence of a pattern is that, by examining a sufficient sample of prior events, you can predict the next one. You can't do this with the digits of π.

This is where human codebreaking is actually rather good. A devout mathematician might look at 20 digits from somewhere in the middle of the first million of pi, recognise their source, and then apply a good algorithm to calculate umpteen more, but that requires information that is not contained in the sample itself.

Apparently some breakthroughs at Bletchley Park were due to humans recognising that .... .... (H H) was significant to Germans and not just a random test signal. Similar flashes of inspiration came from guessing the intent of some mis-spelt words - easy enough if you have prior knowledge of warfare and bad German, but not what you could call pattern recognition, and also recognising that some twit had realised he had mis-spelt some words so sent the entire message again without altering the Enigma settings.

[alancalverd]

As suggested by @alancalverd , we could use a system where Pi seemed perfectly natural and could even be considered as just one unit and written as  1.    However, that would make some other numbers seem strange or "irrational", so we'd just be moving the problem somewhere else.

It would present quite a conundrum for our π-based rotifer. Obviously the things of immediate personal concern, such as his volume (a measure of health) and surface area (determines his ability to transpire and thus maintain that health), would be conveniently rational functions of his radius (his ability to squeeze through a gap) but how can he describe another rotifer and another rotifer and yet more rotifers...if he only has a finite number of fingers to count on?

[reyesryder]

I really don't understand that either!

[Eternal Student]

Hi.

I really don't understand that either!

    That's not a problem.   You can ask for any clarification.    However, quite often a very short reply like the one you have made is just some prelude to advertising something.

Best Wishes.

[Bored chemist]

As suggested by @alancalverd , we could use a system where Pi seemed perfectly natural and could even be considered as just one unit and written as  1.    However, that would make some other numbers seem strange or "irrational", so we'd just be moving the problem somewhere else.

It would present quite a conundrum for our π-based rotifer. Obviously the things of immediate personal concern, such as his volume (a measure of health) and surface area (determines his ability to transpire and thus maintain that health), would be conveniently rational functions of his radius (his ability to squeeze through a gap) but how can he describe another rotifer and another rotifer and yet more rotifers...if he only has a finite number of fingers to count on?

Yay! We are set up for the correct use of the word "incommensurate".