My PI questions - I hope they are valid?

kris kaczmarczyk
1) If Pi has my birthday digit sequence starting at digit at ~ 200k position (I did check, strange experience), can It also contain digital encoded (Ascii) Shakespeare's Hamlet? or Any other human creation ever done now and in the future?

2) Pi square ~ 9.86960440109 , we can calculate exactly the value of these digits "9.8.."
but can I say (?) for example that digit 8 is influenced by the "last digits of Pi", so can I say that looking at 8 I see the consequence of some digits in infinite distance...

3) if our universe is the approximate sphere of 5 billion light years and the unit is Plank distance, can we calculate how many digits of precision of Pi do we need to precisely describe the sphere of the known Universe, then what would be the meaning of digits beyond that number?

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The answer to question 1 is Yes, and further the probability that it contains such an encoding is 1 (which does not necessarily mean it does contain such an encoding).

You might like the short story 'The Library of Babel' by Jorge Luis Borges, which contains an infinite number of books full of apparently random text. The monks that tend the library know that somewhere in the library must be a book or series of books that gives the answer to life, the universe and everything, in a way that is more helpful than Deep Thought's response of '42'. They devote their lives to trying to find it. But of course they never do.

BWV
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These are questions that can’t really be answered with other than guesses.

(1) it’s possible but it’s also possible that it will never occur nor any of the other works of literature in whatever language and coding scheme. Our best guess is that PI has no pattern in its expansion but who knows what we’ll see is selected strings of digits.

It’s akin to the Bible Code book that was popular some time ago where future predictions were discovered in the text of the Bible if you applied some arbitrary pattern of choosing every nth word or nth character or page ...

Of course, coincidences happened but when you prime yourself to find a pattern you have a good chance of finding it.

Carl Sagan in his Contact ended it with someone finding the image of a circle in the digits of pi as an indication that it was somehow constructed to be that way.

(2) i would say no but perhaps @fresh_42 would have a more definitive answer. To affect 8 you would have to generate a sum of 0.04 to push the 0.06 to overflow and that can’t happen as the decimal position gets smaller and smaller. I don’t like my answer here and hope that fresh can explain it better perhaps with a polynomial expansion of the product of ##\pi^2## again another guess.

(3) PI is a purely mathematical number and as such is of interest to a mathematician with all its digits. In contrast an engineer would use a numerical approximation that is relevant to his or her line of work. Same goes for the scientist.

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To expand on @jedishrfu answer to (2):

We can write pi as ##3.141 + 0.001 \theta## where ##0<\theta<1##.
Hence
$$\pi^2 = (3.141 + 0.001 \theta)^2 = 3.141^2 + 0.003141\theta + 0.000001\theta^2 = 9.865881 + \theta(0.003141 + 0.000001\theta)$$

The second term of that must lie in the range (0, 0.003142), since ##0<\theta<1##.

Hence we have:

$$9.865881= 9.865881+ 0< \pi^2 < 9.865881+0.003142 = 9.869023$$

So the first two digits after the decimal must be '86'.

jedishrfu
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1) If Pi has my birthday digit sequence starting at digit at ~ 200k position (I did check, strange experience), can It also contain digital encoded (Ascii) Shakespeare's Hamlet? or Any other human creation ever done now and in the future?

This is an open question in number theory. It depends on whether ##\pi## is a "normal" number or not. See:
https://en.wikipedia.org/wiki/Normal_number
2) Pi square ~ 9.86960440109 , we can calculate exactly the value of these digits "9.8.."
but can I say (?) for example that digit 8 is influenced by the "last digits of Pi", so can I say that looking at 8 I see the consequence of some digits in infinite distance...

When you multiply two decimal fractions, each digit of the answer depends only on the digits up to a certain point. You can see this by looking at the usual technique for long multiplication. Or, by looking at a decimal expansion as, for example:

##\pi = 3 + (1 \times 10^{-1}) + (4 \times 10^{-2}) + (1 \times 10^{-3}) \dots##

From this you can see that the first digit of ##\pi^2## after the decimal point depends only on the multiplication of:

##3.14 \times 3.14 = 9.8596##

If we take the more precise calculation:

##3.141 \times 3.141 = 9.865881##

The second calculation differs from the first only from the second decimal term onwards. The ##9.8## is common. And, in fact, we see now that ##\pi^2 = 9.86 \dots## etc. If we add more decimal places, we do not change the ##9.86##.

3) if our universe is the approximate sphere of 5 billion light years and the unit is Plank distance, can we calculate how many digits of precision of Pi do we need to precisely describe the sphere of the known Universe, then what would be the meaning of digits beyond that number?

This question makes no sense. The Planck distance is not a quantisation of space. That's a common misconception. See, for example:

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

kris kaczmarczyk, QuantumQuest and fresh_42
kris kaczmarczyk
Thank you-
3) @fresh_42 looking at the beautiful book the Zoomable Universe, we have 10^27 Universe and 10^-18 field scale , on next page is Pi ;) - my question was how can we figure out to measure 10^27 with resolution 10^-18 for example (not a Planck but BIGGER) , how many digits of Pi we would need for this resolution;

3b) friend of mine ask me once , i never solved it: from the point in the middle of NYC what is the equal distance I have to walk NORTH, WEST, SOUTH and EAST so I end up 1 meter WEST of the point I have started?

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When you multiply two decimal fractions, each digit of the answer depends only on the digits up to a certain point.
Although this is normally true, there are pathological counter-examples.

Consider an easy one. Take the decimal expansion of ##\sqrt{2}## and square it. The result is 2 (of course).

Now find a non-zero digit anywhere in that decimal expansion. Decrement it and the square of the resulting number will be 1.something.

kris kaczmarczyk
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2022 Award
Although this is normally true, there are pathological counter-examples.

Consider an easy one. Take the decimal expansion of ##\sqrt{2}## and square it. The result is 2 (of course).

Or, the result is ##1.999 \dots##

kris kaczmarczyk, nuuskur and jbriggs444
Gold Member
3) @fresh_42 looking at the beautiful book the Zoomable Universe, we have 10^27 Universe and 10^-18 field scale , on next page is Pi ;) - my question was how can we figure out to measure 10^27 with resolution 10^-18 for example (not a Planck but BIGGER) , how many digits of Pi we would need for this resolution;
It's unclear exactly what you're asking, but (for example) the radius of the observable universe is something on the order of 1061 Planck lengths. If you knew this number to the nearest Planck length and wanted to figure out what (e.g.,) the circumference of the observable universe was to the same accuracy, you would need to know pi to roughly 61 digits, since pi enters into the circumference equation linearly.

jedishrfu
kris kaczmarczyk
It's unclear exactly what you're asking, but (for example) the radius of the observable universe is something on the order of 1061 Planck lengths. If you knew this number to the nearest Planck length and wanted to figure out what (e.g.,) the circumference of the observable universe was to the same accuracy, you would need to know pi to roughly 61 digits, since pi enters into the circumference equation linearly.
Thank you very much, this is exactly what I was asking for. Believing or not this is surprising answer (expected much more digits!) . Now when I think about it, the difficulty lies in measuring Universe with such a precision.

kris kaczmarczyk
This is an open question in number theory. It depends on whether ##\pi## is a "normal" number or not. See:
https://en.wikipedia.org/wiki/Normal_number

When you multiply two decimal fractions, each digit of the answer depends only on the digits up to a certain point. You can see this by looking at the usual technique for long multiplication. Or, by looking at a decimal expansion as, for example:

##\pi = 3 + (1 \times 10^{-1}) + (4 \times 10^{-2}) + (1 \times 10^{-3}) \dots##

From this you can see that the first digit of ##\pi^2## after the decimal point depends only on the multiplication of:

##3.14 \times 3.14 = 9.8596##

If we take the more precise calculation:

##3.141 \times 3.141 = 9.865881##

The second calculation differs from the first only from the second decimal term onwards. The ##9.8## is common. And, in fact, we see now that ##\pi^2 = 9.86 \dots## etc. If we add more decimal places, we do not change the ##9.86##.

This question makes no sense. The Planck distance is not a quantisation of space. That's a common misconception. See, for example:

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/
Thank you very much for the answers.