What Is Pi, and Why Is It So Important?

[rprosperi]

Interesting, though not too technical (for this crowd), article on Pi:

https://www.extremetech.com/science/what...-important

A quoted tidbit of interest for the accuracy/precision oriented folks here:

"Fact: Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen. "

[Steve Simpkin]

(02-20-2024 04:48 PM)rprosperi Wrote:  Interesting, though not too technical (for this crowd), article on Pi:

https://www.extremetech.com/science/what...-important

A quoted tidbit of interest for the accuracy/precision oriented folks here:

"Fact: Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen. "

So Free42/Plus42/DM42 with 34 digits of Pi is probably Ok for most general uses?
   

[J-F Garnier]

(02-20-2024 04:48 PM)rprosperi Wrote:  "Fact: Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen. "

This is a common statement, that just forgets that we don't know the diameter of the observable universe with the same accuracy - maybe 1% if the current theory is correct.
So using 22/7 or 3.14 would be enough to estimate the circumference of the observable universe.
I would just use 3.

J-F

[gregul]

I asked MS copilot (chat GPT-4):[/quote]
what are the last 30 digits of pi ? ;-)

Copilot:
The last 30 digits of pi are 3095295560. This was computed in 2022 by Timothy Mullican, who calculated 50 trillion digits of pi using a cloud computing service. The previous record was 31.4 trillion digits, set by Emma Haruka Iwao in 2019.

[DA74254]

(02-20-2024 05:13 PM)Steve Simpkin Wrote:  

(02-20-2024 04:48 PM)rprosperi Wrote:  Interesting, though not too technical (for this crowd), article on Pi:

https://www.extremetech.com/science/what...-important

A quoted tidbit of interest for the accuracy/precision oriented folks here:

"Fact: Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen. "

So Free42/Plus42/DM42 with 34 digits of Pi is probably Ok for most general uses?

Yes, it should suffice.
I made a post here some time ago, where I compared 15 vs 16 decimal places of PI. Based on the Earth orbit around the Sun, It came down to less than a nanosecond in time of a year and less than the diameter of a single strand of human hair for the traveled distance.

[Matt Agajanian]

Hi all.

This begs my question. If so much precision is needed in certain sciences, why use a physical calculator with its limited precision in the first place? I ask because as much as I love calculators, all the discussion about extreme precision (even at a 32 digit minimum), it seems and sounds like a calculator is useless and is better relegated to paperweight status.

Please enlighten me about the usefulness of calculators in the sciences.

Thanks.

[KeithB]

"The last 30 digits of pi are 3095295560. "
Wait, you did not even get 30 digits! Unless of course it is still waiting for them to be found. Sounds like a way that Captain Kirk would have defeated a robot enemy.

[Johnh]

I can well believe that we can calculate pi to these extreme accuracies now. But I'm more fascinated by how Archimedes did it by polygons inside and outside the circle, of increasing numbers of sides and so approching a circle by geometry. And that's easy enough for us now. But how did HE work the numbers? eg how to calculate square roots in ancient Greece? if he needed to?

In fact, how did they even think about and multiply numbers? Did they have something like the Roman system , but earlier? It's not easy for doing maths with!

[nickapos]

(02-21-2024 10:06 PM)Matt Agajanian Wrote:  Hi all.

This begs my question. If so much precision is needed in certain sciences, why use a physical calculator with its limited precision in the first place? I ask because as much as I love calculators, all the discussion about extreme precision (even at a 32 digit minimum), it seems and sounds like a calculator is useless and is better relegated to paperweight status.

Please enlighten me about the usefulness of calculators in the sciences.

Thanks.

Most times I use a calculator, I do it in front of at least one or even two computers.
I usually use it to calculate intermediate values or most often to validate results I see on the computer screen.
Having a purpose built well designed and robust portable machine for numerical analysis is very convenient and very far from it being a paperweight.

It also helps that these machines are very feature full and on top of that programmable so whatever feature is missing, it can be added.

I usually don’t need extreme precision but it is important to have it to minimise any internal calculation errors.

[Allen]

(02-20-2024 04:48 PM)rprosperi Wrote:  "Fact: Scientists have calculated pi to a precision of a hundred trillion digits. However, mathematicians have estimated that an approximation of pi to 39 digits will suffice for any cosmological calculation humans might attempt. At that level of precision, which mathematicians achieved in 1630, you can calculate the circumference of the observable universe with an error smaller than the diameter of a single atom of hydrogen. "

Nice factoid! Maybe throw in a few extra digits to close the gap between (revised) estimated diameter of the visible universe at 1e27 meters and the plank length at 1e-34 m.

62 orders of magnitude should cover everything except for global sovereign debt (compound interest being most powerful force in the universe - attr. A.E. )

[dm319]

(02-21-2024 10:06 PM)Matt Agajanian Wrote:  This begs my question. If so much precision is needed in certain sciences, why use a physical calculator with its limited precision in the first place?

Having recently been trying to double-check answers on the DM42, I can say it isn't that trivial to exceed the binary-coded decimal representations of calculators. I used an arbitrary precision library to do it, but if you just calculate using the default double precision binary then you get weirdness even with 0.3-0.2. I'm not sure how many digits of BCD is better than double precision binary FP though.

[SlideRule]

(02-21-2024 10:06 PM)Matt Agajanian Wrote:  This begs my question. If so much precision is needed in certain sciences, why use a physical calculator with its limited precision in the first place? I ask because as much as I love calculators, all the discussion about extreme precision (even at a 32 digit minimum), it seems and sounds like a calculator is useless and is better relegated to paperweight status.

Please enlighten me about the usefulness of calculators in the sciences.

a slight reinterpretation from a well known play; “O, thou hast damnable iteration, and art indeed able to corrupt a saint". most handy for recursive, iterative, etc. calculations with initial estimations, approximations, etc. NAY?

BEST!
SlideRule