this post was submitted on 08 Nov 2023
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84
At which scale (real world day to day life situation) when each extra digit of pi become necessary? (kbin.social)
submitted 9 months ago* (last edited 9 months ago) by [email protected] to c/[email protected]
45 comments fedilink hide all child comments
 

xkcd: Coordinate Precision but pi (π)?

I tried looking for some answer but found mostly

  • People reciting pi
  • People teaching how to memorize pi
  • How to calculate pi using different formula
  • How many digits NASA uses

Update question to be more specific

In case someone see this later, what is the most advanced object you can build or perform its task, with different length of pi?

0, 3 => you can't make a full circle

1, 3.1 => very wobbly circle

2, 3.14 => perfect hole on a beach

3, 3.142 => ??

4, 3.1416 => ??

5, 3.14159 => ??

Old question below

In practice, the majority of people will never require any extra digit past 3.14. Some engineering may go to 3.1416. And unless you are doing space stuff 3.14159 is probably more than sufficient.

But at which point do a situation require extra digit?
From 3 to 3.1 to 3.14 and so on.

My non-existing rubber duck told me I can just plug these into a graphing calculator. facepalm

y=(2πx−(2·3.14x))

y=abs(2πx−(2·3.142x))

y=abs(2πx−(2·3.1416x))

y=(2πx−(2·3.14159x))

Got adequate answer from @dual_sport_dork and @howrar
Any extra example of big object and its minimum pi approximation still welcome.

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[–] [email protected] 52 points 9 months ago (6 children)

On the NASA front, I believe I read somewhere that NASA determined that only 40 decimal places of pi are required to define a sphere the size of the observable universe to the accuracy of +/- the width of one hydrogen atom. It seems like you could file that under "close enough."

Just using 3 is certainly too low of a precision -- unless you're writing a major work of religious literature, of course. 3.1 is likewise unlikely to result in acceptable accuracy on a terrestrial scale. I've always used 3.14159 which is conveniently exactly what I can remember without looking it up and it's always been good enough for me. I don't think I've ever in my life needed to scribe a circle much larger than a couple of feet across at any rate.

You may be interested in reading this: https://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/

[–] [email protected] 15 points 9 months ago

And if my maths is correct, 63 decimal places gets that universe circle's precision to within a Planck length

[–] [email protected] 3 points 9 months ago

Thank you, I already skimmed through that article before posting. Maybe I failed to put my question into words properly.

I want examples similar to pool/fence circumference in the article. Along the line of "We're building x, and this is the worst rounding we can go, one fewer digit and it will be off by y"

[–] [email protected] 2 points 9 months ago

When we round pi to the integer 3 … to estimate the circumference of an object with a diameter of 100 feet, we will be off by a little over 14 feet.

It seems like ‘3’ is sufficient for real life. It’s probably more precise that I can freehand draw a circle. If I really did need to measure the fencing for a circle with diameter 100’, it’s within the window of padding I would estimate

[–] [email protected] 1 points 9 months ago

Upvoted for religious text reference. Another way to put it would be if you are OK with being off by 33% for insect leg count, you can use 3 for pi.

[–] [email protected] 1 points 9 months ago (1 children)

Now I'm imagining the Holy Pinity.

[–] [email protected] 3 points 9 months ago

For those still in the dark, I'm referencing the bible in 1 Kings 7, 23-26:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about. And under the brim of it round about there were knops compassing it, ten in a cubit, compassing the sea round about: the knops were cast in two rows, when it was cast. It stood upon twelve oxen, three looking toward the north, and three looking toward the west, and three looking toward the south, and three looking toward the east: and the sea was set above upon them, and all their hinder parts were inward. And it was an hand breadth thick, and the brim thereof was wrought like the brim of a cup, with flowers of lilies: it contained two thousand baths.

Those measurements would only work if either the cauldron were not actually circular, or if Pi were equal to 3.

[–] [email protected] 22 points 9 months ago (2 children)

You can calculate this yourself. For example, if you're working on an object that's 1m in diameter and you use 3.14 to compute the circumference, then you can expect errors of up to 1m * (3.142 - 3.14) = 0.002m = 2mm.

[–] [email protected] 4 points 9 months ago

Thank you. After thinking about it overnight, I realized I asked a wrong question. Your answer still helps greatly and get me more than half way to satiate my curiosity.
Tolerance grade and example objects that require different grade/minimum pi accuracy is what I was looking for.

[–] [email protected] 4 points 9 months ago

This is almost the real answer, only in reverse. Find what the appropriate error bounds of your problem is and use continuity (it probably is continuous) to find what decimal expansion you need. Or you could probably just find a solution expressible in pi and pick the decimal approximation needed. Either way, who cares about pi?

[–] [email protected] 16 points 9 months ago (3 children)

Not a lot, and this is why to speedup thing on some architecture, when working with (unsigned)integer you multiply by 355 then divide by 113 (it's like 3.14159292035)

[–] [email protected] 12 points 9 months ago

I forgot the divide by 113 and now I have a huge house.

[–] [email protected] 9 points 9 months ago (1 children)

TIL there is a closer approximation than 22/7

[–] [email protected] 2 points 9 months ago* (last edited 9 months ago)

There are infinitely many. Any sequence of rational numbers converging to pi contains infinitely many. 22/7 and 355/113 are just particularly good ones for their small denominators. You can find such good approximations ("the best rational approximations of a given size") by truncating the continued fraction representation of pi:

pi = [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,…]
pi = 3  + 1/(7 + 1/(15 + 1/(1 + ... )))

These approximations yield:

pi ~ 3
pi ~ 3 + 1/7 = 22/7
pi ~ 3 + 1/(7 + 1/15) = 333/106
pi ~ 3 + 1/(7 + 1/(15 + 1)) = 355/113
pi ~ 3 + 1/(7 + 1/(15 + 1/(1 + 1/292))) = 97591/31065

and so on. Note that by pure coincidence one of the first few terms is quite large (292), so the difference between the corresponding partial expansions is small (1 + 1/292 is close to 1). That's why 355/113 is an unusually good approximation for such a small denominator.

[–] [email protected] 3 points 9 months ago (1 children)

I’m not sure that’s a great trick. You have to remember 6 digits to calculate an approximation accurate to 8 digits.

How many architectures in 2023 still lack a FPU? They were getting pretty rare when I last worked with this stuff 15 years ago.

[–] [email protected] 2 points 9 months ago (1 children)

you have to remember 11 33 55 and put the bar in the middle. It is mainly small MCU like ATMEGA and co. lacking FPU, and yes old stuff like Z80 ,6502, etc.

[–] [email protected] 1 points 9 months ago

Yeah, I guess the Z80 will never die.

[–] [email protected] 12 points 9 months ago (1 children)

The answer to your question is as simple as it is unsatisfying. Additional degrees of precision in construction are only as useful as your means to use them.

If you using your saw can only cut to an accuracy of 1/8 of an inch, than any precision beyond that point is lost as you are unable to actualize it

However if you are using a saw and you're now at the point of your personal skill that you're measuring to the inside or outside of the mark on a ruler, then it is likely time for you to graduate between more precise form of measurement.

https://youtu.be/qE7dYhpI_bI?si=HCtTbklCA18ZieCh

This video covers a lot of the interesting points around measurement and how we can never truly be perfectly accurate with any measurement of any non-discreet metric.

To give a real world example if you are off by a millimeter diameter when building a car engine cylinder it will likely fail.

[–] [email protected] 1 points 9 months ago

IIRC the 15 digit is the size of a hydrogen atom

[–] [email protected] 8 points 9 months ago

Not a direct answer to the question but one thing not noted in other answers is in computing you often work at a higher precision than you need for your final answer as the errors tend to increase each time you do a mathematical operation.

In the world of reasonably powerful hardware (laptops, desktops, servers, smart phones etc.) we'd typically work with 64 bit floating point numbers which gives pi to 15 digits (I think, not at a real computer now so can't check). because it's simple to do so even though we don't need the full precision.

[–] [email protected] 5 points 9 months ago (1 children)

When someone wants to be more accurate, they add another digit.

[–] [email protected] 9 points 9 months ago (2 children)

Significant Figures: am I a joke to you?

[–] [email protected] 4 points 9 months ago (1 children)

Accurate(closer to the real answer), is not equal to precise(consistent measurements). I think… right?

[–] [email protected] 5 points 9 months ago (1 children)

Yes. I had a science teacher devote a significant portion of a day’s lecture to that exact subject: the difference between accuracy and precision. You got it right.

For example, if you have a digital scale that displays weight to five decimal points, it’s precise. If that scale hasn’t been properly calibrated, though, it will give you a very precise number that isn’t accurate.

[–] [email protected] 3 points 9 months ago

It is kind of weird that real world measurements always have error to them so you only really know something to a few digits of precision… but mathematical constants like pi can have effectively have unlimited significant figures (but it’s kind of pointless to have more because any real world applications only need a rather small amount of them). I feel kind of similarly about integration. It’s nice to find a closed form solution for an integral, which can make certain calculations a lot faster and more accurate… but in reality if you’re just solving an integral or two for an engineering project you’re probably better off just computing it numerically to the correct number of sig figs. There’s something a little sad about that to me.

[–] [email protected] 4 points 9 months ago (1 children)

It's been about 20 years since my engineering degree, but we used to memorize to six decimal places. Anything more than that is never used.

[–] [email protected] 3 points 9 months ago

So, in terms of accuracy required, an old structural book I have includes a "functions of numbers" section which has various things including the square, the square root, the log, and the circumference of a circle with that diameter. It shows pi as 3.142. Outside of alignments, I would expect that to be good enough for most civil engineers.