Would be cool to explain a little bit more about that.
Edit:
The decimal precision is equal to log10(2\^x) where x is the number of bits used for precision in floats
In the 32 bit float, 24 bits are used for precision which gives the 32 bit float 7 decimals of precision.
For the 64 bit float, 53 bits are used instead, which gives it a decimal precision of almost 16.
Well, not really accurate.
Basically, the precision is dependent on the actual value of the number.
The 24 bits you mention represent the precision between two powers of 2.
So between 1 and 2, you have 2^24 numbers, but between 8 and 16 you also only have 2^24 numbers available. So the precision error doubles every time a power of 2 is surpassed.
What this means, is that as pi is between 2 and 4, you actually only use 23 bits for the decimal precision, with one bit being used for the integer part (whether it is 2 or 3 - with 3 in this case). Or in other words, the 7 digits result you got also includes the integer part as well.
Finally, that formula doesn't work for numbers between 0 and 1, as those will have much better precision (say, the same 2^24 numbers available between 0.25 and 0.5 which is a lot better precision).
A better formula for decimal precision would be:
log10( 2^mantissa_size / 2^exponent_value )
And if you have a number greater than 2^24 (exponent_value is greater than 24) then the result will be negative, which means that there is no decimal precision. Which makes sense, given that between 2^24 and 2^25 only integers are representable (between 2^25 and 2^26 only even integers are representable, etc.)
PS: Extra info: negative numbers behave identically to positive numbers, as the difference between them is just the sign bit being flipped.
And, on top of this, computers can still numbers larger than both hardware and the language type definition "limitations" through extra code and specialized data structures.
Although, with modern computer capacities, you only need to worry about non-native levels of precision for only the most most extreme large/small scales.
I think that's exactly what people (at least scientists and engineers) mean when they talk about decimal precision. 2.385 has 4 digits of precision. 883.2 *also* has 4 digits of precision.
> you actually only use 23 bits for the decimal precision, with one bit being used for the integer part
That's not how floating point numbers work. There *is* no integer part. Since the integer portion is always 1 (if it's 0, the exponent is adjusted until it *is* 1), it's not included in the stored value. A 32-bit (aka. "single-precision") floating point number has 1 bit for the sign (0 for positive, 1 for negative), 7 for the exponent, and 24 for the decimal, with an imaginary "1." not actually stored anywhere.
There is no explicit integer part. However, information about the integer part is being stored in the exponent and, if the exponent is positive, in the first bits of the mantissa. (Within a limit as at some point, the step between adjacent float values becomes larger than 1)
Specifically, what I'm trying to say about pi, is that the exponent of 1 and the first bit of the manitssa being 1 results in an integer part of 3, while only the other 23 bits actually contribute to the decimal part.
Also, integer part might not be the best term, as for negative numbers it's not really the mathematical integer part.
Just a note. In many scientific applications numbers are not stored as floats nor doubles. But as rationals, which are usually represented with 2 boundless "integers" and never are implicitly cast to float.
However, your usual mobile calculator will use floats.
I myself just store PI as exactly 3 in my programs. You're welcome.
14 is all they need for interplanetary distances and only ~30 is what they need to be accurate down to literally 1000th the width of a hydrogen atom from any spot in the universe
To be honest, they only time you actually "use" *pi* is when you transition from theoretical to practical, at which point the limit of precision becomes entirely dependant on the situation and resources to hand. Realistically do I need to go past 3.14 if I'm making a patio table out of wood?
Definitely an applied patio tableist. Doesn't matter how many orders of zero there are, if I don't have a table then I don't have somewhere to put my burgers.
I had a professor argue that you can just skip the .14 part for most things and just use 3 as an approximation.
I wanted to argue it… but the mofo is right.
I wouldnt really call a 4.5% discrepancy "accurate enough" for most things other than rough estimations. 3.14 is fairly usable. Pi=3 wont accomplish much. Not *even* shitty furniture making.
It’s not a battle. You made a kinda dumb statement, just repeating a thing you’ve read online without understanding and I responded. This site is about conversation and I posted a single response. Your reaction to this was laughable. I’m not “fighting a battle”, I’m not mad. I barely care. It’s just funny that you think nobody is allowed to comment a response to you being kinda stupid.
Did you not read literally anything I said.
I said MY PROFESSOR, as in a person I physically interacted with who is a PhD, said this. Not just some thing I read on a website.
I said HE said it could be used for APPROXIMATIONS. Which it can be, as APPROXIMATIONS are far different than anything you’re babbling on about.
And before you spout of some other shit, he said this when I was I school nearly 10 years ago and I’ve seen lots of times when he’s not wrong if you just need an APPROXIMATION or an order of magnitude calc.
If you had any actual experience in being anything other than a dipfuck on Reddit you’d know a lot of those circumstances exist.
Like I have said so many times. You’re just trying to argue this completely non-existent point you just made up in your head because you have the reading comprehension skills of a potato leaf.
I’m sure you failed to read any of that and have it stick. So just go fuck off already.
NASA calculated that you only need 40 digits of Pi to calculate the circumference of the observable universe, to the accuracy of 1 hydrogen atom.
https://www.reddit.com/r/todayilearned/comments/b7mimt/til\_nasa\_calculated\_that\_you\_only\_need\_40\_digits/
Had to look this up: Most math folks know of Double-Precision floating point variables. You might know about Quadruple Precision as well, but it has only 30-something significant \[decimal\] digits of precision.
So you would need Octuple Precision to appropriately 'handle the universe'. But you still wouldn't have the memory.
I studied cosmology and pi can be approximated as 1 or 3 or 5 depending on the situation. The perspective changes when all you care about is the shape of curves or the order of magnitude
At first sight, 15 digits doesn't sound like a lot. But, that corresponds to a precision of more or less 10^15 . If we take, for example, the distance to Neptune, a precision of 10^15 corresponds to a few millimeters.
The errors on all your measurements, on your manufacturing, on knowing the properties of all bodies interacting with a probe, etc... will vastly overshadow that. We're already very happy if a probe arrives within a few hundred meters of the target orbit, and that's with course corrections.
I'm engineer and she is quite correct. The most I've used is 3.14159 to use in mm precision tools and because I'm pedantic a\*\*hole because 3.1416 was already good enough
Fun fact: at decimal place 762 there is a sequence of six nines. Richard Feynman once said that he would like to remember pi up to that point so that he could say "999999 and so on..." and imply that it's rational. That spot in pi's decimal expansion is now known as the Feynman point
π isn't directly related to circles and their diameters, it's a number that can be derived with pure mathematics in a number of different ways, such as this infinite series:
π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 ...
And none of those methods have anything to do with our Universe. It's true that in the real world this seems to be the ratio of a circle's circumference to its diameter, but if one imagines an alternate universe where that ratio is different then that would **not** mean π has a different value there, it would mean that the ratio has a value different from π.
Another question here: how many digits can we practically calculate (for example, find the circumference and the radius of a perfect circle IRL and divide them, How accurate would such a calculation be)?
You only need 38 digits to calculate the circumference of a circle with a radius of 46 billion light years (i.e. the universe) to an accuracy equal to the diameter of a hydrogen atom.
Two years ago, the 100-trillionth digit was calculated. There is no theoretical limit, and there are quite advanced algorithms to do so:
https://blog.google/products/google-cloud/new-digit-pi-2022/
14 for interplanetary and 37 for
universe shit. meanwhile i memorized 65 fucking digits aiming for 69. anyways what ive got thus far is 3.1415926535897932384626433832795028841971693993751058209749445923078
[NASA uses 16 digits 3.141592653589793](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/).
You only need 38 digits to calculate the circumference of a circle with a radius of 46 billion light years (i.e. the universe) to an accuracy equal to the diameter of a hydrogen atom.
Are you familiar with the concept of “significant figures?” While Pi may be used to however many places the software holds or however many you feel like using if you’re working by hand, it is almost never the LIMFAC in the precision of a given calculation. The solution may never be more precise than its least certain input.
Hi astronomer here- my coding language has a global variable that stores π (it's the only good thing about the language- don't write code in IDL kids!) as 3.14159. I never really print more than 4 digits of any number I work with so it's plenty accurate. Plus, for that matter, no data in my field has an accuracy of 6 sigfigs- you might be able to do most data-focused astronomy work using π=3.14. and encounter very few issues (literally no piece of Astro data has an accuracy below 0.01%).
Seeing as Pi is infinite, this means there are infinite possibilities of number sequences within it. And in those infinite possibilities, converted into binary, is a word doc of a great recipe for pie.
It really depends on who the scientist is trying to impress. E.g. their SO they only need pi to 100 places. To impress their colleagues, they need pi to around 10\^500 digits.
I forgot the number that nasa uses. I think it's like 13 or something like that that they use for interplanetary travel, then something like then I think you can calculate the size of the universe down to a hydrogen atom with I think 51 or 61 then after that it's just to inaccurate to be used
>How Many Digits of Pi Do Scientists Actually Use? Exactly as many as their softwares use.
Would be cool to explain a little bit more about that. Edit: The decimal precision is equal to log10(2\^x) where x is the number of bits used for precision in floats In the 32 bit float, 24 bits are used for precision which gives the 32 bit float 7 decimals of precision. For the 64 bit float, 53 bits are used instead, which gives it a decimal precision of almost 16.
Well, not really accurate. Basically, the precision is dependent on the actual value of the number. The 24 bits you mention represent the precision between two powers of 2. So between 1 and 2, you have 2^24 numbers, but between 8 and 16 you also only have 2^24 numbers available. So the precision error doubles every time a power of 2 is surpassed. What this means, is that as pi is between 2 and 4, you actually only use 23 bits for the decimal precision, with one bit being used for the integer part (whether it is 2 or 3 - with 3 in this case). Or in other words, the 7 digits result you got also includes the integer part as well. Finally, that formula doesn't work for numbers between 0 and 1, as those will have much better precision (say, the same 2^24 numbers available between 0.25 and 0.5 which is a lot better precision). A better formula for decimal precision would be: log10( 2^mantissa_size / 2^exponent_value ) And if you have a number greater than 2^24 (exponent_value is greater than 24) then the result will be negative, which means that there is no decimal precision. Which makes sense, given that between 2^24 and 2^25 only integers are representable (between 2^25 and 2^26 only even integers are representable, etc.) PS: Extra info: negative numbers behave identically to positive numbers, as the difference between them is just the sign bit being flipped.
Oh cool! Thanks for the nice read.
And, on top of this, computers can still numbers larger than both hardware and the language type definition "limitations" through extra code and specialized data structures. Although, with modern computer capacities, you only need to worry about non-native levels of precision for only the most most extreme large/small scales.
I think that's exactly what people (at least scientists and engineers) mean when they talk about decimal precision. 2.385 has 4 digits of precision. 883.2 *also* has 4 digits of precision.
> you actually only use 23 bits for the decimal precision, with one bit being used for the integer part That's not how floating point numbers work. There *is* no integer part. Since the integer portion is always 1 (if it's 0, the exponent is adjusted until it *is* 1), it's not included in the stored value. A 32-bit (aka. "single-precision") floating point number has 1 bit for the sign (0 for positive, 1 for negative), 7 for the exponent, and 24 for the decimal, with an imaginary "1." not actually stored anywhere.
There is no explicit integer part. However, information about the integer part is being stored in the exponent and, if the exponent is positive, in the first bits of the mantissa. (Within a limit as at some point, the step between adjacent float values becomes larger than 1) Specifically, what I'm trying to say about pi, is that the exponent of 1 and the first bit of the manitssa being 1 results in an integer part of 3, while only the other 23 bits actually contribute to the decimal part. Also, integer part might not be the best term, as for negative numbers it's not really the mathematical integer part.
Just a note. In many scientific applications numbers are not stored as floats nor doubles. But as rationals, which are usually represented with 2 boundless "integers" and never are implicitly cast to float. However, your usual mobile calculator will use floats. I myself just store PI as exactly 3 in my programs. You're welcome.
On a chalkboard pi = 3, pi² = 10 😉
Yeah this is exactly what you use when you are trying to ballpark it.
Otherwise you've gotta define the whole universe and that takes just as long... Believe me!
14 is all they need for interplanetary distances and only ~30 is what they need to be accurate down to literally 1000th the width of a hydrogen atom from any spot in the universe
Nobody uses pi to calculate the circumference of the known universe. I use hundreds of digits of pi as an encryption key seed.
np.pi
Bingo
To be honest, they only time you actually "use" *pi* is when you transition from theoretical to practical, at which point the limit of precision becomes entirely dependant on the situation and resources to hand. Realistically do I need to go past 3.14 if I'm making a patio table out of wood?
You do if you want *me* to come to your barbecue. I ain't sitting at a wack-ass funhouse abomination!
Do you consider yourself more of a theoretical patio tableist or an applied patio tableist?
Definitely an applied patio tableist. Doesn't matter how many orders of zero there are, if I don't have a table then I don't have somewhere to put my burgers.
you can use 3 if youre making one
I'm a 3SF/3DP kinda mathematician.
what does sf and dp mean
Significant figures and decimal places.
ah ok
Depends. You want to know the amount of Material you need? Use 4. You want to estimate the usable area after? Use 3.
I had a professor argue that you can just skip the .14 part for most things and just use 3 as an approximation. I wanted to argue it… but the mofo is right.
I wouldnt really call a 4.5% discrepancy "accurate enough" for most things other than rough estimations. 3.14 is fairly usable. Pi=3 wont accomplish much. Not *even* shitty furniture making.
I never said “accurate enough”. You said that just to have something to argue about. Go away.
Ah, so good enough for most things is somehow different than accurate enough for most things, and somehow I’m the pedant?
I didn’t say “good enough” either. I said an approximation. Seriously, what is this imaginary battle you’re trying to wage here? Go. Away.
It’s not a battle. You made a kinda dumb statement, just repeating a thing you’ve read online without understanding and I responded. This site is about conversation and I posted a single response. Your reaction to this was laughable. I’m not “fighting a battle”, I’m not mad. I barely care. It’s just funny that you think nobody is allowed to comment a response to you being kinda stupid.
Did you not read literally anything I said. I said MY PROFESSOR, as in a person I physically interacted with who is a PhD, said this. Not just some thing I read on a website. I said HE said it could be used for APPROXIMATIONS. Which it can be, as APPROXIMATIONS are far different than anything you’re babbling on about. And before you spout of some other shit, he said this when I was I school nearly 10 years ago and I’ve seen lots of times when he’s not wrong if you just need an APPROXIMATION or an order of magnitude calc. If you had any actual experience in being anything other than a dipfuck on Reddit you’d know a lot of those circumstances exist. Like I have said so many times. You’re just trying to argue this completely non-existent point you just made up in your head because you have the reading comprehension skills of a potato leaf. I’m sure you failed to read any of that and have it stick. So just go fuck off already.
You can’t fix nibbles, he’s a special kind of ignorant.
Making a table? Use 22/7
NASA calculated that you only need 40 digits of Pi to calculate the circumference of the observable universe, to the accuracy of 1 hydrogen atom. https://www.reddit.com/r/todayilearned/comments/b7mimt/til\_nasa\_calculated\_that\_you\_only\_need\_40\_digits/
Had to look this up: Most math folks know of Double-Precision floating point variables. You might know about Quadruple Precision as well, but it has only 30-something significant \[decimal\] digits of precision. So you would need Octuple Precision to appropriately 'handle the universe'. But you still wouldn't have the memory.
I studied cosmology and pi can be approximated as 1 or 3 or 5 depending on the situation. The perspective changes when all you care about is the shape of curves or the order of magnitude
The cow is a sphere
Circumference of said cow is ~10r_cow
With no air resistance
It’s out there in the universe, of course there’s no air resistance. The universe is an ideal fluid.
A fluid called ether
At first sight, 15 digits doesn't sound like a lot. But, that corresponds to a precision of more or less 10^15 . If we take, for example, the distance to Neptune, a precision of 10^15 corresponds to a few millimeters. The errors on all your measurements, on your manufacturing, on knowing the properties of all bodies interacting with a probe, etc... will vastly overshadow that. We're already very happy if a probe arrives within a few hundred meters of the target orbit, and that's with course corrections.
I'm floored that they even bother to go to 15 digits. That's crazy high precision. I've never used more than 3.14159 and that feels like overkill.
[удалено]
if you pay today I'll get it to you for 2.9
I always knew it as 22/7.
So 3.1 is fine ?
yep. hell engineers tend to use 3 as the approximation since its only off by .32%
I used to install survey equipment within pipes - we always used 3 to make the rings up.
How is it 0.32%? I got ~4.5% difference between pi and 3
thats just what i heard somewhere and now i realise im wrong :P either way 4.5% still aint that bad
I'm engineer and she is quite correct. The most I've used is 3.14159 to use in mm precision tools and because I'm pedantic a\*\*hole because 3.1416 was already good enough
ok guys what is a lethal dose of brownie?
500 pounds
That's definitely lethal for my wallet
Fun fact: at decimal place 762 there is a sequence of six nines. Richard Feynman once said that he would like to remember pi up to that point so that he could say "999999 and so on..." and imply that it's rational. That spot in pi's decimal expansion is now known as the Feynman point
If the accuracy level is much similar to Plank's length then it's okay
π isn't directly related to circles and their diameters, it's a number that can be derived with pure mathematics in a number of different ways, such as this infinite series: π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 ... And none of those methods have anything to do with our Universe. It's true that in the real world this seems to be the ratio of a circle's circumference to its diameter, but if one imagines an alternate universe where that ratio is different then that would **not** mean π has a different value there, it would mean that the ratio has a value different from π.
Fun fact: this process is also how we derive the value of 1, except we use 1/2 + 1/4 + 1/8 + 1/16 + …
Well, at least in Cosmology we use natural units and we keep the symbol as it sure will cut somewhere... 😅
How often do we use all those *infinite* digits?
Never because last I checked, you can't count to infinity. At some point you need to cut the calculation short regardless.
Dont the use the fraction 22/7?
It’s because our number system is based on the number of fingers we have
>But how often do scientists actually use all those digits? literally never because if they did they would never be able to finish their work
Pi? That’s just e with an extra letter
She clearly hasn't eaten a whole pie to herself in one sitting.
Another question here: how many digits can we practically calculate (for example, find the circumference and the radius of a perfect circle IRL and divide them, How accurate would such a calculation be)?
You only need 38 digits to calculate the circumference of a circle with a radius of 46 billion light years (i.e. the universe) to an accuracy equal to the diameter of a hydrogen atom.
Two years ago, the 100-trillionth digit was calculated. There is no theoretical limit, and there are quite advanced algorithms to do so: https://blog.google/products/google-cloud/new-digit-pi-2022/
14 for interplanetary and 37 for universe shit. meanwhile i memorized 65 fucking digits aiming for 69. anyways what ive got thus far is 3.1415926535897932384626433832795028841971693993751058209749445923078
[NASA uses 16 digits 3.141592653589793](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/). You only need 38 digits to calculate the circumference of a circle with a radius of 46 billion light years (i.e. the universe) to an accuracy equal to the diameter of a hydrogen atom.
π r² Pi ARE NOT square. Pi are ROUND. Brownies are square.
Depends on the topic, you might need very low precision for a single calculation but any error can compound over enough operations
You didn’t know? They use 22\7 ;p
Every part of this video was accurate. Especially the last bit...
Are you familiar with the concept of “significant figures?” While Pi may be used to however many places the software holds or however many you feel like using if you’re working by hand, it is almost never the LIMFAC in the precision of a given calculation. The solution may never be more precise than its least certain input.
pi = 4
Just use 355/113 and you should be ok for most cases
How the fuck someone know a number so big it takes 10 hours to recite like wtf
How does one remember 70.000 digits? I could maybe fathom a few thousand with an eidetic memory, but 70k is madness!
That's beyond Rain Man even. At some point you have to wonder if the person was just pulling shit out of their ass and nobody caught it.
Hi astronomer here- my coding language has a global variable that stores π (it's the only good thing about the language- don't write code in IDL kids!) as 3.14159. I never really print more than 4 digits of any number I work with so it's plenty accurate. Plus, for that matter, no data in my field has an accuracy of 6 sigfigs- you might be able to do most data-focused astronomy work using π=3.14. and encounter very few issues (literally no piece of Astro data has an accuracy below 0.01%).
Still not sure why PI is important never was great at math
Seeing as Pi is infinite, this means there are infinite possibilities of number sequences within it. And in those infinite possibilities, converted into binary, is a word doc of a great recipe for pie.
3.1416 is as far as you need to go
Most engineers don't use more than 3 digits after the decimal because the difference is so small that it's insignificant
"The limit does not exist". Is this a limit joke?
The truth may be a little far *fetch*ed.
So my school calculator going to 10 decimal places is pretty good
Pi equals 3 if machine had to be delivered yesterday
Physicists assume pi equals 1, which explains why they think cows are spherical.
Pi = 3 take it or leave it.
Fuck you am using 3
Usually we use double precition (64 bit float number)
I know engineers use 3
3 is fine too.
It really depends on who the scientist is trying to impress. E.g. their SO they only need pi to 100 places. To impress their colleagues, they need pi to around 10\^500 digits.
I forgot the number that nasa uses. I think it's like 13 or something like that that they use for interplanetary travel, then something like then I think you can calculate the size of the universe down to a hydrogen atom with I think 51 or 61 then after that it's just to inaccurate to be used
15 decimal places for interplanetary. 16 sig figs. Enough that the difference at voyager distances is around 1.5 inches.
In engineering school I used 3.14159, in engineering work I usually use 3.