# Close Enough for Government WorkMarch 30, 2016 9:26 AM   Subscribe

How many digits of pi do we really need? Thirty-nine.
posted by Eyebrows McGee (72 comments total) 31 users marked this as a favorite

The only reason we need more digits of pi is that it is the most effective "nerd alert" we have yet devised.
posted by jonp72 at 9:31 AM on March 30, 2016 [3 favorites]

How many paintings and novels does humanity need?

Each digit of pi is another chance to show how good your memory is, or how fast your algorithm/hardware is. It's pointless for engineering, but so is everything else if you think about it too hard.
posted by mccarty.tim at 9:32 AM on March 30, 2016

How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places.
Well, which is it? I'm not comfortable with that level of uncertainty.
posted by Etrigan at 9:32 AM on March 30, 2016 [50 favorites]

What a coincidence. I am certain that I will be 39 years old this year. And every year to come!
posted by telstar at 9:33 AM on March 30, 2016 [4 favorites]

There are actually some cryptographic applications that require pi to thousands of digits (verifying elliptic curves).
posted by dilaudid at 9:35 AM on March 30, 2016 [8 favorites]

Can you ever have too much pie?
posted by GenjiandProust at 9:35 AM on March 30, 2016 [14 favorites]

The answer is that you would need 39 or 40 decimal places.

Well, which is it? I'm not comfortable with that level of uncertainty.

I'd interpret that as, "39 gets us an error size slightly bigger than a hydrogen atom, 40 gets us an error size smaller than a hydrogen atom."
posted by explosion at 9:36 AM on March 30, 2016

So let's do 39.5 decimal places and get it exactly right.
posted by It's Never Lurgi at 9:37 AM on March 30, 2016 [30 favorites]

This is really fascinating, and on a blog that I would have never stumbled on in my normal reading. Exactly what the front page is for, thanks.
posted by Think_Long at 9:40 AM on March 30, 2016 [4 favorites]

"Well, which is it? I'm not comfortable with that level of uncertainty."

When we get a spaceship to the edge of the observable universe, I'm sure Congress will appropriate the extra money to do 40-digit calculations instead of 39, just in case.

I thought that was really interesting, though -- of course JPL and NASA must have a standard they use, but that had never occurred to me, and it's very illuminating to realize how quickly more decimal places gets you more precision. (Is this an exponential or multiplicative increase in precision with each added decimal place? I do not math well.)
posted by Eyebrows McGee at 9:42 AM on March 30, 2016

Wow, this is an excellent, real life example to use when talking to students about what precision and accuracy really mean in application.
posted by barchan at 9:42 AM on March 30, 2016 [1 favorite]

For an extra margin of safety we should use 42 digits when calculating the size of the universe.
posted by TedW at 9:43 AM on March 30, 2016 [20 favorites]

"How many digits of pi do we need?"
"42!"

No, doesn't work...
posted by Captain l'escalier at 9:44 AM on March 30, 2016 [12 favorites]

I use a set of binary files that represent pi, e, ln, and other such numbers (one per file) out to ~1 million digits (they are actually direct binary representations, not just ascii encodings of the digits) as inputs sometimes to RNGs, or as keyfiles for encryption. I can't think of anything else where such high precision would be useful.
posted by mystyk at 9:44 AM on March 30, 2016

Wouldn't the precision needed depend on the electron configuration of the hydrogen atom in question?
posted by CBrachyrhynchos at 9:44 AM on March 30, 2016 [3 favorites]

> There are actually some cryptographic applications that require pi to thousands of digits (verifying elliptic curves).

I'm curious about this, because all the elliptic curve stuff I've seen in cryptography has involved curves over large finite fields, where pi doesn't even exist. Can you provide a link?
posted by benito.strauss at 9:48 AM on March 30, 2016 [3 favorites]

My freshman physics professor taught that, due to relativistic deformation of space, eight or nine digits were perfectly adequate in practice at the order of the solar system.
posted by 7segment at 9:49 AM on March 30, 2016

Might be a good test of cell phones, "how fast can your pocket compute calculate a thousand digits of pi?" er, nevermind.
posted by sammyo at 9:53 AM on March 30, 2016

No mention of significant figures, potential round-off due to different exponents, floating-point issues etc ? (or do they use non FP, arbitrary precision math libs )

(and no snark about "what constant do you use to convert between metric and imperial units ? ")
posted by k5.user at 9:54 AM on March 30, 2016

sammyo: " "how fast can your pocket compute calculate a thousand digits of pi?""

Is that pi in your pocket or are you just happy to see me?
posted by chavenet at 9:55 AM on March 30, 2016 [1 favorite]

I appreciate the deliberate literal punny-ness of "government work" in this case (NASA being the subject at hand!), though generally, in my experience "close enough for government contractor work" is a more accurate way to convey "eh, good enough." < /pedantic>
posted by duffell at 10:02 AM on March 30, 2016

1
posted by beerperson at 10:05 AM on March 30, 2016

Can you ever have too much pie?

Is it Key Lime? Then no, no you cannot.
posted by AlonzoMosleyFBI at 10:08 AM on March 30, 2016 [3 favorites]

Is it Key Lime? Then no, no you cannot.

That's a bigger lie than the supposed "moon landing"
posted by Think_Long at 10:11 AM on March 30, 2016 [1 favorite]

Then, the pie is a lie? (Of course any numerical approximation to pi is always a lie.)
posted by Death and Gravity at 10:16 AM on March 30, 2016

But what if you need to calculate a circle larger than the visible universe to the Planck scale?

posted by Rhaomi at 10:20 AM on March 30, 2016 [9 favorites]

When I was 11, we had to memorize 52 digits of pi in order to pass our math class. I never really thought this was an accomplishment worth noting.
posted by inconsequentialist at 10:29 AM on March 30, 2016

visible universe to planck scale: uncertain
posted by telstar at 10:29 AM on March 30, 2016

How many digits of pi do we really need?

All of them.
posted by ZenMasterThis at 10:40 AM on March 30, 2016 [1 favorite]

I should think about half the digits ought to be enough for anybody.
posted by Flexagon at 10:42 AM on March 30, 2016 [6 favorites]

When I was 11, we had to memorize 52 digits of pi in order to pass our math class. I never really thought this was an accomplishment worth noting.
posted by inconsequentialist at 1:29 PM on March 30 [+] [!] [quote]

posted by Chrysostom at 10:44 AM on March 30, 2016 [3 favorites]

Heck, I've memorized all the digits in pi: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

If you want them in order, that costs extra.

/This joke brought to you by the last set of IKEA shelve I bought.
posted by benito.strauss at 10:46 AM on March 30, 2016 [32 favorites]

posted by BitterOldPunk at 10:49 AM on March 30, 2016 [5 favorites]

Fifty two digits is a very weird number of digits to memorize.
posted by jeather at 10:50 AM on March 30, 2016

This is close enough to one of my favorite pulsar astronomer facts:

The pulsar J1909-3744 spins every 2.9 milliseconds, or 340 times per second; the pulsar and its white dwarf companion star orbit their common center of gravity every 1.5 days. The orbit of PSR J1909-3744 is the most circular known in the universe: the elliptical orbit is over one million kilometers across (about 1.5 times the size of the Moon's orbit around the Earth), but the major axis is larger than the minor axis by only 10 microns, a fraction of the thickness of a human hair.

Here's the discovery paper; with ongoing observations, the measured eccentricity[*] of the orbit is 0.00000011, with an uncertainty of +-1 in the last digit.

Now that's a nice "elliptical" orbit!

[*] Eccentricity e = sqrt(1 - b^2/a^2), where a and b are the major and minor axes of the ellipse.
posted by RedOrGreen at 10:50 AM on March 30, 2016 [13 favorites]

He doesn't seem to state why they use 3.141592653589793. The answer, I'd infer, is that's the number of digits provided by an IEEE 754 standard 64-bit double precision floating point value.
posted by rlk at 10:54 AM on March 30, 2016 [5 favorites]

Fifty two digits is a very weird number of digits to memorize.

It's exactly one digit a week for 0.99657768651608 of a year
posted by beerperson at 10:55 AM on March 30, 2016 [11 favorites]

I should think about half the digits ought to be enough for anybody.
posted by Flexagon at 10:42 AM on March 30 [+] [!]

Which half, though?
posted by eviemath at 11:00 AM on March 30, 2016 [5 favorites]

Just one for biblical precision.
posted by stevis23 at 11:19 AM on March 30, 2016

The pulsar J1909-3744 spins every 2.9 milliseconds, or 340 times per second; the pulsar and its white dwarf companion star orbit their common center of gravity every 1.5 days. The orbit of PSR J1909-3744 is the most circular known in the universe: the elliptical orbit is over one million kilometers across (about 1.5 times the size of the Moon's orbit around the Earth), but the major axis is larger than the minor axis by only 10 microns, a fraction of the thickness of a human hair.

That is an extremely weird fact, RedOrGreen -- and an orbit so extremely and improbably circular that it's hard to see how it could have come about except as the result of some sort of circularizing process.

Are there any candidates for such a process? Such as some kind of interaction between the pulsar magnetic field and ionized gas in the vicinity, or greater emission of gravitational waves when the pulsar and its companion are closer?
posted by jamjam at 11:25 AM on March 30, 2016 [1 favorite]

IEEE 754 standard 64-bit double precision floating point value

...is the name of my Kraftwerk/prog-rock mashup tribute band.
posted by Mr. Bad Example at 11:26 AM on March 30, 2016 [4 favorites]

None of them.

We don't really need engineering, either. No poetry or philosophy. Most animals get by just fine without them.

Philosophical precision, the nerdiest precision of all.
posted by oddman at 11:28 AM on March 30, 2016 [1 favorite]

The nearest IEEE double is 3.141592653589793115997963468544185161590576171875 (or 0x3.243f6a8885a3p+0) but yeah in this case you can stop writing digits after 16, as 3.141592653589793 round trips to that value.

The nearest IEEE single is 3.1415927410125732421875 or 0x3.243f6cp+0.
posted by Rhomboid at 11:29 AM on March 30, 2016 [2 favorites]

"an orbit so extremely and improbably circular that it's hard to see how it could have come about except as the result of some sort of circularizing process.

Are there any candidates for such a process?
"

Great, now we have something besides bananas for Intelligent Design proponents to misunderstand and misappropriate.
posted by oddman at 11:31 AM on March 30, 2016 [2 favorites]

Typical of big government to try to hide so many of pi's digits from the people. What is there in the furthest reaches that they don't want us to see? Wake up sheeple.
posted by DanSachs at 11:33 AM on March 30, 2016 [1 favorite]

PI IS EXACTLY THREE!

Very sorry that it had to come to that, but now that I have your attention...
posted by Faint of Butt at 11:42 AM on March 30, 2016 [1 favorite]

I thought that was really interesting, though -- of course JPL and NASA must have a standard they use, but that had never occurred to me, and it's very illuminating to realize how quickly more decimal places gets you more precision.

Note that they use that for interplanetary navigation. I recall that the Apollo program used π=3.1416, which is even more precise than you think (3.141600-3.141593=0.000007, or seven millionths off -- note we are rounding the last digit here) and would use 3.141 often because everything else in the equation had less precision. This was more than enough for Apollo, which was dealing in hundreds of thousands of kilometers, not tens of billions of kilometers, and the errors were less than an inch -- far less then they could measure the position of the Apollo spacecraft to, and far less than the accuracy of Apollo to enter a given orbit.

Yeah, in EC crypto, you need π to a lot of digits, but for the real world, 3.14 is often good enough and 3.142 is possibly good enough to get to the moon with -- but 3.1416 just happens to hit a sweet spot in accuracy in rounding.

Now, if you're dancing around in the Kuiper belt, you'd better be grabbing more digits.
posted by eriko at 11:43 AM on March 30, 2016 [10 favorites]

It's a good question and a great answer.
posted by sfred at 11:52 AM on March 30, 2016

"How many digits of pi do we need?"
"42!"

No, doesn't work...

I think 42 would be a lovely place to round off if you're looking at 39 or 40 anyway. The 42nd digit of pi is 3, but the next two digits are 99. Which means that you can round the 42nd digit up to 4 and essentially get two more orders of precision for free.
posted by 256 at 12:06 PM on March 30, 2016 [7 favorites]

Hydrogen atoms? Talk about close enough for government work. Back when my daughters were memorizing hundreds of digits of π (under the baleful influence of Fox Trot), I figured that if we wanted to calculate the circumference of the known universe after measuring the radius in Plank lengths, you'd still only need 50 digits of π for even the most conceptually precise physical calculation possible.

(I didn't know about the cryptographic applications of thousands of digits, but my daughters are not gonna hear about it from me; they'll have to pick up that info out on the streets.)
posted by straight at 12:10 PM on March 30, 2016 [1 favorite]

"How many digits of pi do we need?"

Just the last one.
posted by HumanComplex at 12:35 PM on March 30, 2016 [6 favorites]

> an orbit so extremely and improbably circular that it's hard to see how it could have come about except as the result of some sort of circularizing process. Are there any candidates for such a process? Such as some kind of interaction between the pulsar magnetic field and ionized gas in the vicinity, or [...]?

Yes, very much so.

Millisecond pulsars are produced by a "recycling" process, where a bloated companion star loses mass to a dead pulsar (a few million years old), and the mass accretion causes the dead pulsar to spin up and be re-born with a much lower (buried?) magnetic field and much faster spin. The mass loss leaves the companion star as a white dwarf, typically, and the mass transfer process also circularizes the orbit - so if you look in the pulsar catalog, for example, you'll find that most known systems have eccentricities in the 0.001--0.00001 range.

(High eccentricities are so uncommon that we published our discovery of an MSP with an eccentric (e=0.44) orbit in Science. Pretty sweet.)
posted by RedOrGreen at 12:42 PM on March 30, 2016 [15 favorites]

Answers to questions I did not know I needed the answer to but clearly, definitely needed to know the answer to are the best answers.
posted by jacquilynne at 12:49 PM on March 30, 2016 [2 favorites]

Great, now we have something besides bananas for Intelligent Design proponents to misunderstand and misappropriate.

I just realized they have claimed both bananas and peanut butter.
posted by CBrachyrhynchos at 12:51 PM on March 30, 2016 [2 favorites]

"How many digits of pi do we need?"
"42!"

No, doesn't work...

42 factorial is way too many digits.
posted by rossmik at 1:04 PM on March 30, 2016 [4 favorites]

> Yeah, in EC crypto, you need π to a lot of digits ...

Okay, I'm going to Snopes this, and in a way that is hopefully a little interesting to folks who "don't math well" but are curious.

1) The curves in elliptic curve cryptography (ECC) are not ellipses. They look rather more like these. Here's the history: When you try to calculate the circumference of an ellipse, it gets ugly. There's no nice expression in terms of plus, minus, times, and divide. You can also throw in pi, square roots and logs and sines, and you still can't find an nice expression. So what math folk do is give the ugly thing a name and start studying that. In this case the functions are called "elliptic integrals" and you can see some of the results of the studying here (It's not pretty either. It'll make you long for the simplicity of regular old trig.)

So after studying these functions for a bit some French or German genius in the mid 1800s discovers that any of these functions (well, not the functions themselves, but something associated with these functions) satisfies an equation like this:
y2 = x3 + a x + b
Maybe not the prettiest equation, but after you've slogged through this mess it's like a drink of cool water. And those are the curves we call "elliptic curves". They originate in studying ellipses, but are quite a bit removed.

2) As for the "a lot of digits". That wouldn't help for crypto, and it comes down to an essential difference in the kinds of calculations physicists and cryptographers do. Physicist do pretty straight-forward calculations. They'll measure 15 values, look up 4 constants, plug them into a few equations, and get an answer. That lets them say precisely what the FFP article says: if we're off by this much in our value of pi, we'll be off by that much in our final answer.

Cryptographers want to mess thing up — in a controlled way that (certain) people can reverse. A great way to do this is to is feedback: plug your data into a function, then take the result and plug it back in to the same function again, and do this hundreds of times. But if evaluating your function introduces some uncertainty, your next evaluation will add on to that uncertainty, and you can quickly find you've completely swamped your original values in a sea of uncertainty.

So how to get around this? Well, in those elliptic curves curves shown above, x and y represented real numbers, and there's some uncertainty in representing real numbers — most of them have an infinite number of digits, and we can only process a finite number of them. So cryptographers stop thinking of them as curves over the real numbers and instead think of them as curves over "finite fields", x and y now represent elements from a finite field, not from the real numbers. Finite fields require a bunch of math to fully understand, but the key thing for us here is that you can write them down precisely, with no uncertainty, and you can do calculations with them and not introduce any error, so our feedback problem just disappears.

There are also a whole suite of finite fields with different properties to fit different sized problems. The smallest one has two elements, but crypto usually uses ones with sizes like 2431. You pay for the nice properties by having to write code to represent them and do math on them — they don't behave anything at all like integers or floating point numbers. Also, none of them has anything that could be considered to correspond to pi.
posted by benito.strauss at 1:09 PM on March 30, 2016 [19 favorites]

The more digits of Pi we have, the more Kate Bush can sing about.
posted by Joey Michaels at 1:41 PM on March 30, 2016 [1 favorite]

The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. Let's say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles.

The more important question would seem to be: "Is 0.621371 enough decimal places when converting miles to kilometer?."
posted by flyingfox at 1:48 PM on March 30, 2016 [3 favorites]

How many paintings and novels does humanity need?

More than 39.
posted by escabeche at 2:24 PM on March 30, 2016 [1 favorite]

Only 39? How else am I supposed to remember http://3.141592653589793238462643383279502884197169399375105820974944592.com
posted by fragmede at 3:45 PM on March 30, 2016

"Is 0.621371 enough decimal places when converting miles to kilometer?."

Well, an inch is defined as 25.4 mm exactly, so there are 1.609344 km in a mile.

So exactly 1/1.609344 miles in a km!
posted by phliar at 4:27 PM on March 30, 2016 [1 favorite]

Wow, philar. I had no idea the inch was defined that way. I thought it had something to do with the distance a plough horse could run in the time it takes for a bushel of sea brine to cool 1 degree or something. I'll take my SI preconceptions and stand over in the Corrected Corner.
posted by flyingfox at 9:01 PM on March 30, 2016 [4 favorites]

FWIW: The inch has been so defined since 1959. This is shorter than the older measurement used in the U.S., which is still sometimes used in surveying, by exactly 2 parts in 1 million. (That is: the newer "international mile" is about an eighth of an inch shorter than the older "survey mile.")
posted by Shmuel510 at 9:22 PM on March 30, 2016 [1 favorite]

I vote we obey Douglas Hofstadter and stop at the six 9s in a row.
posted by The Pluto Gangsta at 9:59 PM on March 30, 2016

Fifty two digits is a very weird number of digits to memorize.

The first stage of an astonishingly difficult and uninteresting card trick.
posted by Segundus at 11:29 PM on March 30, 2016 [3 favorites]

FWIW: The inch has been so defined since 1959.

Amusingly enough, the meter was redefined in 1960 to 1,650,763.73 wavelengths of the orange-red emission of Krypton-86 in a vacuum. So the inch technically changed again a year later.

The meter was redefined again to the current value -- 1/299,792,458 of the distance light travels in a vacuum in one second -- in 1983. This was made possible by a solid reference to what one second is.
posted by eriko at 6:08 AM on March 31, 2016

FWIW: The inch has been so defined since 1959.

Oh, this is why we have statue miles in the US (called survey miles eslewhere). The problem with redefining the inch was that we redefined the mile. The problem with that was land surveys. The redefinition of the inch would have messed up surveys, so for land grants, we use the old definition implicitly by using the old survey mile.
posted by eriko at 6:11 AM on March 31, 2016

You guys are all so punny!
posted by LizBoBiz at 10:20 AM on March 31, 2016

How many paintings and novels does humanity need?

More than 39.

posted by Chrysostom at 10:50 AM on March 31, 2016

How many paintings and novels does humanity need?

More than 39.

41 Discworld novels. I still need more.
posted by Etrigan at 10:55 AM on March 31, 2016

It;s uncomfortable criticizing Pratchett, obviously, but I really felt Discworld dropped off in quality towards the end. Pratchett's health problems certainly played a role, but I think also a case of Steven King Syndrome where big selling authors no longer get ruthlessly edited.
posted by Chrysostom at 12:21 PM on March 31, 2016

The pulsar J1909-3744 spins every 2.9 milliseconds, or 340 times per second; the pulsar and its white dwarf companion star orbit their common center of gravity every 1.5 days. The orbit of PSR J1909-3744 is the most circular known in the universe: the elliptical orbit is over one million kilometers across (about 1.5 times the size of the Moon's orbit around the Earth), but the major axis is larger than the minor axis by only 10 microns, a fraction of the thickness of a human hair.

I think what impresses me about that is:

1. Pulsars are stable enough to permit measurement of their orbit with that degree of accuracy using subtle shifts in frequency.
2. Radio astronomy is sensitive enough to detect those changes.

I can't seem to dial in a radio station on my clock radio.
posted by CBrachyrhynchos at 12:07 PM on April 1, 2016 [1 favorite]