Pi and the size of the Universe - Numberphile
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- Опубликовано: 27 окт 2024
- Our Pi Playlist (more videos): bit.ly/PiPlaylist
Pi is famously calculated to trillions of digits - but Dr James Grime says 39 is probably enough.
More links & stuff in full description below ↓↓↓
An extra note from Dr Grime: "Since pi39 ends in 0, you may think we could use pi38 instead - which has even fewer digits. Unfortunately, the rounding errors of pi38 are ten times larger than the rounding errors of pi39 - more than a hyrdogen atom. So that extra decimal place makes a difference, even if it's 0."
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Наука
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My favourite representation of π is 355/113. It's easy to remember as 113355 and is accurate to six decimal places. It often puzzled my maths teachers when my results didn't match the rest of the class using who were using 22/7. I always hoped that I'd made their lessons a little more interesting for them. At least they never penalized me for doing it.
Yeah sometimes the answers are inaccurate to a decimal place since the answer key doesn't use the exact pi value
yeap
Infact 355/113 is a better approximation of pi than 22/7
@@polpotifywho would've guessed
its called Milü, it has its own Wikipedia
I want this guy to be the next Doctor in Doctor Who. :D
I'd love that show twice as much if that happened
@@gooz1691
How did you calculate that?
@@rudolphguarnacci197 He used Pi to 4 decimal places. :)
Yes, yes, yes !!!
Who?
I memorized the first 100 digits in high school. For some reason, I've managed to retain 75 of them, despite having a crappy memory for important things like people's birthdays, etc.
Hobo Sullivan I'm in 6th grade and me and my friend were seeing who could memorize more in the period (44 minutes). I got 51 digits and beat him!
Hobo Sullivan
And here I am, unable to remember more than 14
"... crappy memory for important things like people's birthdays, etc."
Easy! Just find where those birthdays appear in π (they're certain to be in there somewhere!), and you'll have it!
Fred
I learned about 200 in one hour
@@MrSkinkarde I learned like 5 in a second! It tapered off quickly after that.
What if you used pi to calculate the size of the observable universe to the accuracy of a planck length. Wouldn't that create an accuracy of pi that is all that we could ever need in any way?
William Hunter Yes, and the number of digits needed in base 10 would be 63, to be exact.
I was going to ask the same Q thanks for asking Hunter and thanks for answering Rivera
I also got 63, but keep in mind the universe is expanding.
@@phillipjoubert1119 But does it change the ratio? It expands equivalently to all the sides, doesn't it?
@@phillipjoubert1119 The universe is expanding, but is the visible bit expanding as well?
at what digit of pi would we be measuring to the accuracy of a planck length
+The Collector using PARI I get roughly 63 places total I got that because I knew that d/(10^-35) -> the ratio of the two and 1/ that gives the multiplier to get the ratio which is what the difference in accuracy would have to account for.
***** awesome, thanks dude
I know pi to the 73rd digit!
Sebastian Portalatin you know, and i know you know about the OBSERVABLE universe
+htmlguy88 What's PARI?
Way back some 52 years ago, I memorized Pi to 50 decimal places and can still recite the digits to this day. The only reason I didn't go further was because I didn't have access to more digits in any of my books at that time. I did not use any mnemonic...I simply memorized the digits in groups of 5...I had no particular reason for doing it other than I wanted to do it.
You, sir, could literally measure the entire Universe just with the digits you memorised
You only need 61 digits to find the circumference of the observable universe using planck lengths. Much more fundamental than hydrogen atoms...
Show me the calculations
@@junosoft Yeah type out your calculations in a youtube comment bro
@@EKDupre a link would suffice.
Humans: "Well, we only need 39 digi..."
Pie: "2000 ready and 10 trillion more on the way"
Pi*** dumbfuck
@@stopthecap4317 Gonna need a lil less edge on that.
me in 2000's : right. well at least it work get larger
pi in 10,000's : too late its 19 Vingtrillion digits and were almost at inf
me : at least tau is at 10,000
Could you make a video about infinite series converging to pi and other algorithms used to calculate it to a high degree of accuracy? To me it's astonishing that something like 4-4/3+4/5-4/7+... has anything to do with a circle. It's so neat it really deserves a video.
It doesn't have something to do with a circle directly, it only has something to do with pi which is related to a circle, but if you think like this you could say factorials have something to do with the polar form of complex numbers because you use factorials to calculate e and e is used in Euler's identity.
3blue1brown covered this topic with beautiful explanation
@@juliuszkocinski7478 oh, got any links?
So, I got lost when he said "But one of my favorite pi facts..."
i got lost when he said HI
+Enzo Vargas So you have trouble understanding decimal places and perimeter=diameter*pi ?
yes, i do,
Enzo Vargas Ok then.
+Max Payne this was like best conversation on youtube lol
13.3 trillion digits as of fall 2016.
3.14159265358979323846264338327950288419716939937510582097494459230781640628260899862803482534311706798 is the first 102 digits I think, but I may be wrong.
@@maxdunn4756 pi
@@maxdunn4756 pio
As of Pi Day 2019, 31.4 trillion
@@Tigorahan They really had the 31.4 trillion digits of pi just after the day that 2018 pi day was celebrated, so they had to wait another year without calculating the thing
Im going to use this as my password
Thanks for sharing!
I don't think you're supposed to share that information.
Pretty easy to guess though
No.
So your password is 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868741326047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All this talk of pi has made me wanna eat a piece of pie.
+bigleebrink you made me want to eat a piece of pie
OMG SAME
the ten trillionth digit of pi is....'8'! Mind blown!!! JK.
its 5
you were 3 off
It ends in 420, js
lol first thing i saw
SPECTATlNG He rounded it up. It actually ends with 419.
Same! XD
Wtf this joke is made by 4 years ago lol.
1:13 standard hilarious James moment :D
The amazing thing about Pi is that you can make up any number sequence of any length and it resides within Pi
That's what it means for a number to be 'normal'. We still haven't managed to prove that that pi is normal, we just really really strongly conjecture that it is.
In the same way that a thousand monkeys at a thousand typewriters can recreate the works of Shakespeare, sure.
So you only have two significant figures in the diameter of the universe, try using 8.82*10^26 meters (which would still round to 8.8), and you get thrown off completely, more so even than with using 3.142 for Pi versus 3.1415926535897932384626433832795028841971693993751....
I love how in 1940 700 digits was the standard and had been for 100 years. 70 years later we know pi to 10 trillion digits ... that's crazy
Oh how I wish I had this bloke for my math teacher in school. I had an incredibly amazing professor for accounting in college, which is why I'm very well versed in that subject and I have the ability to learn new concepts quickly, but my math isn't great by any stretch of the imagination. However, every video of this gentleman that I've watched, I've found it interesting and fun to learn.
I was thinking more along the lines of how to arrive at Pi, I understand the unit circle equation and the calculus behind it but it is always cool seeing it explained in simple terms... and maybe with a little history thrown in!
All this talk of pie is making me hungry.
hydrogen atom what about Planck length,
iqbaltrojan How did you get this number? An answer below says it's 63, because (pi-pi63) * universe diameter = 0.1257 planck lengths
andrew sauer It's pretty straightforward. A hydrogen Atom's diameter is about 1/2 angstrom. There's about 6.18 x 10 ^ 24 Planck Length's in 1 Angstrom, so about 3.2 x 10^24 in a hydrogen atom. Just add 24 digits (maybe 25), but I think 24 would do it, so you'd need 63 digits. (39 + 24)
When you're doing π * d - π(sub 39) * d, to what decimal place are you using for the "actual" value of pi?
I understand very little but his enthusiasm makes it endearing and watchable.
It is rounded up, but maybe the fact that you know that next digit is (or rounds up to) a zero gives you a greater degree of accuracy. Kind of like the difference between saying something is about 5cm long and saying it's exactly 50mm. It's the same measurement, but to a greater accuracy level.
I've been using 355/113 for pi for years, and it's as accurate as I need. Can't remember when I last needed to calculate the observable universe...
Love this series but I hate the sound of pen scratching away on card or paper... sends shivers down my spine... much love to all the numbers of the world!
Why the 39th digit? Because the 37th, 38th and 39th digits give you 420.
+Shvet Maharaj the 37-39th digit is actually 419 but after that is a 7 so they put 420 since it's closer (rounding up)
Im dying now
1:05 Actually the computer record was broken around tau day 2014 (no that was not planned). It was calculated to a million billion digits.
+Zion J si envers 2030
Pi is normal if "1" occurs as many times as "2" and "3" and so on, but also "10" occurs as often as "11" or "67". So any N-digit long sequence should occur as often as any other N-digit sequence. You have to specifiy for which base you want normality, usually it will be base 10. I don't know if there are known normal numbers, but it's not known about pi, although all sequences look normally distributed in the first couple of million digits.
The universe does not contain pi. Pi contains the universe
by those way it is possible to build a diffential between a endless circumference and a endend diameter / trunc
lol
Chuck Norris knew that
Explain it ?!!?!.
Does anyone else care that the measurement only had two significant figures. Essentially the measurement is plus or minus >1X10 to the 24 m, or roughly the diameter of 1X10 to the 35th hydrogen atoms.
All science documentary film makers eventually descend to this: they start with noble intentions of treating their audiences as intelligent laymen, then, confronted with the boredom of talking heads, they slip in a photo of a 17C painting to illustrate 'Archimedes'. Personally, I find I am unable to concentrate for the next 30 seconds - my head is just full of "BUT, BUT, nobody knows what he looked like! That's just some old Italian's pot-boiler oil - you might just as well show me Marilyn Monroe".
Does it really matter how they looked like?
There are many, my personal favourite is this: 4 x (1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 . . .) Iterate that sequence long enough and you will 'approach' Pi.
Ok, thank you ! I thought that π(39) was simply the 39 first digits of the number π without taking in consideration the rounding of the 38th decimal.
Thank you again ! :)
Shoulda explained how they got the diameter
estimated age of universe times the speed of light
Last 3 digits in PI is 420. Coincidence? I DON'T THINK SO.
rchan not last 3, pi is endless
Abdullah Mostafa He means of the rounded 39 digit figure, I believe.
aruchan It's 419 rounded up.
Nope its 420 because the next digit is a 7, it's 3.1415926535897932384626433832795028841971 Thats 40 digits after the decimal point. (not rounded up)
last pi's digit is 0
check dat
"Of course everyone loves pi" 😄👍
For example using a property:
Pi = 4(1-1/3+1/5-1/7+1/9-...)
A very simple formula, that goes forever. The more steps you take, the better the approximation. You need to calculate it to the point where the integers are a lot smaller than the digits that you want (so that you can be sure that the digits are correct)
However using just this will be quite slow. Programmers however know many tricks how to optimize their algorithms. I am not a programmer though :(
Yes, I checked that out later. I think what the original post was referring to about "every 10 terms you add, the result is accurate to one more digit of pi," was not every 10 terms, but every power of 10 as the denominator (i.e. ~1/10 gives you 3, ~1/100 gives you 3.1, ~1/1000 gives you 1.14). I wrote a program to do the calculation to 100 million. It basically holds true for each power of 10.
So we would only need ~65 digits of pi to calculate it to a planck length?
I was going to ask that question too! :)
+themanwiththepan What about calculating the volume of the observible universe with this absolute precision?
How exactly do you work out the digits of pi?
I was thinking the same thing.
I don't know how they manage to work out the large amounts of decimals they calculate with computers, but by far the most popular way to do it centuries ago was by making regular polygons around a circle, with its vertexs actually touching it, and with as many sides as they could manage to draw. Another similar polygon was made similarly, in the inside of the circle. Therefore, they could state that the perimeter of the circumference, which was 2*Pi (if it was diametre 1), was between the two perimetres of both polygons.
Jordi Foix
Ahh, ok, that makes sense. Cheers for that.
TheRhinehart86 There are series that sum to pi.
Take a circle and divide its circumference by its diameter
Numbers are serious fucking business.
There are several formulas that we can use to derive pi (google "formulas for deriving pi" if you're interested, it's some fascinating math). Pi is irrational and the digits never end. It's a bit of a bizarre concept, but basically the number can't be expressed as a fraction and so we don't have an exact representation of it. The formulas we have for deriving the exact value of pi can be taken to literally infinite precision, it's just a matter of how far we want to go.
You're right, which is why they should've explicitly stated in the video that we are to assume that the diameter is perfectly precise, which it obviously can't be.
It doesn't exactly make sense to me as to how you're able to the circumference of the universe since we know that the universe is ever expanding and accelerating every second
Priceless23 he means the observable universe
Lil hobo Bob that makes more sense but even that is still expanding
Priceless23
Typically they will make the calculation using the most recent measurements.
They theoretically could potentially calculate the diameter of the universe after it has completely expanded, but that's a bit iffy.
Well actually, quite recently, new measurements indicated that the universe is not accelerating at all.
Dat Guy
There's over 100 years of physical evidence to the contrary.
I'd say it's probably faulty measurement, considering that if you accept special relativity the universe *must* expand.
Pi is a little bit more than 3. Almost no one has any reason to memorize more than that.
+Teddye Kuma So is Sqrt(10) = 3.16227766..., they need to tell that difference, the best simple Pi approximation is 355/113.
I just remembered 3.14 (how hard is it to remember 3 digits?) or if you like fractions, my Geometry teacher always used the approximation of 22/7. Easy to remember, enough to get you through most stuff you don't need absolutely insane precision for, etc.
Hydrogen is not the smallest atom that there is. Helium has a lesser radius.
Research it, it’s an actual fact.
Electron degenerate matter have the smallest radius. Neutron degeneracy will implode into a singularity so they can't be considered matter. EDM can still be classified as bayronic matter.
Hi Josh,
sorry for my perhaps bad English:
One way calculating pi by approximation is to inscribe a regular n-polygon (a polygon with n sides; here: for example a hexagon) in a given circle and then you circumscribe a polygon with the same number of sides around the given circle (here again: a hexagon).
Now you can say with certainty the circle's circumference is smaller than the circumscribed hexagon's circumference and larger than the inscribed hexagon's circumference. 1/3
Pi is the (C) circumference divided by the (D) diameter of a circle. Take any perfect circle, divide C with D and you'll always end up with 3.14192... So basically, you do 5th grade division and, if you do it correctly, you will always end up with a number that is indivisible by your divider (the diameter). So you add a 0 to the indivisible number (whatever is left of your C) and move over one slot over (the next lever of smaller decimals). Rinse and repeat 10 trillion + times.
I don't really get the hype of Pi. Sure, it is a constant and somehow looked upon as a very cool thing, but what about Tau? Tau has been widely used in the mathematics before Pi, and is by far the more usable of them both. Pi is only used because we somehow think it is non replaceable.
This channel has explored the Pi vs. Tau question several times.
The significance of Pi and Tau from a numerical perspective is identical, since they basically represent the same thing. They are both transcendental numbers, both derived from geometry, and both actually represent the exact same geometry. Tau is just twice Pi.
Pi, however, unlike Tau, has more justification in Calculus. The roots of any function satisfying y''+y=0 are distanced exactly Pi from each other. That would be half Tau. There are other similar such results. Once you get to Calculus, Pi just makes more sense.
who else thought that he said he computer that calculated 2k digits exploded?
I think I've had enough pi for today.
Knowing pi to huge numbers of decimal places can be very useful in predicting the motion of chaotic systems, where even small variations in the initial state or the accuracy of the approximations used (such as rounding the 40th digit of pi) can produce very different results to each other.
A question I've always had... how exactly are we able to calculate pi to such mind blowing precision if it is by definition a ratio of measurements which are bound to be rounded and contain some slight error. I'm guessing people aren't calculating pi by trying to take extremely precise measurements of perfect circles, so how are they doing it?
Pi often appears as the result of what infinite series converge to. For example, 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ... converges to pi/4. So you could add more and more of this sequence to obtain an estimate for pi/4. Or, use the series 4 - 4/3 + 4/5 - 4/7 ... to get pi. However, this sequence converges slowly, so someone trying to find many digits of pi would use a strategy similar to this one that yields better estimates for pi in less time.
it is kind of alarming that he used so rough value for the diameter of the universe, though it does not change his point.
1:20 He just wanted to write these digits down!
Going to have to update this now. Since we've detected gravity waves smaller than the diameter of a proton, we'll need a more refined measure of the circumference of the observable universe than merely within the size of a single atom of hydrogen. ;-)
we got the Plank Length. :3
just think, nothing in the universe is smaller than a 2x4
+Mattpoppybros a microscopic one as small as the Planck length
***** if it's not 2x4 it isn't a 2x4.
Mattpoppybros
But 2x4s aren't actually 2x4... (Don't take my word for it, check it out for real.)
Oh, he's referencing Vihart right there in the related videos. I feel silly. I loved that video.
this guy really makes numberphile
Wouldn't Pi end in a real sense at the Planck Constant of digits? Since, of course, you can't get any more accurate...
What would you say?
jjammmees ? what lol
Jack Brooks Laughing out loud.
Jack Brooks I was thinking the same thing
Jack Brooks Pi is just a concept, it depends how big the circle you are measuring is. If you had a 1cm diameter circle than the accuracy you could measure the circumference would end sooner than if you had a kilometre diameter circle.
You can't use all 39 digits of pi if you only 2 for the diameter. Your answer will only be accurate down to a tenth of a meter.
Doesn't really matter because we want to find the difference between using the two pi s
Hoo Dini you didn't make any sense.
Hoo Dini what two
Pis?
Dilip Tien pi_39 and pi
Hoo Dini yea, but you can't calculate the circumference of the universe with more precision than within 1 light year.
How the heck do you even calculate pi? How do we know this isn't made up?
There are several formulae to calculate π:
lim n->∞, π=n(sin(180°/n)).
Basically, π is n multiplied by the sine of 180 degrees over n. n is simply a number where the larger the number is, the more accurate π is.
π=sqrt(6(∞∑(k=1)1/k^2)).
Basically, π is the square root of six multiplied by the infinite sum of 1/1 + 1/2 + 1/4 + 1/9 + 1/16 and so on.
The list goes on.
The Fremd Dinge Guy
*There are several formulae to calculate π: lim n->∞, π=n(sin(180°/n)).
Basically, π is n multiplied by the sine of 180 degrees over n. n is simply a number where the larger the number is, the more accurate π is.*
Yeah. But the first 4 and crucial decimal places you get already with n=240, a polygon of 480 sides. 3/4 of degree. You could still pass your finger btw arc and cord.
And this is an "accurate" measure of Pi?
don't listen to those two above. π=half a circle's circumference devided by its diameter..
for example..
22=half the circumference
7= the diameter
22/7=π
Anime Soul
*don't listen to those two above. π=half a circle's circumference devided by its diameter..*
*for example..*
*22=half the circumference 7= the diameter 22/7=π*
No I don't.
But you can prove that......?
well what he gave is also correct if u havent heard of that why say dont listen to that?
He is just showing how seemingly innacurate something can be but people dont understand the exponential effect of each extra number we use for accuracy. Everybody know that we barely know the diameter accurately, its just impressive how such a small number of decimals can give such an accurate number
If you allow rounding up, then truncated version of the number will alternate between being less than or greater than than the full version, depending on whether the last digit was rounded or not. If you don't allow rounding (which you shouldn't round), then the actual number should always be greater than the truncated version of it.
"circumference" don't you mean "surface area"
It is. To 40 decimal places it would end in 4197, although he makes a good point. 39 decimal places is exactly as accurate as 38 would be, although maybe the fact that you know that next digit is (or rounds up to) a zero gives you a greater degree of accuracy.
I guess the answer would likely be "Yes", since one could arbitrarily choose the "segment" of PI to be as large or as small as one likes. Thus, one could choose an arbitrary segment to be of length 2, for example. Then, all one would have to do would be to find a segment of PI (not counting the very first digit) that was "31". Similarly, this would seem to suggest that one could find any arbitrarily finite segment of PI, such as "31415926", that "begins at the beginning" of PI.
I had not heard of this method of calculating pi before. I have not confirmed if this series actually does converge on the actual value of pi, but I can tell you that it takes many more than 10 terms to get an extra digit of accuracy. I calculated this series out to the 1200th digit with only a 2 decimal place accuracy (using fractions accurate to 30 decimal places, 2401 being the largest number used, I get 3.14242...).
We loooooooooooooooove Dr James Grime! 👍🤓
The exact value of Pi is described for example as a sum of fractions. Check the article on wikipedia or google the Pi.
3.14159 is enough to rock my world.
I think pi is super cool how it pops up in all sorts of places in mathematics.
Hey I am not the winner of the riddle but I still got the prize, thanks for those links, it was interesting!
When you take difference between the true circumference (d * pi) and the rounded one (d * pi39) the space between them is approximately 2.5 x 10^-12, which is smaller than a hydrogen atom. If we knew the diameter of the universe to more significant numbers, we could perhaps say it's 2.5572x10^-12, but it really doesn't change the fact that it's approximately 10 times smaller than a hydrogen atom. I hope that helps.
Oh you're right, my bad. It's meant every power of 10 terms you add. So 10^1 (10) terms would give you 1 digit,10^2 (100) would give you 2 digits, 10^3 (1000) would give you 3, etc. That's why its a pretty inefficient way of calculating it haha.
Just what SirThasik said:
E means “×10 to the power of”
It's pretty common to see a button with a capital E on scientific calculators that does this.
I figure it saves space, and it's no more difficult to figure out than using the carat to denote exponents.
Everyone loves Pi- Vi Hart, Bob Palais, Michael Hart
"620" is the last three! Had to do this many years ago. Still can to this day, the next challenge is to see how fast you can type it into a calculator! Good luck!
Pi is 4 - 4/3 + 4/7 - 4/9 + 4/11 - 4/13... The further you go with the formula, the more accurate the result will be, but it's not exactly simple to calculate in decimal form.
1 - 1/3 + 1/5 - 1/7 + 1/9 ... if you continue that forever, you get pi/4. So if you do 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... forever, you'll get pi. In fact, with every 10 terms you add, the result is accurate to one more digit of pi. So 4 is accurate to 0 places, 4 - 4/3 + ... + 4/21 - 4/23 is accurate to 1 place, etc. This is certainly not the most efficient way of doing it, but it works and it's the only one I know off the top of my head haha
I may be mistaken, but each digit would be stored in 2 bytes, so it would be 20 trillion bytes, or about 18.1 TB
Of course, for that sole purpose, considering digits are numbers from 0-9, you could use 4-bit computing, which would reduce it to 4.5 TB
4:23 35 years of my life :)
You you need 39 digits of PI to calculate the circumference of an orange down to the accuracy mentioned in this video. The diameter length of the circle doesn't matter for the accuracy (as long as you have an accurate diameter length).
Thank you. While I couldn't do the calculation myself, I do understand what you're saying. Having only the geometric picture myself, I couldn't see how pi could be derived without measuring something, and just think how accurate that measurement would have to be in order to calculate pi to seven trillion places.
They don't use real world measurements when calculating it. With the discovery of calculus and things like infinite series, several ways of calculating it's precise value came about. If your interested, wikipedia has several formulas for calculating it.
I used 355/113 if the instructor wanted a fractional value shown in the work. I knew enough digits for other problems. By the way, this was BC (before calculators).
Pi is a ratio (circumference/diameter), like 1/3 or 7/9, finding more digits of pi is simply calculating the ratio to a higher accuracy.
*cont* This is one way of computing pi, and with mathematical reasoning, they find ways to calculate the constant. They don't just choose a random series that looks pretty close, and say that's pi. They have to actually prove it. If you're good enough at maths you can understand how it's all done. These series to in fact converge to pi, and you need not worry about it.
Yes, like getting pi to that accuracy will calculate the circumference with such a small error if the diameter is a time-dependent assumption which was measured with instruments with an accuracy.
continued..the square root of 2 comes into it (which you also have to calcualte, that can be done by trial and error method) it comes in because the square (which is actually inside he circle) will have side lengths of root 2 as it's two other side will be of length 1. The root 2 remains in every iteration
as subsequent side lengths are based on it.
anyway google "calculate pi pythagoras" and first link gives a better explanation than I can here.
No. Some numbers repeat forever because of the number system we use, but pi isn't like that. It's an unrepeating string of digits.
355/113 is so close to the real thing, that a circle with a diameter of 113 miles has a circumference only 1.9 inches short of 355 miles.
Math: (355 - 113*pi)*(5280*12)
James uses wolfram - now I feel justified in my use of it as well.
I am a pi memorizer (amateur). last year I memorized 174 digits in two days (I mean it took me two days to memorized the 174 digits). It is fun to attempt to memorize as much as possible.
There are certain infinite series that sum to pi. For example, 4*(1-1/3+1/5-1/7...) = pi. By adding up more and more terms in the series, you can get a very accurate estimate of pi.
That's a good observation. I guess what they mean is that IF we knew the diameter accurately, then we'd only need the 39 digits of pi to get the circumference to an accuracy of a hydrogen atom.
i took the video as more of an order of magnitude kind of thing, that we don't need 10 trillion digits of pi, since 39 digits is accurate to within the width of an atom for calculations involving the largest distances possible (even if we don't know these distances totally).
That's my thinking anyway.
Yes, I agree. And also the diameter is constantly increasing (exponentially I think?), so any measurement in time dependant.
Imagine traveling back in time and show Ludolph van Ceulen that we can now calculate his life work with a device that fits in our pocket
You are an heartless monster
- I have a small question please: Which number is bigger, is it π or π(39) ? Because if π is bigger than π(39), then the circumference using the number π with infinite digits will be bigger than the circumference using just the first 39 digits of π, and the result of the subtraction will be positive.
Please, I need some clarification !
Thank you.
It is done by making a regular polygon around the circle, they do this with regular polygon of like 1000 angles or 1000000 angles and check what number comes out of the error with the length of the circle. its bassicly always the same and mainly done by computers but it was invented by Euclides if im not mistaken