There seems to be a bit of a tradition of making a special video when your YouTube channel passes a million subscribers.
'It is weighty...' Now I was a bit slow making mine
because - well - you're about to see why. Essentially I wanted to show how big a million is; a way to kind of visualise it. This is where the idea of printing pi to a million digits came from. Of course a million digits of pi have been known for a long time, you can see them on the internet, even buy a book of them; but I wanted to print it like a conventional number, left to right on a single piece of paper. I had no idea what this would look like or even how big it would need to be; so after a few emails and phone calls - and this is cutting a long story short - I ended up travelling to Denmark to a printing company called HSA Systems. These people were amazing and it seems no job was too big for them. Here I am with a guy called Torben who was an absolute star and is the guy who really made this happen from a technical standpoint. For various reasons, including the fact it's a monospace font, we decided to print pi in Courier New. The point size we chose was 8. This means each digit was 40 pixels wide, one pixel is one six hundredth of an inch, so each digit was a fifteenth of an inch across. Now this meant our million decimal places, or a million and two if you count the three and the decimal point, would be 1.05 miles long, nearly 1,700 metres - how is that for some serious Numberphile action? Here we are doing a test print run, and as you can see it's on white paper. Now Numberphile fans will know that we love brown paper, and two reels of special brown paper had been donated by a UK company called Millingtons; but it had not arrived. It was due to arrive, it just hadn't. So it looked like we might resort to using the white paper, which was easier to get our hands on, it was kind of like the backup. But, at the last minute, like the last minute, the brown paper arrived.
- 'You're our hero!' There we go, here it is being mounted and we're about to start printing. By the way this guy is Hugh McPartlan, he's HSA's man in the UK, he's with me here in Denmark, and he's the real reason the whole project really succeeded, you'll see more of him later. I have to say it was amazing how smoothly the whole thing went and a print run took about
48 and a half minutes. It turns out the whole thing hinged on this wheel, the encoder, which measures how far along the paper we are. Now it's just three one hundredths of a millimeter off being a perfect circle which means, I'm told, our spacing was pretty good. And I must say I do find it lovely how circles, and of course pi, were used and were so instrumental in our printing of pi. Now in addition to just the numbers, the decimal places of
pi, we included markers all the way along to count the decimal places. There was a marker every 10 digits. This was so we'd be able to find parts of pi more easily when the time came.
- Okay so I'm looking for 216182...'
- We also had a marker every yard along the paper to give us an idea of distance travelled. Now I would have preferred metres, being bit of a metric man myself, but for complicated reasons it was hard to make this synchronise. It all relates to the fact the technology we were using to print with is in dots per inch. It made our chunks of pi more easily divided into imperial units, we had a lot of problems when we tried to do it in metres. Now here's a quick interview with Jesper, he's the boss at HSA, and another man we owe a huge debt to.
- Well we do not only do one type of printer, right? We do a lot of different printers and printer controllers that were manufactured from
HP technology but also Xaar technology out of England; Cambridge, England. We do both software, we do the printed circuit boards, we do the
printheads, the controllers, everything right? We do a lot of number printing. And that could be, for example, lottery; it could be pin numbers; it could be a unique identifier for products. For example within the the food industry, right, well they would- may require a unique identification of a certain product. And we would either generate the number from- straight from our software or we would load a database and then print the numbers from the database. Well numbers, mathematics, is very much a big part of our day right? Calculating DPIss and pulses per rotation and what have we not so- and I'll say we're not the ones doing the hard mathematics, that's the the guys in Copenhagen who who do the software and also the core- the core of the software doing the hard mathematics. Namely calculating how to get a drop out of the printhead at the right time, right size, etc etc. So- but yes mathematics is a very big part of our our working day that's for sure.
- (Brady: Tell me) what you know about pi? What I know about pi? Okay. Well-
- (It's not a maths test!) There's not a math test (You obviously know the number?)
- Do I have to wear flip-flops and a white sheet now or what? (But like, you've never
print- have you ever) (had to print out pi before?)
- No, never no, We've never done that. We do- we do make use of pi of course in our everyday work, that's for sure, and we know a bit about pi because we have to do a bit of calculations when it comes
to our tooling machines etc.
- (You've got circles everywhere!) We've got circles everywhere,
yeah, and of course we use pi a lot but we have never done 1 million digits pi before on a printing machine, so that's quite exciting.
- (Have you done any number to) (a million digits or am I your first?) Well, you're the first where we've done a million digits that's for sure. (There you go Torben, you're
holding a big piece of pi there!) So after printing our reel, what next? Well, wrap it up of course, and then print another one! We figured this would be a handy backup if something went wrong. If you're interested, the second roll was never actually used and it's still in my office. If you're some wealthy math enthusiast you can even make me an offer if you like - I'll warn you,
postage will not be cheap. Now I also still have the HP cartridges we used to print the whole thing. Amazingly we used only 20% of a cartridge, just over eight milliliters of ink. This is what eight milliliters of ink looks like, or 194 million tiny drops from the printhead. The people at HP are rightly proud of what their stuff can do, it's called thermal inkjet technology. They were also super excited when they found out about the pi project and later they helped me cover some of the production costs. This was really generous of them and I'm very grateful. So, anyway, two rolls were printed and then it was back to the UK for me and the rolls were posted not far behind. Here they are when they arrived at my house. Next was arranging a way to unroll the paper. Now after a wild goose chase, and calling several Air Force bases, I was able to hire the former RAF base at Bruntingthorpe - that's
in Leicestershire in England - which is now used as a car testing track and aviation museum, they've got all these Cold War jets there. I have to say, if you like planes or exotic sports cars, this is a place that will blow your mind. Now a section of their two-mile runway was sectioned off so we could do our thing; it turns out no one else was using the facility that day we had the whole runway to ourselves. But we were there with no real sense of how difficult this might be, no idea how a mile of paper could be effectively unrolled. But that man Hugh, together with a chap named Jon Kenny, devised an ingenious contraption. Here are the designs they made, and here is the real thing after it was made by Jon. Now I've got to admit I had a mild heart attack here thinking the
paper wouldn't fit Hmmm, not sure about that. But it got on and we were able to start. Numberphile regular and pi enthusiast Matt Parker came along to be our tour guide; I should probably thank him for giving up a whole day but I think he enjoyed himself far too much as it is.
- This is very exciting. Now the unrolling was okay, and even the rain was tolerable though a nuisance, but the biggest problem was wind. The wind was mercifully very light this day but it still played havoc with the paper. Luckily I'd brought some bags of sand; we were able to use this to weigh down the paper. But the sand was running out fast. We sent for more cones which was stored elsewhere on the site, they helped, and then I went for a drive to the end of the runway and found something else that helped, these cone bases which were nice and heavy but not too tall so we could stack them in the back of the car. Now while the runway crew worked really hard on the rollout, Matt and I spent some time further back recording pi facts. Now much of this was cut from the main video, that's just how things go, we needed to keep the video short, but now I'm going to show you more of them. Enjoy these and when they're finished I'll tell you more about how the day ended. Pi - what a number. Abs- it's unparalleled in mathematics. And, as we all know, the digits never end; they just keep on going forever and ever which is incredible. And you can download huge numbers of these digits and some people, insanely, will print off tens of thousands of them and make like a poster. Not Brady, no. At Numberphile he has printed off a million digits. So this is the beginning and this is a mile; so you can see they're unraveling it further up there. there is one mile of a continuous piece of paper-
Alright so it is raining a little bit, but that's fine, that's the great thing about pi, it
doesn't change if it gets wet. So we're going to begin our exploration and-
- (Wow, pi is the same when it's raining?) I know, who knew! It's moisture invariant.
So here I've marked the first 10 decimal places that only use three different digits. Okay here we've got 1565 51565666111; and that is a run of 1,2,3,4,5,6, 7,8,9,10,11,12,13,14 - 15! - which only uses three different digits, so there you go. I mean-
so here at Numberphile we're answering all the questions that no one's ever asked. Ok so we've come a fair way, where are we now? We're up to a 121 yards, so we're at 11 squared yards, and we're working away along. And I mean famously people try and memorise pi. And so I've got it down to, you know, the first couple of digits
and - oh, one of our support vehicles is coming up Brady, you might want to pop out of the way, no it's not. He stopped, he's getting out. 89,000- If you take the alphabet; so A becomes 1, B becomes 2, C becomes 3 and so on you can turn words into letters. So my name is Matt, MATT; M is the 13th letter of the alphabet, A is the first, T is the twentieth - so I'll be looking for 13012020 for 13 1 20 20; and you
can- oh we've gone past it. So that 02, that
is a - I'll put up here - that is a B; work my way backwards. So before that is a 12, that is a L, before that is an 8 that's an H, before that is a 15 that's an O, before that is a 23 that's a W, before that is a 5
that's an E, before that is an 8 that is
another H - ahh I'm writing in a puddle!
Before that is a 14 that is a N, before that is a 25 is a Y; and before that is a 61 so ok so that's not- so that there that is nine digits, well, nine letters so it's 18 digits, and it says YNHEWOHLB.
- (That's not a word!) Well, it's a word. It's not a word that makes sense, like it's not a word in the English language but it is a string of letters. And actually if you want the longest word that makes sense it's over here somewhere, we can go and find that. Corot. That is the first five letter word which is actually a recognised word in the English language. Corot was a French landscape and portrait painter as well as a printmaker in etching - there you go! Or Corot I guess it should be. If we go all up to 380353 we come across 'Grede'; not 'greed' as in 'greedy', grede: GREDE; as in a leader in the development and production of lightweight ultra high-strength cast ductile iron components. So it's a company called Grede. After that's Bruel at 466574 digits. Bruel is a French singer, actor, and professional poker player. And then later on Kuthu which is a 2004 Indian film; and then at the very end, about nine digits before we finish, is via fj - which I think looks a little bit like via Fiji. Not impressive- you look underwhelmed to be honest. A lot of people think that any word, if you look far enough, will be in pi somewhere. And so far all the ones we've checked are but it's not guaranteed. So pi, we think, is something called a normal number. However, we don't know for certain if it is a normal number, we just think it is. And if it is a normal number it means any string of digits is equally likely or appears with equal frequency compared to any other string of digits of the same length. And so
if it is normal that's true, but we don't know. So whenever someone says, any word or any expression or any phrase you ever come up with it somewhere
in pi, you can say ah, we think so but we don't know for certain. So the vast majority of real numbers are normal but it's very hard if you're given a number as an example to prove that that number someone's given you is definitely normal. In fact we've done that so far for no numbers. No naturally- like, no numbers we've just found have since been proven to be
normal. But we know that they're all out there, we know most numbers are normal. In fact, the complete works of William Shakespeare appears in more numbers than it doesn't. So I mean- so people say
that pi- it's not that exciting when it comes to pi, in fact my name Matt appears in the first billion digits of root two, it's in the first billion digits of e - I've checked all these - and it's in the first billion digits, I hate to say it, of the golden ratio. So in fact if you pick a number at random, odds are my name will be in there probably in the first billion digits. If you actually do want to find your name in a
normal number, there was a number called the Copeland–Erdős number.
And what Copeland and Erdős did was they got the number- they started zero point and then it's all the prime numbers in a row. So it goes 2 and then 3 and then 5 and then 7 then 1 and 1 for 11 and you just put them in their base 10 form in there. That number is proven to be normal because it was generated to be normal. And so if you have the Copeland–Erdős constant in there somewhere, is guaranteed to be the works of Shakespeare, your name, what you're going to have for lunch tomorrow; anyway you want to encode it is in there somewhere. I mean, it's also got the complete list of everything you're not going to have for lunch tomorrow, so it's not very predictive. Pi will never fall into a recurring pattern. So so no point will it become recurring but you may have incredibly long strings, maybe infinitely long - not infinitely
long - you're looking at very very long strings of digits or predictable bits or- I mean 3.141592 it may appear in there somewhere else. And so any chunk I look at here that's going to appear somewhere else but not in any kind of predictable fashion. Pi is what's called a transcendental number which means you can't write it down in a nice neat equation. So you can have equations for pi, I mean somehow we calculated the first million digits, but that's with an equation where you always need more and more terms to get more and more accuracy. If you have a polynomial with coefficients that behave themselves you're never going to get pi out as an answer, there's there's no repeating finite way you can write down pi. Now while we've been talking I've noticed that the people putting down pi have gotten substantially smaller; in fact they're probably most of the way to the horizon at this point so we should go- we should go catch them up.
If I label that you'll see here 176451 and the digits 314159, there it is. 314159. That is the longest bit of the beginning of pi in pi in the first million decimal places; so there it is. 314- I'm exhausted, I shouldn't have run this far.
You know what's really interesting right? People say pi is random and this is a million random digits so a lot of what we've been finding are things you'd expect
anywhere in randomness. But I've never seen this bit of paper before in my life right? You said 'Matt look up some things in pi'; I just did it at home on my computer, but yet when I come here to a million random digits I knew exactly where that was going to be. I came
along, there it is. Pi, yes it's random, but predictable and we can calculate it and it's always the same and that's amazing,
I need a lie down. There it is, half a million. So that there is the half a million digits of pi. If you don't include the 3 at the beginning the half million digit is a 2,
there it is, right there. Far, far in the distance a dedicated team of people with a trolley with the pi like reel on it gradually unrolling and, I mean, it's hard to tell how much is left; if only we had some kind of mathematical constant to calculate how much paper is going around it as it unravels - who knows? But they're doing an amazing job and they're putting it down faster than we're talking about it which I find a little bit disturbing; because they're all business. Like, they've got a well oiled machine, cones moving, things going down; whereas we're just nerding out on 'hey look three 7s!' and then they're disappearing off into the dist- we probably
should either go and help them or make them some tea. You can see, I'll underline it there, it's a bit small, that's 11492- Okay, so before that, if I- if we'd stop putting the pi out here right, if that had been the end of the piece of paper, every number from 1 up to 11491 is in there somewhere. So 11491 is the biggest number for which every smaller number is somewhere in there; and then suddenly 11492 appears. And now you
think, well, you know, where's 11493? We've already had it. In fact we've already had loads. The next number, well- in the entire mile of pi, as of now, the biggest number for which every number smaller than is in here somewhere is 14522. There is no 14523 anywhere in the first million digits of pi, it is the smallest number which is not in the first million decimal places.
- (Are you sure? You gonna check?) I've checked! I- I think I've got those digits right, I assure you if I've gotten them wrong people will leave a comment under the video. Okay, so we've found 11492, the next number up Brady is a bit of a special treat for you, so we need to go up into the- it's on the order of 857,187 from memory - don't quote me on that, let's find it. Okay we've actually got a drone here filming which
is actually amazing and as you can hear they're incredibly subtle, you wouldn't know you're being filmed by a drone right now.
Modern technology man... Five million, six hundred and thirteen thousand, four hundred and eighty seven; d'you recognise that?
- (I feel like I should) You should, that is the
biggest Brady number that appears in the first one million decimal places of pi. So there you are, that's- I honestly- that's seven
digits long. I was I was astounded when I saw that appear. So, I mean, there's every four digit numbers in
here somewhere; most of the five digit numbers; reasonable chances on six - once you hit seven it starts getting a lot less likely. And so I thought, you know what, I'll just check the Brady numbers. And sure enough there it is. I mean we're not that far from the end now, you can see our crew just down there, I think they've almost
finished. And so we're what- we're 85.7%
of the way through the first million digits, sure enough, a seven digit Brady num- I hate- I to encourage the ongoing discussion and research into Brady numbers but I'm very impressed there's one just there.
- So, we'll film the last little bit-
- It took three hours to unroll the first half mile of paper, but then the team got on a roll with the new cones and the cone bases and a better system and better weather; the next half mile was all done within about an hour. Now I did a bit of editing of
the video finale when you watch the main film, but here's how it played out without me interfering with the editing. Okay we're four hours into this and we are nearly at a million digits. So they've been coming out at a rate of a quarter of a million digits an hour, which is approximately 4167 digits a minute. And we're- we're almost there. So, I mean you guys have done the hard work, do you want to wind off the last of this? And a million digits is there! All right, so if we- so if we just take that off there, all right. So that is- there is wha- it's a 1, in case you're wondering. So we- - Just tip it towards me a bit, yep.
- It's a 1. So there you go, we have just- I mean, I think a token round of applause, well done guys, we have just hit - that is the quietest round of applause I've heard in my life! Too tired to applaud. So, I mean that is absolutely amazing; and Brady, you have just set the world record for the most digits of pi ever printed out on paper. Although by simply adding a three I will now have the world record for the most digits of- 30 - is it a nine or a seven? Two digits is good enough. So I've now got the world record for a million and two decimal places of pi listed out. So that is absolutely fantastic and now the very quick process of rolling it all back up again before an aeroplane lands.
- Cool, all right. I should also thank James Hennessy and Ed Fielden for doing much of the camera work. I was very much focused on the stuff with Matt but most of the great shots you see in the final film came from James and Ed. The drone shots, I mean they were just amazing, I think I could look at these all day. Here I've marked the start and the end point of our 1.05 miles. Finally it was time to clear up-
- And I'm- I'm not even 0.1% of the way through!
- Before we started there'd been plans to reroll the paper back onto its reel but that was clearly misguided; and besides the rain and wind had wreaked havoc, finally weakening and then starting to rip some sections of paper. So I sent out two of my helpers, Josh and Jacob, to salvage some of the best paper snippets. I still have these ones as souvenirs, it includes the start and the finish of course. Some other people there like Josh, Jacob, and even Matt I think, also snipped out their birthdays where they could find them within pi. The rest of the paper was simply collected, chopped up, screwed into giant balls and sent to that recycling bin in the sky. And with that the job was done. Now I have to say, the whole task was probably much harder than expected but also more rewarding. And driving around on a runway is awesome. The whole thing wouldn't have been possible without these people. I think if the team was any smaller on the day we may well have failed. Everyone really chipped in and thank you to all of them. And thank you to you for watching this and watching all the videos here on Numberphile. Hopefully you're among our million subscribers and will continue to be for years to come. Okay you're gonna start seeing sand, that is effectively ballast, paperweight.
