# Tag Archives: big numbers

## Sideband #28: 2 ^ 64, ‘K!

I’d planned to do this later, probably for Sideband #64, but in honor of my parents 64th wedding anniversary (2 parents, 64 years, okay!) this numerical rumination gets queue-bumped to now.

Just recently I wrote about 64-bit numbers and how 64 bits allows you to count to the (small, compared to where we’re going) number:

264 = 18,446,744,073,709,551,616

That’s 18 exabytes (or 18 giga-gigabyes). Just to put it into perspective, if we were counting seconds, it amounts to 584,942,417,355 years; more than 500 billion years! (That’s the American, short-scale billion.)

Now the number, 64, is a significant one to computer programmers. For one thing, it’s 8×8, a full chessboard. Among programmers, the numbers 2, 8, 64, 256  and 1024 may be the most significant, although all powers of two have some significance.

The number 2, of course, is central to computer programming. A bit–the fundamental information unit–has two values: zero and one. (Hence my second-favorite bumper sticker regarding computer science: Calm down! It’s just ones and zeros!) Everything is based on this, which is why powers of two have the significance they do.

The number 8 is also central to computing in that it’s (by far) the most common grouping of bits. The well known computer word, “byte,” by default means eight bits. (And it does get its name from being a “bite” of  data. The half-sized package of four bits, a half-bite of data, is known as a nybble.) In fact, the history of computing contains six-bit bytes and nine-bit bytes, so the term “octet” is used to remove ambiguity. But these days, byte pretty much means eight.

Eight bits allows you to count up to 256, which is why it’s a significant number in computing. But these days 256 isn’t as useful as it used to be. In the early (American) days, 256 was more than enough codes to represent all the letters (both upper- and lower-case), all ten digits, all the punctuation symbols and plenty left over for control codes and drawing symbols. There was even room for some non-American characters, such as ñ and ö and Ω.

28 = 256; 82 = 64

Two to the power of eight is 256; eight to the power of two is 64. And 64 is the number of squares on a chessboard. It is also the number of bits used as a single package of data in many computers. (A 64-bit package is 8 8-bit bytes.)

And as mentioned above, a 64-bit number counts a good deal higher than an 8-bit number.  If you’re interested in the progression, it goes like this:

one byte (eight bits): 28 = 256
two bytes (16 bits): 216 = 65,536  (“64 K”)
four bytes (32 bits): 232 = 4,294,967,296
eight bytes (64 bits): 264 = 18,446,744,073,709,551,616

(Programmers tend to double things, which is why the powers go from 8 to 16 to 32 to 64.)

Which at least brings us to the point of this post! Two to the power of 64 counts to a rather big number, but it’s a number that has a name: 18+ exabytes. What if you had 64 K (65,536) bits?  How high would that count? First, here’s a  handful of ways to notate it:

2 ^ 64K
= 2 ^ (2 ^ 16)
= 2 ^ (2 ^ (2 ^ 4))
= 2 ^ (2 ^ (2 ^ (2 ^ 2)))

And here it is written out (ready?):

2003529930406846464979072351560255750447825475569751419265016973
7108940595563114530895061308809333481010382343429072631818229493
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5152690293170026313632894209579757795932763553116206675348865131
7323872438748063513314512644889967589828812925480076425186586490
2411111273013571971813816025831785069322440079986566353715440884
5486639318170839573578079905973083909488180406093595919090747396
0904410150516321749681412100765719177483767355751000733616922386
5374290794578032000423374528075661530429290144957806296341383835
5178359976470885134900485697369796523869584599459559209070905895
6891451141412684505462117945026611750166928260250950770778211950
4326173832235624376017767993627960993689751913949650333585071554
1843645685261667424368892037103749532842592713161053783498074073
9158633817967658425258036737206469351248652238481341663808061505
7048290598906964519364400185971204257230073164100099169875242603
7736217776343062161674488493081092990100951797454156425120482208
6714586849255132444266777127863728211331536224301091824391243380
2140462422233491535595168908162884879899882736304453724321742802
1575577796702166631704796972817248339284101564227450727177926939
9929740308072770395013581545142494049026536105825409373114653104
9433824843797186069372144446008267980024712294894057618538922034
2560830269705287662137737359439422411470707407290272546130735854
1745691419446487624357682397065703184168467540733466346293673983
6200040414007140542776324801327422026853936988697876070095900486
8465062677136307097982100655728510130660101078063374334477307347
8653881742681230743766066643312775356466578603715192922768440458
2732832438082128412187761320424604649008010547314267492608269221
5563740548624171703102791999694264562095561981645454766204502241
1449404749349832206807191352767986747813458203859570413466177937
2285349400316315995440936840895725334387029867178297703733328068
0176463950209002394193149911500910527682111951099906316615031158
5582835582607179410052528583611369961303442790173811787412061288
1820620232638498615156564512300477929675636183457681050433417695
4306753804111392855379252924134733948105053202570872818630729115
8911335942014761872664291564036371927602306283840650425441742335
4645499870553187268879264241021473636986254637471597443549434438
9973005174252511087735788639094681209667342815258591992485764048
8055071329814299359911463239919113959926752576359007446572810191
8058418073422277347213977232182317717169164001088261125490933611
8678057572239101818616854910850088527227437421208652485237245624
8697662245384819298671129452945515497030585919307198497105414181
6369689761311267440270096486675459345670599369954645005589216280
4797636568613331656390739570327203438917541526750091501119885687
2708848195531676931681272892143031376818016445477367518353497857
9242764633541624336011259602521095016122641103460834656482355979
3427405686884922445874549377675212032470380303549115754483129527
5891939893680876327685438769557694881422844311998595700727521393
1768378317703391304230609589991373146845690104220951619670705064
2025673387344611565527617599272715187766001023894476053978951694
5708802728736225121076224091810066700883474737605156285533943565
8437562712412444576516630640859395079475509204639322452025354636
3444479175566172596218719927918657549085785295001284022903506151
4937310107009446151011613712423761426722541732055959202782129325
7259471464172249773213163818453265552796042705418714962365852524
5864893325414506264233788565146467060429856478196846159366328895
4299780722542264790400616019751975007460545150060291806638271497
0161109879513366337713784344161940531214452918551801365755586676
1501937302969193207612000925506508158327550849934076879725236998
7023567931026804136745718956641431852679054717169962990363015545
6450900448027890557019683283136307189976991531666792089587685722
9060091547291963638167359667395997571032601557192023734858052112
8117458610065152598883843114511894880552129145775699146577530041
3847171245779650481758563950728953375397558220877775060723394455
87895905719156736

Now that’s a big number!

A long, long time ago on a USENET far, far away, I was part of a debate that started with the idea that, even if we had disk drives with 64-bit addressing, people would still fill them up with videos, images and whatnot. The idea grew from some of us old-timers reminiscing about our first brick-sized 5-meg hard drive and how we thought, “Gee, I’ll never fill that up!” (And look how that turned out; I have single image files that wouldn’t fit on that drive!)

The premise was that, even with seriously gigantic hard drives, we’d still manage to fill them and need more, more, more. And a couple of my computers with many hundreds of spinning gigabytes are showing much less free space than not. On the other hand, hard drives currently are in the terabyte range, which seems very spacious (I haven’t come close to filling mine). Plus there is cloud computing, which removes the burden of storing your own data, and there’s the idea of streaming video and music rather than owning it. This all appears to, perhaps, put an end to the tendency to fill the available space. Or even need much of it.

So it may be moot now, but a dozen years ago it was a debatable point (this was USENET; everything was a debatable point) that even a drive with 64-bit addressing might end up being not enough. We’ll leave off the detail that disk space is never actually addressed at the byte level, but at the sector or cluster level, so the available bytes are actually quite a bit more than the address space suggests. It turns out to be a detail of no significance.

Given the see-saw history of storage filling and capacity growth, it did seem possible. Until one really looks at the numbers. (In fact, I wrote my own arbitrary-precision calculator decades ago for the very purpose of exploring such topics.)

With 64 bits, you can count to 18,446,744,073,709,551,616. The size of that number alone should give you pause. It’s beyond terabyte, beyond petabyte, it’s 18+ exabytes! And while large data storage organizations might need that kind of storage, it seems rather a great amount for any individual.

Still, five megabytes seemed pretty big once, and each jump in storage seemed to offer an all but unfillable wealth of space, but each time we did find ourselves wanting more, more, more. So it’s possible that 18 exabytes might somehow be too much for an individual user with a passion for owning movies.  Let’s run some numbers.

I started by considering a very long movie: Gone with the Wind. It runs 238 minutes; nearly four hours. (This was before Lord of the Rings or other long epics came out.) I bumped that up to just over 4.5 hours to give me a nice round 16K seconds per movie. I assumed a frame was 8K pixels by 4K pixels (which at the time seemed ultra-high fidelity). I also assumed 32-bit color, 60 frames per second and eight-channel, 24-bit audio with a 96 kHz sample frequency.

This gave me 131,979,144,069,120 (uncompressed) bytes per very long movie. That’s nearly 132 terabytes for a single movie. You might think you could fill up an 18-exabyte disk pretty easily at that rate.

You’d be wrong.

It would take 139,770 movies that size to fill your disk. And they would take you over 600,000 hours (72.6+ years!) to view. And that’s assuming you could watch 24 hours per day, every day of those 70-some years.

I suppose you could fill such a disk (at least in theory), but what would be the point? For that matter, can you even imagine over 100,000 movies you’d want to see? (Forget the 100,000… can you imagine the 39,770??)

And there is the matter of exactly how you would fill it up. Downloading the data would be a bit of a problem. Assuming a 1-gigabit internet connection (125 megabytes) running full-out, each (uncompressed) 132-terabyte movie takes over 12 (24-hour) days. (If that seems improbable, keep in mind the movie we’re considering streams at a rate of 8 gigabytes per second. Stuff that down your Netflix streaming connection!)

Bottom line: filling your disk with the 139,770 movies would take 4,671 years!

Which all tells you something about the bit rate of real world information as well as the degree of data compression involved in our media.