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The fifth hyperfactorial: 5⁵×4⁴×3³×2²×1¹=86400000 milliseconds is exactly 1 day (twitter.com/fermatslibrary)
361 points by slbenfica on May 16, 2018 | hide | past | favorite | 189 comments

It was useful for sailors to be able to divide the day into shifts, so our day is the most divisible number of hours long.

1 x 2 x 3 x 4

That way you could have half days, quarter days, or third of days.

An hour is divided into 60 minutes: 3 x 4 x 5

The word "second" means a "second division by 3 x 4 x 5"

This article (from downthread) suggests that 60 comes from the Sumerians, who had a base-60 number system, and 24 comes from the Egyptians, who had a base-12 number system. Although it agrees that 12 and 60 are conveniently divisible in the way you describe, I'm not finding any evidence to connect this specifically to sailors, or even to people dividing up shifts. Rather it seems likely that the civilization's numeric base was chosen first, and then it was applied to timekeeping.


Shipboard life was one of the first large scale applications of precise timekeeping. You need it for navigation and for organising shifts and jobs (though I think navigation is the major player here).

For most people at that time there was no need for even hourly timekeeping, much less minutes or seconds. You woke up in the morning and ate your breakfast, milking the cows took however long it took, then you gathered the eggs from the hens, probably around that time you had lunch, then depending on the season you'd do whatever was needed in the fields until the sun started setting and then you ate supper.

But with navigation you had to know how far to sail in any given direction to not get knocked off your course.

I'm certain that the use of 60 and 12 base systems came from the Sumerians and Egyptians, but their application for timekeeping is probably influenced heavily by sailors.

It seems like even without precise knowledge of absolute time, precise knowledge of intervals would be useful; you could evaluate a time-trial to know if you’re improving as a footman/chariot rider; you could evaluate the maximum speed of a horse; you could measure height by dropping things off of the sides; etc.

So I would expect that, even if we didn’t have hours, we probably had second and minutes fairly early on.

Portable timepieces were specifically invented for seafaring, at least a century after it was obvious that they were needed for navigation.

I suspect most of the applications you described could be serviced by an hourglass and scale, which (I think) are much older than precision timekeeping.

You don't need any particular units for that, though. Any periodic oscillator you have at hand would work. (And with an analog clock you don't need to name the subdivisions, just look at the clock). You only need standard units with nice divisions for mass-coordinated scheduling.

How will you develop an equation that maps time-to-fall to height without some consistent unit for time?

Just keep the same periodic oscillator around for all of your experiments? Of course, if someone else wanted to reproduce those experiments, she would be using a different oscillator than you, but the same one for all of her experiments. She's probably dropping things from different heights anyway. The theory doesn't actually depend on units of time. The specific value of a constant like G does depend on that, but by the time engineers need to use that they have probably built real clocks.

Or mass-coordinated record-keeping for something like the Olympics. How do you know whether this year’s best runner beat last year’s best runner?

> Or mass-coordinated record-keeping for something like the Olympics. How do you know whether this year’s best runner beat last year’s best runner?

That sort of timekeeping a modern affectation. In the ancient past, it was sufficient to just be the fastest in whatever was the most recent contest. If you wanted to see if this year's best runner could beat last year's best, then you set up a race between them...

Have them race each other?

Ummm, on a farm you milk the cows and gather eggs and move the herd BEFORE breakfast. Then you tend to the crops for several hours before lunch, if you have lunch.

I am not a farmer, so I apologize if I described it incorrectly. My point still stands though; order of operations matter but not the timing of them.

Can you explain why?

Modern cows are milked twice a day, so it's more convenient morning and evening than e.g. noon and midnight. I don't know if the cows of long ago were like that, because the whole concept of a "dairy" is sort of modern anyway. I'm similarly suspicious of this "navigation" explanation. Timekeeping is important for determining longitude, but ancient mariners didn't know how to do that.

The cows udders are full and they are mooing at you and the longer you leave eggs the more shit that can happen to them.

Not the OP, but farmers tend to get up really early and have long working hours to boot. (I worked on and managed a dairy farm for a few years early in my career, plus have read about farmer life, is how I know.)

As an aside, I remember and like the point in a children's book I read as a kid, where they vacationed on a farm, and saw that the farm workers took their time about their work, did everything at a (somewhat) slow and measured pace, and still got a lot done (and well) by the end of each day. Very applicable to the modern software field, IMO, instead of the noise, flame and fury, often quickly descending to ashes, that we see a lot of nowadays.

>As an aside, I remember and like the point in a children's book I read as a kid, where they vacationed on a farm, and saw that the farm workers took their time about their work, did everything at a (somewhat) slow and measured pace, and still got a lot done (and well) by the end of each day. Very applicable to the modern software field, IMO, instead of the noise, flame and fury, often quickly descending to ashes, that we see a lot of nowadays.

JFYI, in Italian there is a saying is "col passo del contadino" that roughly translates to "at the peasant's pace" to indicate someone who is working at first sight slowly but never stops and at the end of the day has done the same or more work than someone else's that works fast but takes several pauses.

>"col passo del contadino"

Great saying. Will look it up in Google Translate. I know a little Spanish, and Italian is somewhat similar, I guess. But didn't know those words, except can guess / figure out meaning of passo (pace?) and del (of).

Passo is actually more literally "step" in the sense of walking, "fare un passo avanti" means "making a step forward", and you would say "ha un buon passo" to mean "he is walking at a good pace", but you can use it also (like in English) outside strictly walking, coming from the very same Latin etymology (in the case of English via French):


OT, but still originated in the farms, this is another great saying:


Getting it now, thanks for the info and links. I guess then that's where the term compiler passes (strides, per the etymonline link) over source code came from too :)

Yep, and you also walk across a border with a passport and, should your heart have some arythmia, you may have a pacemaker implanted, all from the same common root.

In Italian "passo" is used also for other things, it means also the pitch of a thread (of a screw or bolt) and the wheelbase (of a vehicle).

Just checked it in Google Translate, for English, Spanish and German:

  col passo del contadino (IT)

  with the pace of the farmer (EN)
  con el ritmo del granjero (ES)
  mit dem Tempo des Bauern (DE)

I think in English a good equivalent is "slow and steady wins the race".

>I think in English a good equivalent is "slow and steady wins the race".

It is very similar,yes, but that seems more like coming directly from Aesop's the tortoise and the hare, that in Italian would be "Chi va piano va sano e va lontano".

Cows with full udders are like humans with full bladders, only they have no way to relieve themselves. Cows with partially empty udders just stop giving milk if you don't milk them.

If I had to guess, get the most labor intensive part of the day done while the sun is still low in the sky and not as intense. Otherwise, you'd have to wait until around 3 or so before it starts setting.

Hilariously one of the most labor intensive parts of the day, bucking hay, is specifically done during the hottest part of the day. (The hay is too wet in the morning.)

Like natural selection at a cultural level.

This is just theoretical appropriation!

Both could be true. Those particular numerical systems would have started in those cultures and would have existed amongst many but other groups would have adopted the ones that made most sense in their field and consequently solidified their usage.

Kinda like any open standard; ultimately it’s a popularity contest.

Numbers like 24, 60, and 360 are 'highly composite numbers'. Here is a GREAT video on these 'anti-prime' numbers :https://www.youtube.com/watch?v=2JM2oImb9Qg

These anti-prime numbers were studied first by Ramanujan (of course) and conditions are that:

1) The prime factors must be consecutive (2^n X 3^m not 2^n X 5^m)

2) The powers of the prime factors must be deceasing as the prime number increases (2^4 X 3^3 not 2^1 X 3^5)

3) The numbers end with the the highest prime to the highest prime (2^4 X 3^1 not 2^3 X 3^2). Except for the numbers 2 and 4.

More here: https://en.wikipedia.org/wiki/Highly_composite_number

>These anti-prime numbers were studied first by Ramanujan


Ramanujan was easily one of the greatest mathematicians in history. Full Stop.

Entirely self-taught, his insights were incredibly novel and have helped human advancement immeasurably. Incredible stories about this great person abound. Sadly, he died at 32 from a curable disease, though misdiagnosed. He was deeply religious, a devout Hindu and vegetarian. His legacy is truly astonishing despite his few years. What he could have accomplished, had he lived, is a tragedy of the first order to mathematics and the human quest for truth.

Here is a good biography if you are interested: https://www.amazon.com/Man-Who-Knew-Infinity-Ramanujan/dp/14...

>Here is a good biography if you are interested: https://www.amazon.com/Man-Who-Knew-Infinity-Ramanujan/dp/14....

Thanks, will check it.

The movie was also enjoyable and heart wrenching... I watched it on a flight to India!


Thanks for this! I loved this explanation and I had never heard that divisibility is the reason we have 24 hours in a day and 60 minutes in an hour. After reading your comment, I found this article in Scientific American that gave additional detail: https://www.scientificamerican.com/article/experts-time-divi...

I believe divisibility is also the reason we have 360-degrees in a circle.

And 12 inches in a foot, etc. Perhaps in centuries to come, the US will be vindicated for its metric system resistance :)

In metric land we just tend to standardise measurements of things to fit into easily divisible numbers and keep our internally consistent system of units.

E.g. IKEA cabinets are 60cm wide, and so are most appliances like fridges. Double-wide things are 120cm etc. Pre-cut lumber is sold by 120cm lengths, sheetrock is 120x240cm etc.

All the easy division, plus the fancy units.

This has to do with Preferred numbers and ISO 2848: https://en.wikipedia.org/wiki/ISO_2848

Ahhh, thank you. I knew there had to be some standard behind their preferred number use - just didn't know what it might be.

The number of Ikea things that you can mash together and have work is awesome.

So, you’re saying that europe is about to switch away from meters to “ikeas” as the canonical unit?

With brexit, it should be easy for the brits to lead the charge.

I for one welcome the day when all the dimensions of all manmade objects are either products of numbers in the set {5, 3, 2, 1} or the set {1/2, 1/3, 1/5}. It will drastically simplify real world long division and factoring, and also lead to tighter packing of shipping containers.

Knowing apple, they’ll screw it all up when they drop some port and shrink some device from 1/6000 to 1/6001 thick.

Metric system is beautiful. Here’s a little poem

1 litre of water weights 1 kilogram and fills a 10cm cube to its brims. At 0 centigrade, it freezes, and at 100 it steams.

Let’s try imperial units.

1 gallon of water weighs 8.34 pounds and fills 6.8in cube to its brims. At 32 farenheit it freezes and 212 it steams.


1 fluid ounce of water was supposed to weigh 1 ounce. But municipal differences have foiled that equivalency over time. The goal with Metric is that this won't happen - we'll see how long it takes for one group to claim their meter is 1.1 official meters.

it's "curious" that there isn't a standard metric time. We are all happy with our Imperial clocks, i guess.

Some science fiction authors like Charles Stross I think (or was it Vernor Vinge) have used kiloseconds, megaseconds, etc. as units of time for space faring civilizations.

There were attempts, the latest was https://en.wikipedia.org/wiki/Swatch_Internet_Time

I know you're joking, but I hope not, actually. Although it's fun to divide by different numbers like 2, 3, 4, 6, 12, etc., on the whole, I prefer the decimal system, because it is much easier (and hence faster) to calculate things mentally, using uniform powers of 10.

As a simple example, when I need to mentally convert between millions and billions (used in the West) and lakhs and crores (used in India), it is so easy, because I convert everything to powers of 10 (e.g. million = 10^6, lakh = 10^5) and then multiply or divide as necessary, which really just amounts (pun not intended) to addition or subtraction of powers, using simple algebra.

40 million / 2 lakh = 4 * 10 * 10^6 / 2 lakh = 4 * 10^7 / 2 lakh = (4 * 10^7) / (2 * 10^5) = 2 * 10^7 / 10^5 = 2 * 10^2 = 200.




I also prefer the Western numbering system (groups of 3 powers of 10) over the Indian system linked above, due to its uniformity (for the same reason as metric over Imperial).

Nice thought, but not in my book!

Once you need to work with real precision, the US system is even more awful.

From an exactly 12" wide board, you cannot get three 4" pieces, since the saw kerf width is nonzero. It gets worse when you work with "standard" dimensions. A common 2x4 is actually only about 1.5" x 3.5" (and I don't think even the big sawmill blades take that big a kerf), and similar discrepancies all over.

And converting drill bit sizes? Don't even start; it's either tables or a calculator.

Whenever I get metric projects it's always such a pleasure. I wish we'd get our act together and switch...

But... none of those issues have to do with imperial units. If you have a 30cm wide board, you can't get 3 10cm wide pieces either. And 2"x4" in lumber sizes refers to the size of the lumber prior to drying, treating and planing, all of which shrink the wood a bit.

What is it about metric projects that makes those issues go away?

Good point - it's not so much that the issues of accounting for the kerf go away, they're just easier to deal with in decimal.

I usually am using materials like structural foam, carbon fiber, or metals, just used wood as an easy example.

And I'm using higher precision than most carpenters, so typically measuring with a micrometer which reads out in 0.001in (or 0.01mm) units. You can't just go from 1x12in to 3x4in pieces, you have to knock off maybe 5/64in for a kerf, which is 0.078125in so to get 3x 4in wide pieces, I'll need a board which is 12+5/32...

Lots easier to work with 1.984mm so for my 3x 10cm pieces I'll need a 304 mm wide board.

The bottom line is that once you get precise, the fractional regime which is indeed often easier to divide out by 1/2, 1/3, 1/4, 1/5, 1/6... etc. turns out to be no use at all because you can never actually use those nice fractions, and you're in a decimal regime with lots of fractional conversions (some micrometers even have a fractional inch readout mode, but it turns out to be not all that useful.)

Once you're in decimal, it's WAAAAY easier to be in metric all the way.

How's that working out for inches->feet->yards->miles. Seriously, there are some good things about the imperial system but divisibility isn't exactly one. It's more relatable and human, probably yes.

Up to about a yard, divisibility for imperial distances is pretty good, but you're absolutely right that past yards, it doesn't make much sense. However, for most day-to-day, hands-on tasks (with the major exception of driving), those shorter distances are sufficient.

I find this same principle is perhaps better exemplified by weights/volumes commonly used in cooking. Since many sizes (1 cup = 8 fl.oz, 1 pound = 16 oz, 1 gallon = 128 fl.oz, 1 fl.oz = 2 tablespoons, etc) are powers of 2, scaling recipes up or down on the fly is extremely easy.

On the other hand, given an accurate metric scale, all recipes can be converted to weights, allowing scalar conversions. However, I suspect the proliferation of accurate scales in home kitchens is a relatively recent (nascent?) phenomena.

In a way, there are parallels with a type system. Yards, a measure taken from a human limb, are for fabric, furlongs are for arable land, and miles (a thousand paces) for roads. Prior to the need for exercises in teaching arithmetic to the masses, it might have seemed rather pointless (if not exactly a type error) to express the measurement of one thing in the units for another.

In fact, land wasn't really measured in units of area as much as it was measured in units of productivity (ie bushels), which was much more useful to know

An acre (which is still used in the US for land measurement) is roughly the amount of land a team of oxen can plough in a day.

And a league is the distance a person can walk in an hour. The actual distance doesn't actually matter; two places can be 5 leagues away and be at different distances. However, it'll take you 5 hours to get to any of them, be it because different quality roads or elevation changes

Really? an acre is quite small. i find it hard to believe that a team of oxen can't do much more.

And distances (on land and maybe sea too) were long ago measured in interesting units like maybe an arrow shot (distance an arrow can travel) and such like (from my memory of reading legends as a kid, both Indian and probably Western or other ones) :)



Metric recipes are quite easy to scale too. 8 desiliters / 3=2.6 desiliters, not 7.2 arbitrary units.

People often point to that decimal portion and say it makes metric impossibly hard, but to anyone used to the system, it reads “‘bout half”. Most home measurement problems are not super exact science.

>Up to about a yard, divisibility for imperial distances is pretty good, but you're absolutely right that past yards, it doesn't make much sense.

Seems to work well with miles

miles ft 1/10 528 1/11 480 1/12 440 1/15 352 1/16 330 1/20 264 1/22 240 1/24 220 1/30 176 1/32 165 1/40 132

miles yds 1/2 880 1/4 440 1/5 352 1/8 220 1/10 176 1/11 160 1/16 110 1/20 88

One of the problems here is that given that our "home" numeric base is 2 times 5, we just aren't going to get both convenient unit conversion via moving the decimal point (or appropriate term in base of choice) and convenient divisibility.

So I suppose if metric just isn't signal-y enough for you (in the general sense of "you", not JackFr), you can join the call for the world to convert to duodecimal: http://www.dozenal.org/

This problem will solve itself when we apply appropriate sexual selective pressure to evolve 6-digit hands and feet.

"Hey honey, this is going to sound weird, but hear me out..."

That's tremendous -- never knew such an organization existed.

It's surprisingly easy once you get the hang of it. I started counting in dozenal for fun, and now when I think "7, 8, 9" my brain finishes the series with "X, E, 19".

It's not common enough to use with other people, but I use dozenal in my personal projects and journal whenever possible.

Well, if you don't skip all the intermediate units there is actually a nice, highly composite chain all the way up. Admittedly, the numbers are variable... but

22 yards in a chain 10 chains in a furlong 8 furlongs in a mile.

It would have been nice if numbers had developed into a base-12 instead of a base-10 system. But it's even nicer that the number system (now 10-based) and the measurement system (now metric) are in sync.

They in fact did, many years ago. This is discussed elsewhere on this very page.

Yeah, I was aware of that. I could have been more precise; it developed multiple times, but didn't stick around to the present.

But base 10 is an archaic, arbitrary or divisibility based number system itself

Every base is base 10!

Decimal is natural hands-based system. It's perhaps archaic, but neither arbitrary nor particularly divisible compared to its neighbors. (It's better than undecimal (9+2), but same as novemal (9), and worse than octal (8) or duodecimal (9+3))

USA isn't some sort of "metric-system hold-out". We sell soda in liters and wrenches in millimeters, among other things. And in the UK, cars are still rated in miles per gallon, as well as using Stones for human weight.

FWIW, according to WP, "the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year." https://en.wikipedia.org/wiki/Degree_(angle)

From many other sources, it's suggested that the length of the year is rounded to 360 for divisibility reasons, possibly because of the Babylonians' sexagesimal number system.

Where do you think the sexagesimal number system came from?

I think it came from 60 being highly divisible. Why do you ask?

There are other highly divisible numbers. 60 was probably chosen because of its approximate relationship to the length of the solar year.

> There are other highly divisible numbers. 60 was probably chosen because of its approximate relationship to the length of the solar year.

I had heard & read the reason was more one of counting convenience, more directly related to divisibility. What sources do you have suggesting that the length of the year is the "probable" reason?

Poking around, I see several suggestions that it might be related to the number of months, which has nothing to do with the number of days. I also found this opinion, which seems to contradict the idea that the number system is related to days in the year:

"Several theories have been based on astronomical events. The suggestion that 60 is the product of the number of months in the year (moons per year) with the number of planets (Mercury, Venus, Mars, Jupiter, Saturn) again seems far fetched as a reason for base 60. That the year was thought to have 360 days was suggested as a reason for the number base of 60 by the historian of mathematics Moritz Cantor. Again the idea is not that convincing since the Sumerians certainly knew that the year was longer than 360 days. Another hypothesis concerns the fact that the sun moves through its diameter 720 times during a day and, with 12 Sumerian hours in a day, one can come up with 60. [...] I [EFR] feel that all of these reasons are really not worth considering seriously."


That completely ignores the fact that ancient communities used the lunar calendar with periodic corrections every few years. 360 is equal to 12 lunar months, to the precision required of the ancients who practiced the insertion of entire leap months to compensate for drift (because the phase of the moon served as a more reliable clock than anything else that was available).

>"It was useful for sailors to be able to divide the day into shifts, so our day is the most divisible number of hours long."

Is there a source for specifically "sailors"? Also:

  1 x 2 x 3 x 4 x 5 = 120
120 hrs would be even more divisible (including by 5 and 10). So really they should have used 120 hrs per day consisting of 12 minutes each instead of 24 hours of 60 minutes each.

But then you can't count to 120 on one hand, whereas you can count to 12 by pointing to the joints of your fingers with your thumb. Also since you have four fingers and three joints on each finger it makes it easy to count in groups of those numbers, by counting up or across the fingers.

Back when I discovered how flexible the number 12 was this was something I thought would be awesome. (120 "hour" days). So much better than the metric proposal for 10 "hour" days (so called "decimal time").

120 hour days would be pretty cool, then you could say things like "Oh my god, it's almost 90."

”It was useful for sailors to be able to divide the day into shifts, so our day is the most divisible number of hours long.”

To be able do divide a day into halves, thirds and quarts you only need 12 parts in a day, not 24.

Also, according to Wikipedia, the 24 hour clock started with the Egyptians dividing the night into 12 parts around 2800 B.C. (https://en.m.wikipedia.org/wiki/Hour#History).

That “12” may have come from the ability to split it into halves, thirds and quarts, but I couldn’t find evidence for that.

Wikipedia's source says that the "12" hours was simply derived from the 36 (12*3) already in use in the annum calendar (36 10-day weeks in a year).


It's a bit strange to argue which use-case came first, as there are many use-cases of varying important to varying people, that all have the same natural solution.

Yeah, at first this seems like a shocking coincidence, until you realize that days are divided up into completely man-made units, not in any fundamentally natural way. If someone discovered something similar about the number of days in an Earth-year, that would be different.

It's quite surprising that in 2018, February still has 28 days. You'd think we'd bump it to 30, and pull two other 31's in a day.

> That way you could have half days, quarter days, or third of days.

The day needs to be divided into eighths in order to justify the factor of 2. Otherwise, the factor of 4 is able to handle both quarters and halves.

It's not just the divisibility of the day, it's the divisibility of the compass that ties into it. There are all sorts of shortcuts in the math.

Wow, that's interesting, thanks for sharing. I've always wondered why date, time and even other units of measure were not calibrated in the decimal system from the beginning. I had read about the controversy when Britain adopted the metric system, that may have been when I started thinking about this.

They actually tried to introduce decimal time[1], as well as a decimal week[2]. They didn't stick though.

If you are interested on the topic of the history of the metric system, check out The Measure of All Things[3], an absolutely fantastic read

[1] https://en.wikipedia.org/wiki/Decimal_time

[2] https://en.wikipedia.org/wiki/French_Republican_Calendar

[3] https://www.amazon.com/Measure-All-Things-Seven-Year-Transfo...

Thanks, sounds good, will check those out.

Also, there were exactly 30 days in a month and 360 days in a year until more precise accounting screwed up the math.

I guess it was useful for other people, too! Thank you so much for that explanation!

24 hr per day, 60 min per hour, 60 s per min, 1000 ms per s

    24   : 2 3 4
    60   :   3 4 5
    60   :   3 4 5
    1000 : 2   4 5
   count:  2 3 4 5

Babylonian number system was based on 60 (which is why we have 60 minutes in an hour and 360 degrees). 60 is dope because it has divisors 2,3,4,5,6,10,12,15,20,30 which is a lot of divisors.

Bee tee dubs Babylonians were like way ahead of their time math-wise. They were aware of Fourier for example.


Bonus points for combining "dope" with mathematics.

But quickly negated by "bee tee dubs" which is a cryptic mannerism that is actually longer than its original ("by the way" via btw). It's like whiskey tango foxtrot, just without the cool spoken alphabet.

In reply to ad-hominem's dead comment: word.

couldn't resist:

there are people who make a whole living doing that :V

60 is a preferred number for many card game decks as it works for 2, 3, 4, 5 And 6 players. 52 has difficulty with even deals for 3, 5, and 6 players.

But it works great for 13 players, so there's that.

this assumes the entire deck being dealt, which is really rare

But is that rarity cause or effect?

Bridge, Hearts, Pinochle, Spades, BS/Cheat, and President off the top of my head all deal the full deck. The first four tend to be played only with 4 players because it's an even deal, which is too bad

You can throw away the As or Ks to get a lovely (2^4 * 3)-card deck.

Or make a piquet deck like for 500 or Euchre.

Fourier analysis, not Fourier the guy. :P

Knowing about Fourier the guy would have been double dope.

The fifth hyperfactorial: 5⁵×4⁴×3³×2²×1¹=86400000 milliseconds is exactly 24 hours. 24 hours is approximately 1 earth day.

It would be very suspicious if our day was such an even number of milliseconds. It isn't, but still pretty cool! Things like this are what leap years are for, after all.

Or does that count? Is that caused by a partial day in the revolution of the earth around the sun or a failure of a day to fit into exactly 24 hours? Or both? Are we talking solar or sidereal days?

How so? Hours, minutes, seconds, and milliseconds are all man-made units. It’s the failure of 24 hours not fitting exactly into a day, not the inverse.

Leap years have nothing to do with milliseconds in a day- it’s days that fail to fit into a year; days and years are defined by the orbit and rotation of the earth. Leap seconds, however, are another story...

I know they are man-made units, but I don't think the men who made the second could measure the length of a day and split it into 86,400,000 units accurately.

Leap years account for a year not being exactly divisble into days. Leap seconds account for a day not being evenly divisible.

Unfortunately, the definition of one second as 9,192,631,770 energy transitions of the Cesium atom factorizes in a less pretty way:

2 x 3^2 x 5 x 7^2 x 47 x 44351

Previous discussion from last time the same OP made a tweet with the same content.


5 months for an exact repost by the same person getting a similar amount of upvotes/comments. That's a new one for me...

It's not a pure coincidence. Coefficients used for time measurements (12, 24, 60) were deliberately chosen to have as many (small) integer divisors as possible.

I was going to make a comment to this effect -- I think this is by design.

This is a logical alternative to our metric system. We use powers of ten for increasing units because we operate in base-10. Older civilizations created larger units by multiplying a smaller unit by either an existing member of the set, or the next largest factor. So you end up with a sequence like 2, 6 (3x2), 12 (3x2x2), 60 (5x3x2x2), 360 (6x5x3x2x2), 2520 (7x360), etc.

In effect, this is akin to saying a kilogram is 10^3 grams. It's novel to use because we were not taught to think that way. I bet a Babylonian would find this tweet to be kind of obvious.

Ehhhhh, sort of? But more: "surprisingly close to one day", because if you want to do correct time-keeping, a real day (or rather, a sidereal day, the time in which the earth makes one full rotation wrt "fixed" stars) is currently 86164090.7 milliseconds long.

You could also look at the solar day (the time it takes the earth to rotate such that the sun appears in the same place), in which case a day is actually a little longer than 86400000ms.

That's basically because our definition of second is no longer based on the rotation of the earth. If it was it would be exactly that.

An equally uninteresting coincidence is that 10! seconds is 6 weeks

And a related mnemonic: The number of seconds per day, in modern programming language notation, is 864e2 (That's 8-6-4-2 with an 'e' inserted before the last digit).

I generally prefer to have a SECS_PER_DAY constant, or write (24 * 60 * 60) to make the value clear, but when code golfing, and as a mnemonic, I remember that SECS_PER_DAY=86400=864e2

Well, a civil day anyway. A solar day is 86400002

Apparently, 86400000 was exact two centuries ago:


off by 2

4 hard problems in computer science

Hence, a very poor approximation to pi is given by 365 × 5⁵ × 4⁴ × 3³ × 2² × 1¹ / 10^10.

Why would "the number of milliseconds in a year divided by 10 billion" have anything to do with pi?

I feel like I was just nerd sniped, but this drove me crazy until I thought about it for a bit.

If we treat Earth's orbit as a perfect circle, then the number of milliseconds in a year would be its circumference. To get to pi then, we just need to divide that by its diameter or 2*its radius. In addition, we have the circumference in ms so we want to convert that into a distance or the radius into ms so we need the speed the Earth is rotating around the sun.

The average radius of the Earth to the Sun is 149,600,000 km so the diameter is 299,200,000 km. Earth's average orbital speed is 30 km/s or 0.03 km/ms. Combining these two numbers to get ms, (299,200,000 km / 0.03 km/ms) = 9,973,333,333.333 ms, which is very nearly 10 billion.

This isn't a reason or an explanation, just a statement of the arithmetic.

The siblings comments to yours are right - it's nearly random chance (give or take conservation of momentum during accretion of the solar disk into planets).

sure, but this is still just a consequence of the definition of "second" and a coincidental relationship between orbital period and day length. you can readily see that no such relationship holds for any of the other planets in our system.

unless i am grossly misunderstanding something, this is just an interesting tautology, similar to why torque and power curves for ICEs always cross at the same rpm.

Oh cool. My thought process went pretty much the same way. Essentially we're just using time as units and dividing circumference by diameter.

The nice denominator is what makes this interesting at all though, so the question sort of boils down to why Earth's orbital radius is such a round number of milliseconds.

... the diameter is 299,200,000 km ...

The distance light travels in a kilosecond.

It's pure coincidence, but "pi seconds is a nanocentury" is a useful mnemonic.

Don't think Hacker News notifies you of sibling comments, but it's not actually pure coincidence. See my answer at https://news.ycombinator.com/item?id=17084159

no real reason. it's just that 86400000 / 10^2 ~= 0.01 and 365 is relatively close to pi shifted two places.

there's no direct relationship between orbital period and length of day.

Don't think Hacker News notifies you of sibling comments, but there is actually a reason. See my answer at https://news.ycombinator.com/item?id=17084159

3.1536 - thats way to inaccurate to be of any interest imo. Even 22/7 is a way better approximation

Though nothing beats Milü in its simplictry and accuracy:


355/113: >An easy mnemonic helps memorize this useful fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits.

Why "hence"?

It is a pretty great coincidence, but there are things that helped improve the chances of this:

Numeric bases are often chosen to have many divisors. The numeric bases 60, 12 and 10 are used in time, which have many 2's, 3's and 5's as divisors.

So if you multiply them all, you get exactly such a product. The only coincidence is how nicely the powers line up.

Utter coincidence, and it only works at all if you throw in the milliseconds term, which makes it seem forced.

A lot of our numbering systems are inherited from early mathematics that dealt mainly with ratios of low whole numbers. And so selecting bases with many prime factors made the rational math easier. When your base is 60, it's easier to divide by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.

You might as well divide up the mean solar day into 10! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 2^8 * 3^5 * 5^2 * 7 = 3628800 chunks, which are each 1/42nd of a second. Or maybe make the 7 represent the 7 days in a week, and use 518400 chunks per day, each 1/6th of a second. You could divide up your time by so many whole factors.

In related news a (typical) year is 10^2 + 11^2 + 12^2 == 365 days.

Also known as 13^2 + 14^2.

This isn't much of a coincidence. We have days, hours, and minutes that are designed to be divided into 3's, 4's and 5's, and then there are lots of factors of 10's in the fifth hyperfactorial, getting us down to milliseconds. It's fun that it happens to work out to the hyperfactorial, but if it didn't, it was always going to be just a couple of additional or fewer 2s, 3s, and 5s away.

Speaking of factorials, this post I had written a few years ago may be of interest:

Permutation facts:


It mentions many kinds of factorials and other interesting types of numbers too.

6 weeks? Unless I'm radically misunderstanding that multiplication.

Yes, 6 weeks. Not sure why I typed 7.

Despite being well aware that 60 was chosen to be evenly divisible by small numbers, I find it a really cool coincidence that the exponents fell into place so perfectly.


The reason the number 108 recurs in Hinduism and Buddhism is that as the third hyperfactorial it was esoteric knowledge discoverable by sacred geometers.

(3³×2²×1¹ = 27×4×1 = 108)

Lol. Its just because its a highly divisible number. Saying basic multiplication is "esoteric" shows how little you expect of ancient civilisations. They estimate the circumference of the fuckin earth and you say its "esoteric" to think 108 is holy.

I would say that considering a number as "holy" is esotric by definition.

And perhaps in our lifetime the next hyper factorial (6^6 * 5^5 * 4^4 * 3^3 * 2^2 * 1^1) will be a lifetime (127.8 years (approximately)

So is 5^5 * 4^4 * 3^3 * 2^2 * 1^1 * 0^0

There is no consensus for what 0^0 means.

That's why they are called emoticons, just conveying emotions ;-)

10 factorial seconds is 42 days.

And 10! seconds is exactly 6 weeks.

It would be impressive if days and seconds weren't defined arbitrarily.

You can divide a day in any arbitrary manner. This is just a coincidence.

yeah but as other comments explain, this isn’t some numeric “property” as such, giving insight into anything. It is by definition.

Furthermore, a day is not even a day. We need leap seconds to sync up the solar day.

A "real" day is 86164100 milliseconds (23h 56m 4.100s)

And what makes stellar day real, and mean solar day not real? Having day defined in terms of the star Earth is orbiting around has a benefit of having more or less consistent time for noon. Having a day defined in terms of the mean sun instead of the real one has a benefit of having noons at exactly 24h apart instead of variable inter between noons of the real sun (see https://en.wikipedia.org/wiki/Equation_of_time).

... what makes stellar day real, ... ?

It's not really real, just side-real.

Oh man this was the pun of the week. Well done.

You won the internet today, sir.

I think there is a misunderstanding somewhere in here. m0skit0's comment was contrasting sidereal day (23h 56m 4.100s) to the original post concerning a mean solar day (24 hr 0m 0s). rimliu's comment seems to be saying that mean solar day has advantages over apparent solar day (each day has a different number of seconds that average to 24h 0m 0s over a year). I love the equation of time, but it is not related to sidereal time as discussed.

Small correction to m0skit0: wikipedia says the sidereal day is 23h, 56m, 4.0905s to account for the 26,000 year procession of the March equinox.

I also mention measuring days with regard to the start we are orbiting (Sun and solar days) compared to the distant stars (sidereal days), so no misunderstanding here. Once again it is about having Sun at more or less its highest point in the sky at noon. With sidereal time it would constantly shift and if you had sun highest at noon today, if would be highest at midnight half year from now.

"Real" is not real.

A solar day is still not exactly 86,400 seconds long.

I never understood this until now. Thank you.

almost like the person who thought up time keeping, thought about how it would be most useful.

signs of The Architect

Err, we created the "seconds in a day" metric, by picking the time span of seconds.

The earth's rotation takes a constant K time. Whether the time is 86400000 seconds or 50 seconds or 642552 seconds, is based on whatever arbitrary value we pick to mean "a second".

It's longer than that though, which is why we need leap seconds.

Yeah, it would be surprising if it was 100% regular too -- it's the rotation of a (very slightly but still) changing body of mass in space.

The Architect was off in terms of the ACTUAL time it takes for the earth to rotate - 86,164,100 milliseconds. What a "stellar" mistake for numerical perfection. Did The Architect screw up the math?

Our "original" definition of seconds was in term of our earth's rotation so it is exactly OPs number. We just changed our definition to a more constant second that slightly varies from our old definition.

But "milli" implies the metric system's version of a second, which is defined as per the decay of a caesium-133 atom. I doubt very much The Architect wanted us to mix our units of measure, unless The Architect was referring to that window of time where "milli" existed, but we hadn't yet changed the second to a uniform length of time (as opposed to one that would be affected by the variable rotation of the earth). If so, we are already past His golden age.

If I ever meet the referenced fictional character from the Matrix trilogy I'll be sure to ask.

There was never a Matrix trilogy, there was one very good movie and that was it.

I assume you're referring to a god here. I'm curious as to why you think this has anything to do with a god. Did it manipulate us to choose highly divisible numbers for timekeeping? Or is it responsible for 24 * 60 * 60 * 1000 being equal to the fifth hyperfactorial?

Is it evidence of The Architect every time two seemingly unrelated numbers are equal, or is there something about these two in particular that points to The Architect?

It was a Matrix joke. Lighten up

we're only 15 years past the release date of the 2nd movie... unless you go with XKCD's theory/view about the matrix - https://xkcd.com/566/

Next year, that xkcd strip will have been published closer to the original release of The Matrix than to the current day. I can't believe I've been keeping up with xkcd for so long.


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