Base 60 (sexagesimal) - Numberphile

The ancient Babylonians used a number system with base 60 (sexagesimal).
More links & stuff in full description below ↓↓↓
Tablet image courtesy of Bill Casselman and Yale Babylonian Collection - more at www.math.ubc.ca/~cass/Euclid/y...
This video features Thomas Woolley - he tweets at #!/thomasewoolley
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[@georgemissailidis7581]

"So the number we are talking about is 60....it is quite a big number.... ...." And the previous video I watched was about TREE(3)

[@racheline_nya]

awesome idea. base TREE(3).

[@jimi02468]

@@racheline_nya With base TREE(3) you could have just two digits of pi and it would be more precise than the 22 trillion digits that have been calculated in base-10

[@TCG-Collector77]

George Missailidis me too, haha :)

[@randomalbum9879]

He means it's a big number for a base. After base 36 you run out of numbers and letters so u would have to start using new symbols

[@Xnoob545]

@@randomalbum9879 but if we used an easy to learn pattern to make new symbols, it could be pretty convenient

[@raydredX]

I'd love to see many numerical representation systems from various languages. Not sure they're very knowledgeable about them. I love languages, I love linguistics, I love math. That's why I know a bit about it. Many people in Math don't like languages that much. But maybe! It'd be awesome! Hopefully numberphile's reading this.

[@CadaverSplatter]

The Babylonians followed the Sumerians in this. The system alternated between 10 and 60 depending on the numeric place. So for the single digits you went 1-10, but for their 10's place, they went up to 60. In their "hundreds place" they would use up to 10 units of sixty, and, the pattern continued in alternating between 10's and 6's in that way.

[@smartereveryday]

Is this called cuniform?

[@charlett3736]

1 like, 0 comments, from 8 years ago, by SmarterEveryDay, what have i stumpled upon

[@oliverolsen1120]

Weird

[@DerUbermonke]

Huh

[@ishdx9374]

8cocks ago

[@PraytoJah]

cuneiform* hehehe

[@yogiiification]

A more in depth video about the Babylonians and their counting/numeracy system from Brady would be fantastic!

[@macronencer]

I was hoping you might mention the fact that the Babylonians had a floating point system (which I find amazingly interesting!).

[@numberphile]

well, there is always more coming!

[@Xnoob545]

@Taurus Capricorn Yoyo

[@rolantv9605]

@Taurus Capricorn yoyoyoyo

[@fenderat1713]

@Taurus Capricorn yo

[@jonahrasnake498]

@taurus capricorn Happy 10th Birthday 9-15-2011

[@anabulatovic7306]

"you can cook half of a meal" 😂😂😂😂

[@MrHwilRRR]

LOL! You know what he means though. :3

[@shruggzdastr8-facedclown]

Ana Bulatovic: ...with Eric the half a bee! (a Monty Python reference, if you're unaware)

[@Megacooltommydee]

Thanos approves.

[@phodd]

They wrote numbers the same way we write time, so each of the base-sixty 'digits' is written in base-ten, but carries to the left occur after that. There's an example in the video on the Babylonian tablet: The sequence 1 24 51 10 is actually a base-sixty fraction that we'd write 1.24:51:10 if we followed time-style formatting and translates as approximately 1.414213 decimal. It's the square root of two... which is the length of the diagonal of the square it's written against!

[@AlderDragon]

Love your videos, Brady. I'm subscribed to all of your channels, but I think this one is my favorite :)

[@jonyeahyeahhhhh]

This was superbly clear, when you said they used their knuckles, I immediately counted to 15 by including my thumb, so I had the 30 x 2 for 60 but your method worked also which I think highlights the power of using base 60.

[@KevinHilt]

Brady's just so smart that he factored in time for the ads.

[@numberphile]

Like the number 60 - why not check out the sixtysymbols channel by the same film-maker as numberphile!? :)

[@prettyigirl1]

The symbol he drew for 57 is a unique symbol for 57. If he drew another number next to it (say 25) the resulting number would be equal to : 57 * 60 + 25 = 3445 This is the same as our base ten: 36 = 3 * 10 + 6

[@andizle123]

Wow, this connected so many different things i never knew had anything to do with each other! Truly fascinating!

[@Time2Splitt]

Thought the link between 60, calendar, length of day a bit confusing/lacking. But that's to be expected with a quick interview, and the video is much appreciated. But it seems it's such a huge missed opportunity! To teach about the origin of our time & calendar system! I understand that's a lot of work though (getting the right animations, narration, etc), and again, the video's much appreciated anywho! And if anything it makes you go and research it - hopefully finding a well organised youtube video that pedagogically well organised, explaining the origin of time & calendar.

[@kofi-tawiahagyeman]

Great videos! The sexagesimal system wasn't devised by the Babylonians. It was actually created by the Sumerians of Sumer Civilization in the 4th millennium BC, many many centuries before adoption by the Babylonians.

[@eurovisioncyan9550]

2520 is divisible by 1,2,3,4,5,6,7,8,9,10.

[@romanski5811]

But 5040 (=2*2520) has 60 factors.

[@blueyvids931]

@@romanski5811 wat

[@romanski5811]

@@blueyvids931 The video is about base 60.

[@blueyvids931]

@@romanski5811 watwatwT

[@romanski5811]

@@blueyvids931 Sorry, I can't help you.

[@raydredX]

That's how we usually represent bases but there can be more "artistic" ways to represent numbers. The Babylonians apparently used a mix of 60 and 10 based system. There's also Romans for example which use a mess of symbols with base ten. III=3 IV=4 V=5 VI=6 for example.

[@barkspawn]

Doesn't this mean they used base 10? If the ones column counts up to 9 then 'rolls over' and they put the 10 in the next column... that's the definition of base 10 isn't it??

[@andrewxc1335]

+Barkspawn I think he misspoke. They probably would have used lots of 12, the same as their hands. This would be base-60 (since 12s look different than 1s, they are not mixed-base, or base-12).

[@eyescandeceive]

+andrewxc1335 he definitely misspoke there

[@Qladstone]

Indeed how entrenched we are in base-10 that even when trying to think in other bases we revert to our base-10 structure of thought; there is probably nothing inconceivable about a base-12/60 system besides our mere unfamiliarity and lack of imagination.

[@twincast2005]

No, it's just how they wrote their numerals; doesn't make it base 10. It's base 60 because the actual "roll-overs" aren't at powers of 10 but at powers of 60. And yes, they did use marks for 10, not 12. Makes me believe the knuckle-counting came after they'd decided on base 60 for convenience (and made it possible to popularise it among the populace) while their numerals would be a remnant of a _pre-_ancient decimal (finger counting -> tally marking) system.

[@Wasserkaktus]

Nope, how we write base 10 is only possible with the numerals we adopted from the Arabs.

[@sdp8483]

I love this channel, very interesting. Keep it up.

[@badhombre4942]

The Babylonian counting system evolved around an economy based on poppy. Ten plants are efficiently stacked 1, 2, 3, 4 in a 60 degree arc segment, of which 6 completes a circular plot. This is reflected in the symbols; for the digits 1-9 plants and then 10 is folded into the arc segment. These plots can then be efficiently stacked with hexagonal packing, where each is surrounded by 6 others. So, from planting, harvesting and trading, they could very efficiently count their product.

[@katm9877]

Can you share a source? That's a very interesting theory....

[@AluminumStudios]

That was really fascinating. Thanks for doing this one! (and all of them)

[@ThisisTurbulence]

Never really thought about numbers. This video is extremely interesting

[@PandaBasher]

Very interesting video, and I like this guys style. Clear and concise.

[@sufiahussain3241]

Thank you so much for this video, this really helps explain the sexagesimal system to me.

[@GyanPratapSingh]

Do one on 5318008 Brady, it looks awesome when you enter it in a calculator and read it upside down.

[@XXBiBiX]

Every time I watch one of your videos... I want more :D

[@ThePeaceableKingdom]

I have read speculation (perhaps this is "out-of-date" and no longer a current speculation among experts...) that the Babylonian system arrived at base 60 as a way to integrate various multiple earlier counting systems across its territory, as 60 could reconcile various systems based on 5 (hand), 10 (fingers), 12 (knuckles), etc...

[@leightonjulye]

The seven-day week being approximately a quarter of a lunation has been proposed (e.g. by Friedrich Delitzsch) as the implicit, astronomical origin of the seven-day week. Problems with the proposal include lack of synchronization, variation in individual lunar phase lengths, and incompatibility with the duodecimal (base-12) and sexagesimal (base-60) numeral systems, historically the primary bases of other chronological and calendar units

[@AdlerMow]

But it compatible with a 13 month year, each month with 4 weeks. This gives you 364 days. If you add "a day out of time" (so that it does not enter in the week count), you have them have a year where every given day of the month will correspond to a day of the week (4th july will always be a tuesday, for example). Seems much more rational to me than having months of variable lenght.

[@MPythonGirl]

While it is easier to divide 60 evenly easier, it means that instead of having a 10x10 multiplication table like we do, it would have to be a 60x60 multiplication table. They did simplify them, by doing the multiples of 1-9 and then 10-60 and adding the tens and ones places. It doesn't even really work as a base system because they alternate groups of tens and groups of six. The current theory (according to my "History of Math" class) is that it grew from combining two older number systems.

[@asherdossetter4480]

60 is also divisable by 10. It's divisable by most numbers from 1-10 with the exception of 7,8, and 9.

[@acecombatmerc]

By God, Man !!! Brilliant !!!!!!. I love your videos. I hated math in school, but your videos make me want to go back to school but in Nottingham. It must be nice to know so many great minds.

[@imkluu]

Fascinating, always wondered why minutes, seconds were sets of 60

[@lordoffacts]

I love your videos.

[@numberphile]

thank you

[@Xnoob545]

4TH LIKE

[@praneethmashetty591]

7TH LIKE, 2ND REPLY

[@mariafe7050]

you're welcome

[@L_f_h_yt]

9th like fourth reply

[@technik-lexikon]

1:31 I learnded this fact recently and it absolutely blew my mind

[@e2theeyepie]

WHOA! That's freaking awesome! :D

[@legaralith]

It seems that way at first, but apparently all the way to 59 was basically in the same units place (like ones, tens, or hundreds are for us) and that 57 represent a unique digit like our 5 or 9. At 60 they jumped a units place and started new, which is a big deal because this is the first known system where the units place matters and one doesn't need unique symbols for each progressive units place advance

[@audreyteachesmath8317]

out of curiosity, of the babylonians were so heavily reliant on the number 12 when finger counting, why do we think they flipped their stylus to make the different mark for 10 instead of flipping it for 12?

[@josephrobertsjr]

I believe he made a mistake there. He actually depicted 67

[@diamine75]

First, the sexagesimal system is from Sumerian, not Babylonian. There's some difference. Funny ideas, but I think there's more than this into the sexagesimal system.

[@projectpearblossoms]

Awesome Video.

[@thesimulationist]

This is a better explanation than we did in our podcast

[@Mixa_Lv]

For the same reason it would be lenient if a deck of cards had 60 cards instead of 52.

[@Thebigkeoghowski]

The second the numberphile logo appeared at the start, my phone's battery was at 60 😅

[@ChrisP286]

Thanks! Really helping my h/w!

[@kenierstad]

Thanks this was really cool!

[@christopherverdery1294]

This video is a great segue into dozenal mathematics. Sixty is a superior highly composite number, but too large to work with, but twelve, the next smaller one, isn't. We even use that knuckle counting.

[@joshbuckler]

There probably should've been. I know of a couple modern attempts at a base-60 with unique symbols for all numbers. DeVlieger's Arqam is one (you can Google it), and Dozensonline Systematic Symbols is another. But these modern versions are also written the way we're used to writing numbers: with a pen/pencil on paper. Babylonians couldn't make such intricate symbols because they used a reed stylus and a clay tablet: basically limiting them to those little triangles.

[@oddrey52]

This is very interesting! I have wondered many times before, why sixty? I have always thought it was rather random but now I see it was actually a pretty good choice on the part of the Babylonians.

[@j.moonstorm3158]

Could you guys do a video on the advantages and disadvantages of kaktovik inupiaq numerals

[@evkin117]

thanx for teaching me something new

[@av733]

Good video.

[@CASHSEC]

Wow. So clear. Still need to cross this to understand the plimpton 322 tablet. These Babylonian kiddos had such insight.

[@ductuslupus87]

This channel has such a mad on for brown paper. I don't blame then, it's awesome.

[@Herobrine-fm3bh]

it's 1.8% longer than 360 days.

[@0623kaboom]

it is also the astrological calendar length ... and not a seasonal calendar which they also had that was 365.25 days long ... a very important fact he missed entirely

[@cjvan713]

A survival trick using an analog watch. Point the hour hand at the sun and halfway point between 12:00 and the hour hand is in North South line. In southern hemispheres, 12:00 at the sun in between 12:00 in the hour hand is the north south line.

[@Leibowitz]

yes, i'm actually really surprised the didn't use base 12 in their writing of numbers, considering their way of counting...

[@ghelyar]

Real mathematicians still use blackboards and chalk anyway, but the brown paper and marker pens make a sound that sends shivers down my spine.

[@osrevad]

Did you know: The number 360 is the smallest number that can be divided by every number from one to ten (except seven). This makes it great for circles, because you can divide it by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. That's a ton!

[@gregorycawthorne]

These only got on the top most viewed, because sxephil recommended them!!!!

[@MichaelEdwards]

The clock shows half a day or 12 hours and like the video says 60 is 12 groups of 5 as displayed in the hand counting system thus making 1 minute equal to 1 60th of an hour and therefore 1 second is 1 60th of a minute. But I do like the new information on the way Babylonians counted up to 60 on two hands.

[@usernumber1337]

They devided the number 60 similar to how we divide 1000 today. 6 * 10. The nuber 60 would visually look like 100, and 61 like 101, etc.

[@ThePheonix123]

finally! a question i have asked all my life but no one was able to explain!!! =P

[@rakshitdave5792]

Same here😂

[@XJoonasX]

64 Can be written 2^6(2*2*2*2*2*2) So it only has number 2 as a common factor meanwhile because 60 is smallest number divided by 1,2,3,4,5,6 so it can be written 1*2*3*4*5 (1,2,3,2^2,5,2*3) so it has 1,2,3,5 as unique common factor. 4 vs. 1.

[@jamesghost538]

They lived longer because of this system, it changes reality

[@GLENC0C0]

The wikipedia page shows some contrast between the thumb and the other four digits. It has it's differences and advantages/disadvantages.

[@PikalaxALT]

These numbers are what I will call minimal factorables. They are the smallest numbers that are divisible by the first n integers. For example, the first 15 minimal factorables are: 1,2,6,12,60,60,420,840,2520,2520,27720,27720,360360,360360,360360,... Notice that there are repeats. Some integers can be decomposed into a product of primes and prime powers - for example, 12=(2^2)*3. Because we are looking for the smallest numbers that have the factors 1,2,...,n, we can repeat the previous one

[@shonkhonil]

nice explanation.......

[@Aassymcass]

The Babylonians were pretty cool!

[@FountainMath]

WOAH MIND BLOWN! However, questions arise such as "When was the 0-9 Number system introduced?" What if it were introduced after this? OR It had already been in use long before this, just that the Babylonians using it creatively/outside the box way.

[@Nilguiri]

My friend Curt is a stage manager in the UK and he insists on using feet, inches, and fractions of an inch. It's easier to calculate in your head than metric. 12 inches is also a multiple of 2,3,4,6 It's the same idea.

[@casterospell]

this movie is super good for studying!!!

[@swapnil4613]

I just have discovered you. You're such a cool guy. Take love from Bangladesh. ❤🇧🇩

[@stonium69]

Wow, you are totally right, even about 1 (2^0).

[@GaryHurd]

A circle divided in half (with a "count" of 360 degrees) gives straight 180 degree lines, and in quarters a right angle of 90 degrees. The application to piling up mud bricks, and laying lines for the first cities are obvious. They also had a 10 day market calendar, and you can imagine the interesting combinations. We owe the mathematics of trigonometry to the Sumerians and their circle obsession.

[@jamesscholl2729]

Thanks :)

[@MehulKamdar]

Interestingly, Indian children were taught to count on their knuckles but in base 60 until my father's generation, following which we began to learn to count on our fingers. As a boy, I would find it strange that my father counted on his knuckles in trying to teach me, while I just couldn't get the hang of it. And then, of course, I could calculate mentally and didn;t need to bother. Thank you for a great video!

[@GaryHurd]

Actually, the place to start is Sumerian astrology. Their cosmology believed the circle was the "perfect shape," that the earth was a circle (disk), and the planets traveled circles. They observed the ~365 days of a year. Since 365 lacks symmetry it can't be "perfect." 360 has periods of 60, and 90. The Sumerian idea that a temple Holy Day didn't "count" as a day was the answer. The obvious Holy Days (the equinox, and solstice days, + one or two more) made the "real" 360 day year work.

[@zodiacmansions]

Egyptians did the same thing using Holy Days (holidays), along with using multiple calendars simultaneously, the most obvious the Civil calendar 3 weeks of 10 days, 1 30 day month.

[@celotehkibe]

You mentioned difficulty of using base-12 because of many symbols to draw but so is base 10 if you are using tally, for example. What if we make 12 symbols just like we make 10 symbols 0-9?

[@MinecraftHedgehog99]

Please please please do a video about sqrt -1!

[@ErichoTTA]

Thanks, I didn't know that.

[@EnderDio]

You, weren't but the original comment got blammed, and I was sticking up for him.

[@gregg4]

So to write 57 they would write 5 sets of one symbol and 7 sets of another symbol. Doesn't it seems like they were using base 10 in a way?

[@gurumullen]

Iv had that revelation before...then I usually have an inexplicable craving for a packet of cookies ;)

[@smaflenna]

I saw this was asked before, but I'm also confused at how this is not a mix of base 10 and base 60. Everything I've looked at says the numbers were written in tens and ones. (5 tens and 7 ones for example) as in the base ten system. Doesn't that indicate a base ten system?

[@nomenclator4339]

How it was determined that a second lasts that amount of time?

[@gregoriomarquezreyes9111]

+Martin Sosa Dias it's merely because these ancient civilisations chose to separate the day by 12 parts, and then by 60 parts, followed by another 60. There's nothing more complicated about it than that. Our GPS and time systems are remnants of the Babylonians and it seems like during the metrication of the world these were left untouched. Because of this, pilots and captains fly/sail in knots and travel in nautical miles, because a nautical mile is a minute of one degree of a circle, and a knot is a nautical mile per hour. So if you travel 60knots you've covered 1degree of 360 on Earth. Their is a metric clock(10:100:100), and a metric GPS system called gradians(400.00). Because the day in a metric clock is divided by 100,000 seconds and not 86,400seconds in our current time system, a metric second is 0.864seconds. So just think a second as really 1/86,400th of day.

[@nomenclator4339]

thanks !!

[@Sensanty]

However, the second DOES have a definition in this day and age where we know about atoms and their oscillations! It's a bit of a weird definition, certainly not what you'd think. A second is defined as exactly 9,192,631,770 periods of a certain frequency of radiation from the caesium-133 atom. So, that means that the protons and neutrons in the Cs-133 nucleus will get attracted and deflected that many times, every single second (to put it into INCREDIBLY vague terms). It's the basis of how atomic clocks work. For everyday life, the definition of 1/86,400th of a day works fine. But for things like satellites, where it's vital that absolute accuracy is observed, that definition doesn't cut it, because the duration of a day fluctuates every single day, plus time actually travels slower in space when contrasted to the surface. The duration of 9,192,631,770 fluctuations remains constant every time, however!

[@Qladstone]

Well at some point they had to measure the number of oscillations over some time period T and match it to the number of seconds in that time period T in order to come up with that definition; and in that they first required some measure of the seconds, however approximate it may have been.

[@0623kaboom]

from the long count calendar 360 day cycle was easily split in 60 .. and that was again easily split in 60 and so on as small as they wanted ... base 60 numbers are extremely ancient numerical systems that modern science has discounted as cumbersome but considering the ancients did more stuff with that system than we can do with modern day science it was obviously a far superior system when compared to todays limited base 10

[@BrotherAlpha]

I use a similar counting trick to count to 100 on my fingers. The fingers on my left hand are worth one, the thumb is worth five, the fingers on my right hand are worth ten, the thumb is worth fifty. (Technically, that only gets you to 99, but that's close enough.

[@Awwoh]

You should make a video about Euler's constant, e.

[@thehearth8773]

One way to say it would be that they used a combination of base 5 and base 10, but the simplest description is that they weren't using a number system that can be considered as having a base.

[@DeoMachina]

I find ancient math pretty interesting, but how on earth did WE figure out how the Babylonians counted?

[@omardjperez]

@numberphile I know this is old but can you make a follow up video on the Yale Babylonian Collection's Tablet YBC 7289? Is Pythagoras discredited for the Thereom? Please 🙏

[@OverAddictedGamers]

can you do a video on 2520?

[@Jakiladd]

Hoooray! i live in nottingham aswell :D

[@GaryHurd]

Babylonian, and Hebrew numerology held the number seven signified "completion." This magic number has obvious astronomical associations as it is the number of visible "planets," remembering that the ancients thought that the sun and moon were planets circling the Earth.

[@MichaelEdwards]

If you hold your hand at arms length and stack one on top of the other from horizon to zenith there are 9 hands. If you do that for a full circle there are 36 hands. If you divide that into decimal (because we like 10s) you get 360 divisions for 360 degrees. I believe this to be the reason there are 360 degrees in a circle.

[@Ellenki_007]

Please also explain about sexagesimal functions.

[@PerfectionReincarnated]

I have a smart way to count in hexadecimal. Each knuckle in the fingers and thumb,E. And the palm of the hand is 10.

[@AvielMenter]

I considered that but decided against using base 3 for two reasons: first, as a computer scientist, I find base 2 much more intuitive. Second, it's sometimes hard to tell between a 1 and a 0 in base 3.

[@numberphile]

I think there is a good chance!