Cuneiform Numbers - Numberphile
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- Опубликовано: 14 мар 2021
- Alex Bellos discusses how numbers were written in ancient Cuneiform.
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More from this interview on Numberphile2: ruclips.net/video/n-sxOVSZc-Q/видео.html
Alex Bellos books on Amazon (including the Language Lovers Puzzle Book which features cuneiform): amzn.to/3czJjXl
More Alex Bellos on Numberphile: bit.ly/Bellos_Playlist
Do you have any videos of Euclid(ēs)?
don't forget the 12 note music scale is from noncommutative phase as Fields Medal math professor Alain Connes points out. The ancients were more advanced than we are.
Is that the Adam Savage in your patreon list or a different Adam Savage?
I made 4 clocks and the world doesn't know what they look like or how they work but I do, not joking message me I'll show you but I want numberphile to see them
Comment if you have more than 3 brain cells
The world: "...meters, feet, inches..."
Mathematician: **half an iphone**
banana for scale
you mean:
Americans: feet, inches
rest of the world: meters :D
@@huawafabe
Eh, no, there are a few edge cases besides us folks in the U.S. who also use feet and inches.
Overall, yes, most of the world uses metric.
Mathematicians have no use for units. 8D
@@huawafabe Supposedly it's the rest of the world but 100% of complaints are by a snobby Western European or a disgruntled American. SI is really a French thing at its heart. Commonwealth people are generally fluent in both. Chinese, Indians, and I assume much of the rest of the world have their own traditional market units that are common in daily life, with metric taking over in formal settings.
"Actually for doing math its much much better than the Roman system"
I would hate to see a system worse than the Roman system.
That obscure medieval number system maybe
Unary.
I mean binary is awful for manual maths
@@FrostedCreations Maybe you just learnt it. But you can easily do big numbers computations (both additions and multiplications) with binary. I mean it's like decimal but a bit less efficient (more subcomputations but easier subcomputations). Roman is not suited for any computation at all.
@@RedwoodRhiadra Even then, unary can sum and subtract better than the Roman system. :p Use physical sticks. Heck, that's even how we teach children. Roman system really is that awful that unary is better.
One of my favourite curses I've ever heard was "may you be stuck with Roman numerals."
what did u do to get that thrown at you
He ridiculed Cassius Tiberius's goat smh
Harsh
Greek numerals were even worse.
@@raystinger6261 Hebrew (and Phoenician) numerals are pretty bad too!
I had an Nepali friend, who could count to 60 with her both hands. On her left hand she was using her thumb as a cursor touching each of her finger flanges on her other four fingers to count to 12. In her right hand she was counting the multiples raising her thumb after the 1st twelve set. She told me, she thought, that everyone counts like that. Ever since she showed me that, the invention of a sexagesimal system just makes sense to me. I can imagine that for the great traders of the bronze age, who were obsessed with contracts, counting and documenting goods, it was pretty important to be able to count at least to 60 on your hands.
I can count to 1000 on both hands no problem. Just use binary.
I made up a way to count to 100 easily. Count up to five on your right hand starting from the thumb, then folding down the fingers starting at the thumb again get you to nine with just the pinky up. Ten is then of course the thumb on the left hand, and so on.
@@davidharmeyer3093 that is wrong as some combinations would be extremely difficult to perform unless your fingers are broken.
As I know that in South Asia counting by touching your phalanges is still common practice, I too thought it might be connected to that. But I did not know, they would use the five-finger counting method to keep track of the second position. That is awesome, because you can any time distinguish between first and second numeral position counting either way around. And I guess, you could switch to phalanx-counting on the second hand, if you needed larger numbers up to 156.
Thank you for the enlightenment!💡
@Andrew Friend One of the three parts a finger is made up of.
0:13 * Flashbacks to Irving Finkel absolutely DESTROYING Tom Scott in the royal game of Ur *
That is such fun video (as anything with Irving Finkel)
In another video he taught Matt and Tom to write cuneiform.
The RUclips universe is colliding!
First thing I thought too. I actually just re-watched that video recently, It's really awesome!
Have you seen on RUclips the talk he gave about his involvement with a project to recreate the Ark? Finkel thinks that early depictions of the Ark show that it was circular, and he got involved with a project to recreate it, albeit on a smaller scale. Towards the end of the project he went out to the site---in Pakistan, I think. When he was there, lots of the local workers wanted their picture taken with him. Being a friendly guy, he complied, but he asked one of the project members why. He was told that the workers thought he was descended from Noah.
The Sumerians chose 60 as their base because they used their hands to count:
The phalanges of your fingers (there are 3 per finger) were used to count up to 12 (4 fingers x 3 phalanges). Then to count past 12 you use a finger on the other hand. So when you run out of fingers on the other hand you get 5 x 12 = 60.
Interesting, I’m watching this on my tablet
Tablets have gone a long way
8:39 because hexagons are the bestagons...
Ahh, another CGP Grey fan, I see
Cue Thatched Villagers
I never understood how people came to use the duodecimal system until I visited the Museum of Natural Sciences in Brussels to see the Ishango bone and its prime numbers. There, they showed that by moving your thumb along the twelve phalanges of the same hand, it makes sense to tally to twelve. If you use your other hand to move your thumb each time twelve is reached, you then count to a gross (144). While counting steps during a walk, I noticed it is much more robust to count to twelve on one hand and then use full fingers on your other hand, making the sexagesimal (12x5) system quite natural. Try it.
Did anybody else think
“I need a new sheet of brown paper”
when he picked up a fresh lump of putty?
No, nobody else.
What if the ten in this system were not chosen because of fingers, but because it’s one more than a square number, so they could make a square of ones before going to the ten symbol?
That's what I thought when he started putting them down
base 17. it's the future
@@reedh3950 base 2 is a prime number base, works alright for computers ;)
Can we split the difference and say base 9.5?
@@marklonergan3898 Base phi, take it or leave it
i dunno why but the way these symbols look surprisingly sci-fi for something so ancient
They really do!
I wonder how much of that is sci-fi taking inspiration from ancient societies (whether intentional or not)
It's the aesthetics of sci-fi movies. Doesn't mean the future would actually look like that. Looks rather communist i.e. brutalism.
You also have 12 finger joints (excluding thunbs). Makes sense for base 12. Five fingers on another hand, and now you have base 60.
Numberphile, thank you for the brilliant content you guys always manage to bring. There is no other channel like you. You always make me rediscover the beauty of mathematics.
I just had one request, can you also cover some well-known yet unthought ideas of number theory, perhaps like your video of Chinese remainder theorem...topics we basically know, yet we fail to dig deeper and analyse well sometimes.
he Sumerian number system, originated out of an economy based on poppy. The digit symbol is a poppy plant which are arranged in a 60 degree sector [ symbol for 10 ], ordered as 1,2,3,4 to give 10, and 6 sectors completes the circle. So, from production to trade, this system provided for very efficient accounting.
This is so interesting, especially since it's a mix of base-10 and base-60!
I've been wanting this video for years! Thanks for making it happen!
"One of the curators of the cuneiform collection at the British Museum"
Why not just say his name known around the world, Irving Finkel?
My first thought as well.
Why it was stolen from Iraq in the first place?
@@heliocentric1756 you can't steal from a people who don't exist
@@heliocentric1756 Who said the tablets were stolen? Some were, of course, but many others were taken with the permission of the Ottomans.
@@edwardcrews3174 you could steal artifacts that belong to the country and the people within the country and also the closest people to Babylon now a days are the chaldean and Assyrian people so you could say it is there’s if anything
They used base 60 because they counted in sets of 5 on one hand with their fingers and used their thumb against the knuckle digits on the other hand to keep track of sets of full finger counts. This practice is still common in many middle eastern countries today, and goes has been done for many thousands of years in the region.
I tend to believe this is the reason, especially that I use this counting method myself.
7:00 - With Roman numerals, you can add a horizontal bar over a symbol or group of symbols in order to multiply that group by 1,000. Two bars is a multiplication by 1,000,000, and so on.
It's been a while since I read about it, however, and it may be a more recent, nonstandard addition to the system in order to expand its usefulness.
Oldest number system that we’re aware of. Who knows how many other systems didn’t leave behind an artifact trail
That's what I thought. I suppose "aware of" covers it, but historians often say "the oldest" and forget to add "currently known". Embarrassing when someone happens upon the Antikithera mechanism...... The other nit I'd pick is that this is not about number systems, but specifically written ones. Orally, people must have had a way of saying "none" before "zéro was invented ".
@@raykent3211 I'd say that is very difficult to have an oral number system. If you don't need to write them, you don't need to operate on them. Even the Incas, with no writing system, had a "written" number system. (Well, woven, but it's the same).
Current pure oral cultures, often have very simple number systems, of the kind of: one two three , many and a lot. And things like that.
There isn’t anything older than the Sumerians
Not that survived the Flood
@@framegrace1 Sure, but we don't know if other older groups used less permanent methods to write down their numbers. Clay tablets can survive quite a long time in the right conditions. Symbols carved into wood or woven into fibers, not so much.
@@jonathanwilliams1065 Göbekli Tepe is 4500 years older than the Sumerians, with parts of it dated to at least 9130 BCE. The largest pillars at Göbekli Tepe weighs over 50 metric tons. The building site is over 22 acres, which is roughly the size of the are where the pentagon in the USA is. So it's not some small site by any means. Realistically there is no way a civilization that is capable of making the tools and plans to build a large scale architectural project could exist without maths. Who that civilization was and what their history was we do not know, but the Sumerians are much more modern in the big picture. They are part of our modern era, but certainly not the progenitors of civilization. The understanding that their civilization basically sprung up overnight from nothing and contained within it all the things we have in the modern day (math, taxation, contracts, land ownership, prostitution, etc...) just shows that they were merely one civilization in a long line of civilizations that all learnt and borrowed from each other across time. Modern archeology is conservative and ignorant to an absurdist degree.
60 is a highly composite number and makes it easier to do calculations without a calculator as it’s so divisible.
Wow!! Its usually a huge wait in-between uploads. Long may this speed of upload continue, i love a numberphile vid.
Alex' "horizontal" looks more like diagonal to me. I have ofcourse watched Tom Scott's video with Irving Finkel.
I remember hearing something about counting up to 12 on one hand using the knuckles of your fingers (excluding the thumb). I wonder if the use of base 60 might have been an extension of that, counting up to 60 on both hands, using your off-hand to count multiples of 12.
What an incredible insight. My mind is blown. Best finger counting system ever.
You can count all the way to 144 using this method on both your hands.
@@JamesDavy2009 You can probably do better than that if you use positional number systems; an easy method of medieval finger counting gets to 10,000 using just the fingers in the two hands in base 10.
Excellent video. This channel's content really is top notch. Keep up the good work guys!
Thanks
Could 60 be related to 60*6=360 being very close to the number of days in a year?
I think that the degrees of a full angle are related to the days of the year but I don't get where 60 comes from
No, the idea of 365 days in a year originated from the Egyptian empire, which used a base 10 counting system.
this comment is actually first was this video unlisted before XD
@@incription well 360 degrees are also counted in a base 10 system.
You can count to 12 with one hand, using the thumb to count phalanges on the other 4 fingers: 12.
Fascinating video, thank you for the great explanation!
This is so interesting! I always wondered how this system worked.
How can you tell what decimal place the first column is starting in??? If you have four columns of tens or three columns. Could be one decimal, two decimals, zero decimals. It works fine if you never go into decimals because you always know that position is the ones. And count up by sixty from there. I don’t understand how the same number of columns could have different answers depending on if the right most column is a decimal or not.
That was the weakness of Babylonian numerals-with no trailing zeros and no radix point, there was no way of telling what power of 60 the most significant figure represents.
Reading it out in 2021, probably more complicated than to a Babylonian used to the system and either wrote the number or has a much more clearer picture of what the number is referring to. It usually requires context and other things you know about the number beyond the supporting notes. Just like you and I know that when I say John was born in 78, you know it means the year 1978 CE, or that let's meat at 2 o'clock means pm or 14:00, not 2 am the next day.
@@louisvictor3473 “Let’s meat.” Nice.
When they wrote the fractional part in the space beside the tablet at 7:15 I was thinking - how would they do that?
You see mostly smaller tablets in the British museum since they were transported there!
This was very educational. Thank you!
Great presentation, The fact that they could use their multiplication tables to do fraction arithmetic is worth looking at. For example to do 1/2 x 1/3 they used 30 (half of 60) x 20 to get 600 then casting out a 60 they got 10 that is 1/6 of 60! No one could do fractional arithmetic at that time Egyptians just memorized the most common fractional operations.
Incredible that the first writing system started up around 3000 BC, and by 2500 BC the Pyramids were made.
Great info ... we had a cuneiform exercise at a language seminar I attended a few yrs ago
Wow! Thanks for the video and the history lesson.
Loved this, thank you!
still count in 60s for the really important things, where and when (space and time)
His explanation with the hexagon doesn't make any sense. The angle is 60 degrees because they choose that number. It may explain why it's 60 degrees but not why they choose that number.
What he means is that a base number that is divisible by 6 makes sense. But base 12 would have worked fine as well. It's the combination of base 5 or 10 (natural because of fingers) and base 12 (mathematically logical) that leads to base 60.
I get the hexagon, its easy to transcribe a hexagon into a circle using a compass and the radius, i guess the 60 comes from breaking each on the triangle into 10 subdivisions.
@@willpettit1022 if they had devided each angle in 10, it would be 10 degrees instead of 60.
@@Pfooh that's the second explanation he gives. I'm talking about the first one.
You can count in duodecimal (base 12) with your thumb and the segments of your fingers all the way to 144.
Interesting. Reminds me of the Mayan counting system, but instead of a base ten inside of base 60, the mayans used a base 5 within a base 20 system. Something natural about counting on fingers I guess, that such similar systems spring up independent of one another.
The Sumerian LANGUAGE actually counts "one, two, three, four, five, five-plus-one, five-plus-two, ..., ten, ten-plus-one, ten-plus-two, ..., fifty-five-plus-four, sixty." However, the intermediate fives aren't reflected in the script, as far as I know.
@@zh84 ah that’s a fair point, I hadn’t considered differences in spoken versus written language. I suppose I can’t speak on how the Mayan language does counting, I just read a bit about their written counting system. Cool point!
Counting 12 units on one hand using finger segments and the 12s on the other hand just using the 5 fingers is quite a nice way to count as it you can show someone the number without confusion about which hand is which and happens to limit you to 60.
Very interesting video, great start to my Monday! :)
I had a professor once who decided we were just gonna watch RUclips videos about cuneiform maths all class. Useful? Nah. Interesting? Yup.
More videos on the (ancient) history of maths, please!
*The base 60 system comes from the 12 digits on your righthandfingers, using your thumb to count them, times 5 fingers on your left hand.*
Great content!
I’ve always wondered if the 60 comes from 5 fingers on one hand and 12 sections on the non-thumb fingers of the other hand. Use one hand to count by 12s and the other with your thumb to count 1s.
hm... (ignoring the 5 fingers on the other hand part) wouldn't counting with those 12 sections actually be base 13?
Simpler: 60 is the smallest number divisible by 2, 3, 4 and 5
I didn't fall for Alex's 60 trap, as I've given a few talks myself on the Babylonian sexagesimal system ;) Showing people how they counted to 60 with their hands (knuckles) never fails to intrigue!
How do you count to 60 in your hands? I know how to count a base 12 system by using the thumb to mark the position and the three sections in our remaining four fingers as the thing you count/touch.
@@SKyrim190 So yes, it builds on the original base 12, 12 per hand system - but then expanded across two hands (it gives more credibility to base 5 and base 12 being united giving rise to the sexagesimal system). They would use each knuckle on one hand to count 12, and then count the number of 12 on the other.
@@skarrambo1 it may have been more common for people in those days to have twelve fingers opposed to ten these days.
A small correction to the video, Roman numerals did not add more and more "M" for big numbers, they added a macron to the symbol to multiply it by 1000. For example to write 3000 one could either write MMM or īīī. An C with a line on top of it would represent 100 000, etc. And this was not the only way they used to represent bigger numbers, Roman numerals where horrible for calculations but they could express numbers just fine.
This was useful. Thank you!
Wow, this is amazing! 👍🏻😀
Imagine some aliens count by 7, 11 and 13s, xenohistorians are gunna be so confused.
Counting by prime numbers implies some interesting things about their physiology
@@AstroTibs Not just physiology, how do you do fractions and stuff with that?
This is awesome!
You do a Cuneiform video and don't include Irvine Finkel?
As much as I'd love it, I don't imagine he'd want to zoom call
Part of the base 60 system's usefulness, seen with the Mayans, is that it could be counted easily on the fingers, using a thumb to count the other 12 digits and the other hand as a counter when you run out of digits
The Mayas counted in base 20, not base 60.
big thanks
I like knowing how to pronounce things properly. Thank you for the "cuneiform" clarification.
7:00 actually if you overline (or bracket) Roman numerals it means it is multiplied by 1000
Fascinating
60 as explained in the video is 5 times 12, which is a one handed counting method using your 5 fingers as well as the twelve knuckles of your fingers when counted by pointing at them with your thumb. I think this method is still used in some countries, can't really remember which ones though.
Come to think of it, Babylonian numbers have slightly less character density than Arabic numberals. All horizontals are base6 while the verticals are still base10.
When you have 85 for example, that's 1 verti, 2 hori and 5 verti, which is about three digits long as it goes from base10 to base6 and then back to base10.
I had to look up this video to understand the x3600 and x60 system, in order to solve a puzzle in a game called Chinatown Detective Agency. Cheers for the info!
You've skipped over the most amazing thing with babylonian math. The lack of decimal numbers made accuracy quite different.
Great 👍 information
⭕️❤️
I'm just glad we have calculators.
Each finger has 3 knuckles x 4 fingers = 12, and then you can use the thumb to count the twelves until 60. You can use your hands as an abacus.
From an astronomy point of view, part of the reason they might've liked the idea of dividing the circle into 6ths is that (if I'm remembering correctly) at the latitude of the Babylonians, the Sun rose at 30⁰ north of East on the Summer Solstice, set at 30⁰ north of West that evening, rose at 30⁰ *south* of East on the Winter Solstice, and set at 30⁰ south of West that evening. This divides the horizon into almost exact 6ths, a point that certainly would not have escaped their notice. (It's also just a coincidence of their latitude and definitely does not work much farther away from there - except of course for the same distance south of the equator).
Even this ancient base60 system seems so developed. It almost seems like that the Babylonians may have been aware of using fingers (base10) but still they chose base60 for calculation purposes. Really wish we had more info about ancient mathematics. Great Video!
One argument I once heard for the base 60 system was that you can count to 60 using your fingers rather easily and using this you can count higher (obviously) with your fingers than if you just did the standard base 10. (Additionally to all the nice maths features mentioned in the video.)
It goes like this:
Choose a hand, say the left hand. Ignore the left thumb because it has only two segments while the others have 3. Now using your right thumb, count the finger segments (12). Once you run out of segments, you continue with the index. And so on and so forth until you get to 60 which is where you run out. But you could even count to 120 if for the first 60 you had your thumb touching your palm, now you can spread it with the other fingers and start again.
9:40 - five fingers in right hand, twelve knuckles in left hand, and thumb in the left kept track of knuckles in left. Folded fingers kept track of numbers in right. Accountants in Tamilnadu used this counting system when I was young.
Watching the earth rotate fwds in time as the clock counted backwards through BC dates was a fun little brainteasing moment
It would have been nice to show an example of adding, subtracting, multiplying, and dividing using cuneiform too. One other thing I heard about the reason for 60 was that 12 being a multiple of 60 was countable on one hand -- using your thumb as a pointer... count the three joints in your pinky.. then the three in your ring finger ... there are twelve. so it made for easy counting by twelves on your hand in the marketplace.
I think I recall a documentary where they explained that Sumerians used a base 12 number system because is the base you can use to count the biggest number and 'store it' using your hands. I think they used each non thumb finger join of one hand as units, allowing them to count to 12, then they used the other hand's finger joins to store 12's, allowing them to count and store up to 144.
One can count way higher than 144. Imagine each finger as a binary digit. Bent means 0, straight means one. With your ten fingers you can count from 0 to 1023. If you allow three positions for each finger (straight, bent a little, bent a lot) then you can count from 0 to 3**10-1, which is 0 to 59048. Some numbers might be awkward to represent on your hands if you try it right now, but if you grew up using that system I'm sure it wouldn't be a problem.
@@fudgesauce you're right. I don't think Sumerians considered binary back then, though. Do you have any thoughts on why base 12?
That's so cool.
If you have only one hand free, you can use your thumb to count on your phalanges. You have 3 per fingers, and four fingers if you exclude the thumb you are using as a pointer. So you can use one hand to count to 12.
Dr. Grimes explained the same system in their base 12 video, check that out if you haven't already 😃
Alex Bellos: 21st century Martin Gardner, I think. Great stuff!
very useful for accounting, but it's no protection against buying a shipment of low grade copper!
We must have seen the same video about the sumerian complaining of bad shipment.
@@riadsouissi actually, i had been familiar with Ea Nasir and Nanni before, but it was the topic of a lot of memes on twitter yesterday lol. such a funny story, eh?
I like the book in the background
I've heard the Imperial measurements system persist, or was a big influence on the industrial revolution, because base 12 let you have more meaningful and immediately understandable divisions. The 5x12 idea is an interesting one.
Didn't know Micheal Sheen was into recreational maths.
Another theory behind 60 is that you have 12 finger pads on one hand. So, if you want to count something big-ish, you can count your finger pads with your thumb and then fold fingers on the other hand: 12x5=60. It is basicaly base sixty countdown. I am using this method for years and it is pretty convinient
There are 12 knuckles on the fingers, not counting the thumb. From there, it's easy to work out base-60, especially if you combine it with the "easiest to divide a circle into six parts" hypothesis. Or, for that matter, to get to it by multiplying by five, though that raises the question of why they'd count the thumb for multiplying but not for the getting to twelve.
8:30 The bestagon
triangle is better since it's stronger
@@trimethoxy4637 Triamgles are stronger when they form 120 degree angles, and when they do that, they also form the hexagon, the bestagon.
10:03 12 is not just a divisible number, but the base for Roman fractions.
4:50 nice
According to QI, the Babylonias had a base 12 system when counting the finger segments on one hand while using the thumb as a pointer. The other hand is used to keep track of those 12s by raising the 5 fingers of the other hand. 5x12=60. This is why 60 was used.
I always thought that hexagon in a circle bit had some influence. :)
I didn't know that about the size of tablets. Man, they were writing TINY
Surely the curator you are referring to is The Resplendent And Magnificent Wizard Supreme Irving Fenkel?
Base 60 is natural. Take your right hand, palm facing you. With your thumb you count segments on the little finger, ring finger, etc. until you get to the index finger. Each of the 4 fingers has 3 segments, getting to 12 in total. Then you just raise one finger on the left hand and continue counting the segments on the right hand from the start. On your left hand you have 5 fingers. So 5 times 12 equals 60.
Also, you can very easily count to 12 on one hand. If you face your palm towards yourself, and use your thumb. Start on the pinky and count the knuckles of your 4 fingers. I've heard that's how babylonians counted. 60 divides by 12 into 5
I always thought 60 because there are about 360 days in a year, give or take.
I was going to argue, but as I was putting my thoghts down, I was all: "Wait a minute there, Batsman ...".
I also think that's why they divided the circle in 360°-and once that's done, the hexagon construction in the video kicks in, and you end up with a system which is useful for most applied sciences of the day, practical because of the phalanges trick, and also convenient when it comes to divisors.
That's why we have about 360 days in a year, not the other way around 🙂 same goes for 360 divisions of the circle, the 60 minutes in an hour and 60 seconds in a minute, the list goes on. Babylonian mathematics is still going strong!
@@marcushendriksen8415 Not really? The number of days in a year is completely independent of how we count time. Conceptually, a day is the time the earth takes to make a rotation around itself, and this happens a fixed number of times (about 365) while the earth completes a full rotation around the sun.
There better be memes about Ea-Nassir and his terrible copper.
Could you add a link to the journal article shown on Babylonian astronomy in the description? Thanks.
You suggested that the tablets are quite small because they were to be held in the hand and marked. Isn’t it more likely that they are small because large tablets would be prone to cracking when they dried?
It’s almost like French numbers in that it goes up to sixty but then you have to use different words.
French is like this (Cuneiform numbers being much much older), but actually French is base 20, so...
@numberphile I'm shocked that you did not notice that in base 60 it is much easier to spot high prime numbers (Sieve of Atkin)
The base 60 aspect of cuneiform reminds me of the Lindybeige video wherein he talks (a lot lol) about the history of English coinage and why 240 pennies make a pound.
“Dovahkiin! Dovahkiin! Naal ok zin los vahriin...”
Not just every day but every second :D
i heard once that the babylonians estimated there to be 360 days in a year, which is also simply 6*60
They were great astronomers, surely they figured out at least 365 if not the quarter too.
@@royschreiber1 I'm pretty sure they had 12 lunar months, and then an extra month every few years so the months will still line up with the seasons. It's called the Metonic Cycle, the Hebrew calendar still uses this system. So there were generally 354 days in a calendar year, but they knew it should really be 365.
The Egyptians had 12 months of 30 days each. They later added 5 days to align with the solar cycle and those were holidays.