@@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
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.
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.
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!
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.
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
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.
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??
+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).
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.
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.
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.
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.
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.
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...
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.
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!
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.
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
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.
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
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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
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.
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.
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?
You're referring to the mathematical framework that enables us to formulate and analyze theoretical concepts! There are several mathematical systems and tools used to develop and explore theoretical ideas, depending on the context and field of study. Here are a few examples: 1. Mathematical Logic: A formal system used to express and analyze mathematical statements, arguments, and proofs. 2. Category Theory: A branch of mathematics that studies the commonalities and patterns between different mathematical structures, providing a framework for abstracting and generalizing concepts. 3. Type Theory: A system for expressing and analyzing mathematical concepts using types, which can be seen as a formalization of mathematical structures and relationships. 4. Model Theory: A branch of mathematical logic that studies the relationship between formal theories and their models, enabling us to reason about theoretical concepts. 5. Differential Geometry: A mathematical framework for describing and analyzing geometric objects and their properties, crucial for theories like General Relativity. 6. Operator Algebras: A mathematical framework for studying linear operators and their algebras, essential for Quantum Mechanics and Quantum Field Theory. 7. Homological Algebra: A branch of abstract algebra that studies the properties of algebraic structures using homology and cohomology theories. Some specific mathematical systems used to develop and analyze theoretical concepts include: 1. Hilbert's Formalism: A formal system for mathematics, which provides a foundation for mathematical logic and proof theory. 2. Zermelo-Fraenkel Set Theory: A formal system for set theory, which provides a foundation for modern mathematics. 3. Non-standard Analysis: A mathematical framework for working with infinitesimal and infinite numbers, developed by Abraham Robinson. 4. Causal Dynamical Triangulation: A quantum gravity theory that uses a discretized spacetime, similar to lattice gauge theory. These mathematical systems and tools enable us to develop, analyze, and refine theoretical concepts, making them essential for advancing our understanding of the world.
+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.
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!
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.
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
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?
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!
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.
Yes and no. 12 has been brought up (see Numberphile's "base 12" video), but not as a replacement for 60. As for replacing it: We know that 60 is the smallest number to be divisible by all numbers 1-6. The smallest number to be divisible by all the numbers 1-7 is 420. A base 420 counting system is ridiculous. If you just mean in terms of dividing time, then also remember that 60 is good because 60 x 60 x 24 seconds is 1 day, whereas the aforementioned 420 doesn't fit as nicely into a day.
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.
@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 🙏
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.
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
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.
Albert Einstein's work revolutionized physics, and his mathematical techniques were instrumental in developing his theories. Some of the key mathematical concepts and tools he used include: 1. Differential Geometry: Einstein employed differential geometry to describe curved spacetime in his theory of General Relativity. He used tensors, manifolds, and Riemannian geometry to model the curvature of spacetime. 2. Tensor Analysis: Tensors were crucial in Einstein's work, particularly in General Relativity. He used tensors to describe the curvature of spacetime, stress-energy, and the Einstein field equations. 3. Riemannian Geometry: Einstein relied on Riemannian geometry to describe the curvature of spacetime. He used Riemannian metrics, Christoffel symbols, and the Riemann tensor to model spacetime geometry. 4. Partial Differential Equations (PDEs): Einstein's field equations in General Relativity are a set of nonlinear PDEs that describe the curvature of spacetime. He used techniques like separation of variables and perturbation theory to solve these equations. 5. Group Theory: Einstein used group theory to describe the symmetries of spacetime. He employed Lie groups and Lie algebras to study the symmetries of his field equations. 6. Calculus of Variations: Einstein used the calculus of variations to derive his field equations from the principle of least action. He minimized the action integral to obtain the Euler-Lagrange equations. Some specific mathematical equations and formulas associated with Einstein's work include: - The Einstein Field Equations: Rμν - 1/2Rgμν = (8πG/c^4)Tμν - The Mass-Energy Equivalence: E = mc^2 - The Photoelectric Effect: E = hf - φ - The Brownian Motion Equation: x(t) = √(2Dt) These mathematical tools and concepts were instrumental in shaping Einstein's groundbreaking theories, which transformed our understanding of space, time, and gravity.
The presentation here, and your comment, concern computation. However, an important problem to the folks of yore was dividing angles. 60 was handy for that.
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.
So it wasn't really Base 60 was it? It was actually a strange hybrid of base 10 and base 60. According to the video (timestamp 2:50), assuming it's correct, the way they drew 57 was using base 10. That's probably why we don't use it any more - because it's not in a metric format - the rollover to an extra digit as the count increases is not consistent.
Yes, the Moon has resources that could be used to support a human settlement or city. Some of these resources include: 1. Water ice: Deposits of water ice have been found in permanently shadowed craters near the lunar poles. This ice can be used for life support, propulsion, and in-situ resource utilization (ISRU). 2. Regolith: The lunar soil, or regolith, can be used as a construction material for building structures, radiation shielding, and even 3D printing. 3. Helium-3: A rare isotope of helium that could be used as fuel for nuclear fusion, providing a clean and efficient source of energy. 4. Rare earth elements: Deposits of rare earth elements like neodymium, dysprosium, and cerium have been found in lunar regolith, which are essential for advanced technologies. 5. Solar energy: The Moon's surface receives abundant solar energy, which can be harnessed using solar panels. 6. Metals: The Moon has deposits of metals like aluminum, iron, and titanium, which can be extracted and used for construction and other purposes. To create a city on the Moon, we would need to: 1. Extract and process resources: Develop technologies to extract, process, and utilize the resources mentioned above. 2. Transport materials: Transport materials and equipment from Earth or other sources to the Moon. 3. Construct infrastructure: Build habitats, life support systems, energy generation, and other essential infrastructure. 4. Establish a reliable food supply: Develop a sustainable food supply, such as through hydroponics or in-situ resource utilization. 5. Address radiation protection: Develop adequate radiation shielding to protect both people and electronic equipment. While we have the resources and technological capabilities to establish a lunar city, it would require significant investment, innovation, and collaboration.
The "Æ" symbol has a rich history and multiple meanings across various contexts: 1. Phonetics: In phonetics, "Æ" represents a vowel sound, like the "a" in "cat" or "hat". 2. Latin: In Latin, "Æ" is a ligature of the letters "A" and "E", used to represent a diphthong (a combination of two vowel sounds). 3. Old English and Norse: In Old English and Old Norse, "Æ" was a distinct letter called "ash", representing a vowel sound similar to the "a" in "cat". 4. Danish and Norwegian: In modern Danish and Norwegian, "Æ" is a letter in their alphabets, representing a vowel sound like the "e" in "pet". 5. Elven script: In J.R.R. Tolkien's fictional Elven script, "Æ" is a symbol used to represent a vowel sound, as Grimes mentioned. 6. Typography: In typography, "Æ" is often used as a stylistic element or ligature in fonts. In the context of Grimes and Elon Musk's son's name, "Æ" represents a connection to art, fantasy, and the elven script from Tolkien's mythology.
I can read you comment in my e-mail, but it says,"Comment removed Author withheld" I think you got in trouble by numberphile .... and numberphile I haven't seen a video about the golden ratio, I love math thanks for all the videos
Actually, the Babylonians didn't choose base 60 because of its many factors, it was created because two different cultures (one that uses base 10 and the other uses base 6) were combined, and they created the base 60 system so everyone from both cultures could understand the number system. When you look at how they wrote numbers, you can see how it feels like a combination of base 10 and base 60.
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.
![Blox Fruits Dragon Rework Update [Full Stream]](http://i.ytimg.com/vi/EqDAp8udhm0/mqdefault.jpg)
"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)
awesome idea. base TREE(3).
@@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
George Missailidis me too, haha :)
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
@@randomalbum9879 but if we used an easy to learn pattern to make new symbols, it could be pretty convenient
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.
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.
well, there is always more coming!
@Taurus Capricorn Yoyo
@Taurus Capricorn yoyoyoyo
@Taurus Capricorn yo
@taurus capricorn Happy 10th Birthday 9-15-2011
Is this called cuniform?
1 like, 0 comments, from 8 years ago, by SmarterEveryDay, what have i stumpled upon
Weird
Huh
8cocks ago
cuneiform* hehehe
Like the number 60 - why not check out the sixtysymbols channel by the same film-maker as numberphile!? :)
thank you
4TH LIKE
7TH LIKE, 2ND REPLY
you're welcome
9th like fourth reply
A more in depth video about the Babylonians and their counting/numeracy system from Brady would be fantastic!
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!
I was hoping you might mention the fact that the Babylonians had a floating point system (which I find amazingly interesting!).
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.
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
"you can cook half of a meal" 😂😂😂😂
LOL! You know what he means though. :3
Ana Bulatovic: ...with Eric the half a bee!
(a Monty Python reference, if you're unaware)
Thanos approves.
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.
Love your videos, Brady. I'm subscribed to all of your channels, but I think this one is my favorite :)
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??
+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).
+andrewxc1335 he definitely misspoke there
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.
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.
Nope, how we write base 10 is only possible with the numerals we adopted from the Arabs.
Wow, this connected so many different things i never knew had anything to do with each other! Truly fascinating!
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.
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.
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.
Can you share a source? That's a very interesting theory....
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...
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.
2520 is divisible by 1,2,3,4,5,6,7,8,9,10.
But 5040 (=2*2520) has 60 factors.
@@romanski5811 wat
@@MaineMappingreal
The video is about base 60.
@@romanski5811 watwatwT
@@MaineMappingreal
Sorry, I can't help you.
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!
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.
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
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.
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
Brady's just so smart that he factored in time for the ads.
Never really thought about numbers. This video is extremely interesting
Thank you so much for this video, this really helps explain the sexagesimal system to me.
60 is also divisable by 10. It's divisable by most numbers from 1-10 with the exception of 7,8, and 9.
Very interesting video, and I like this guys style. Clear and concise.
Do one on 5318008 Brady, it looks awesome when you enter it in a calculator and read it upside down.
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.
I love this channel, very interesting. Keep it up.
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.
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.
Fascinating, always wondered why minutes, seconds were sets of 60
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?
I believe he made a mistake there. He actually depicted 67
Every time I watch one of your videos... I want more :D
Amazing, I wonder why I am watching it after 12 years of it's upload and my mind is blown by it
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.
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.
That was really fascinating. Thanks for doing this one! (and all of them)
Wow. So clear. Still need to cross this to understand the plimpton 322 tablet. These Babylonian kiddos had such insight.
The Babylonians were pretty cool!
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.
This is a better explanation than we did in our podcast
For the same reason it would be lenient if a deck of cards had 60 cards instead of 52.
1:31 I learnded this fact recently and it absolutely blew my mind
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.
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.
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.
Could you guys do a video on the advantages and disadvantages of kaktovik inupiaq numerals
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
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.
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.
Real mathematicians still use blackboards and chalk anyway, but the brown paper and marker pens make a sound that sends shivers down my spine.
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?
This channel has such a mad on for brown paper. I don't blame then, it's awesome.
You're referring to the mathematical framework that enables us to formulate and analyze theoretical concepts! There are several mathematical systems and tools used to develop and explore theoretical ideas, depending on the context and field of study. Here are a few examples:
1. Mathematical Logic: A formal system used to express and analyze mathematical statements, arguments, and proofs.
2. Category Theory: A branch of mathematics that studies the commonalities and patterns between different mathematical structures, providing a framework for abstracting and generalizing concepts.
3. Type Theory: A system for expressing and analyzing mathematical concepts using types, which can be seen as a formalization of mathematical structures and relationships.
4. Model Theory: A branch of mathematical logic that studies the relationship between formal theories and their models, enabling us to reason about theoretical concepts.
5. Differential Geometry: A mathematical framework for describing and analyzing geometric objects and their properties, crucial for theories like General Relativity.
6. Operator Algebras: A mathematical framework for studying linear operators and their algebras, essential for Quantum Mechanics and Quantum Field Theory.
7. Homological Algebra: A branch of abstract algebra that studies the properties of algebraic structures using homology and cohomology theories.
Some specific mathematical systems used to develop and analyze theoretical concepts include:
1. Hilbert's Formalism: A formal system for mathematics, which provides a foundation for mathematical logic and proof theory.
2. Zermelo-Fraenkel Set Theory: A formal system for set theory, which provides a foundation for modern mathematics.
3. Non-standard Analysis: A mathematical framework for working with infinitesimal and infinite numbers, developed by Abraham Robinson.
4. Causal Dynamical Triangulation: A quantum gravity theory that uses a discretized spacetime, similar to lattice gauge theory.
These mathematical systems and tools enable us to develop, analyze, and refine theoretical concepts, making them essential for advancing our understanding of the world.
How it was determined that a second lasts that amount of time?
+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.
thanks !!
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!
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.
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
yes, i'm actually really surprised the didn't use base 12 in their writing of numbers, considering their way of counting...
The second the numberphile logo appeared at the start, my phone's battery was at 60 😅
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?
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!
I think there is a good chance!
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.
Yes and no. 12 has been brought up (see Numberphile's "base 12" video), but not as a replacement for 60.
As for replacing it: We know that 60 is the smallest number to be divisible by all numbers 1-6. The smallest number to be divisible by all the numbers 1-7 is 420. A base 420 counting system is ridiculous.
If you just mean in terms of dividing time, then also remember that 60 is good because 60 x 60 x 24 seconds is 1 day, whereas the aforementioned 420 doesn't fit as nicely into a day.
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.
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.
@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 🙏
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.
it's 1.8% longer than 360 days.
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
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.
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?
I love your videos.
Albert Einstein's work revolutionized physics, and his mathematical techniques were instrumental in developing his theories. Some of the key mathematical concepts and tools he used include:
1. Differential Geometry: Einstein employed differential geometry to describe curved spacetime in his theory of General Relativity. He used tensors, manifolds, and Riemannian geometry to model the curvature of spacetime.
2. Tensor Analysis: Tensors were crucial in Einstein's work, particularly in General Relativity. He used tensors to describe the curvature of spacetime, stress-energy, and the Einstein field equations.
3. Riemannian Geometry: Einstein relied on Riemannian geometry to describe the curvature of spacetime. He used Riemannian metrics, Christoffel symbols, and the Riemann tensor to model spacetime geometry.
4. Partial Differential Equations (PDEs): Einstein's field equations in General Relativity are a set of nonlinear PDEs that describe the curvature of spacetime. He used techniques like separation of variables and perturbation theory to solve these equations.
5. Group Theory: Einstein used group theory to describe the symmetries of spacetime. He employed Lie groups and Lie algebras to study the symmetries of his field equations.
6. Calculus of Variations: Einstein used the calculus of variations to derive his field equations from the principle of least action. He minimized the action integral to obtain the Euler-Lagrange equations.
Some specific mathematical equations and formulas associated with Einstein's work include:
- The Einstein Field Equations: Rμν - 1/2Rgμν = (8πG/c^4)Tμν
- The Mass-Energy Equivalence: E = mc^2
- The Photoelectric Effect: E = hf - φ
- The Brownian Motion Equation: x(t) = √(2Dt)
These mathematical tools and concepts were instrumental in shaping Einstein's groundbreaking theories, which transformed our understanding of space, time, and gravity.
The presentation here, and your comment, concern computation. However, an important problem to the folks of yore was dividing angles. 60 was handy for that.
finally! a question i have asked all my life but no one was able to explain!!! =P
Same here😂
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.
These only got on the top most viewed, because sxephil recommended them!!!!
if they were using base 60, shouldn't there be a unique symbol for 57? ie different symbols from 1 to 60.
Thanks! Really helping my h/w!
So it wasn't really Base 60 was it? It was actually a strange hybrid of base 10 and base 60.
According to the video (timestamp 2:50), assuming it's correct, the way they drew 57 was using base 10.
That's probably why we don't use it any more - because it's not in a metric format - the rollover to an extra digit as the count increases is not consistent.
Is it really base 10? they are using symbols for 10, but the "column values" are not in base 10, but base 60.
So did the babylonians use both base10 and base60? The comment at 3:23 seems to imply so
Yes, the Moon has resources that could be used to support a human settlement or city. Some of these resources include:
1. Water ice: Deposits of water ice have been found in permanently shadowed craters near the lunar poles. This ice can be used for life support, propulsion, and in-situ resource utilization (ISRU).
2. Regolith: The lunar soil, or regolith, can be used as a construction material for building structures, radiation shielding, and even 3D printing.
3. Helium-3: A rare isotope of helium that could be used as fuel for nuclear fusion, providing a clean and efficient source of energy.
4. Rare earth elements: Deposits of rare earth elements like neodymium, dysprosium, and cerium have been found in lunar regolith, which are essential for advanced technologies.
5. Solar energy: The Moon's surface receives abundant solar energy, which can be harnessed using solar panels.
6. Metals: The Moon has deposits of metals like aluminum, iron, and titanium, which can be extracted and used for construction and other purposes.
To create a city on the Moon, we would need to:
1. Extract and process resources: Develop technologies to extract, process, and utilize the resources mentioned above.
2. Transport materials: Transport materials and equipment from Earth or other sources to the Moon.
3. Construct infrastructure: Build habitats, life support systems, energy generation, and other essential infrastructure.
4. Establish a reliable food supply: Develop a sustainable food supply, such as through hydroponics or in-situ resource utilization.
5. Address radiation protection: Develop adequate radiation shielding to protect both people and electronic equipment.
While we have the resources and technological capabilities to establish a lunar city, it would require significant investment, innovation, and collaboration.
Exelente explicación muchas gracias❤
I just have discovered you. You're such a cool guy. Take love from Bangladesh. ❤🇧🇩
The "Æ" symbol has a rich history and multiple meanings across various contexts:
1. Phonetics: In phonetics, "Æ" represents a vowel sound, like the "a" in "cat" or "hat".
2. Latin: In Latin, "Æ" is a ligature of the letters "A" and "E", used to represent a diphthong (a combination of two vowel sounds).
3. Old English and Norse: In Old English and Old Norse, "Æ" was a distinct letter called "ash", representing a vowel sound similar to the "a" in "cat".
4. Danish and Norwegian: In modern Danish and Norwegian, "Æ" is a letter in their alphabets, representing a vowel sound like the "e" in "pet".
5. Elven script: In J.R.R. Tolkien's fictional Elven script, "Æ" is a symbol used to represent a vowel sound, as Grimes mentioned.
6. Typography: In typography, "Æ" is often used as a stylistic element or ligature in fonts.
In the context of Grimes and Elon Musk's son's name, "Æ" represents a connection to art, fantasy, and the elven script from Tolkien's mythology.
It's odd that their number writing has 10's (the triangles) and 1's, when their finger-counting has 12's and 1's.
Awesome Video.
I find ancient math pretty interesting, but how on earth did WE figure out how the Babylonians counted?
I can read you comment in my e-mail, but it says,"Comment removed
Author withheld" I think you got in trouble by numberphile .... and numberphile I haven't seen a video about the golden ratio, I love math thanks for all the videos
Actually, the Babylonians didn't choose base 60 because of its many factors, it was created because two different cultures (one that uses base 10 and the other uses base 6) were combined, and they created the base 60 system so everyone from both cultures could understand the number system. When you look at how they wrote numbers, you can see how it feels like a combination of base 10 and base 60.
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.
So are you saying that they had another symbol for 60 as well as 10 and 1?
thanx for teaching me something new
1:45 what about the thumbs though?
You should make a video about Euler's constant, e.
@ 2:12 Doesn't each degree represent a day, therefore each minute is 1/1440th of a degree not 1/60th?