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https://www.theregister.co.uk/2017/08/25/plimpton_322_babylonian_trig_table/

Forget trigonometry, cos Babylonians did it better 3700 years ago - by counting in base 60!
Tablet beat Pythagoras by a millennium, may explain how the ancients built pyramids

Those of you who can remember trigonometry can feel free to forget it, because ancient Babylonian mathematicians had a better way of doing it – using base 60!

That's the conclusion of a new paper, Plimpton 322 is Babylonian exact sexagesimal trigonometry, in the new issue of the journal Historia Mathematica.

The “Plimpton 322” from the title is a clay tablet inscribed with cuneiform script. Discovered in the early 1900s, the tablet has been of interest to mathematicians for years because it describes Pythagoras' theorem, yet is thought to have been created around 1800BC, more than a thousand years before Pythagoras was born and started tinkering with triangles.

The rest of the tablet's four columns and fifteen rows of numbers have been debated for some time, but in the paper Daniel Mansfield and N.J. Wildberger, both of the University of New South Wales' School of Mathematics and Statistics, suggest it was a trigonometric table.

Mansfield explains his work in the video below:

*vid @ link*

Mansfield and Wildberger have also penned a piece at The Conversation in which they explain that the Greek version of trigonometry defines triangles by their angles, but Babylonians cared only for the ratios of the lengths of each of their sides (depicted above, or here for m.reg readers).* They also opine that the Babylonian way of doing things may have saved generations of students plenty of sweat and frustration, as calculating triangles with ratios is consistent and simple compared to wrangling cos and sin functions.

The pair also feel their work explains how the ancients could build things like step pyramids in Babylon, and perhaps even Egypt's pyramids as our knowledge of that ancient nation's maths has also shown us they had a fondness for ratios. ®

* You can see Plimpton 322 itself here, where you will also learn that commercial reproduction of images depicting the tablet is forbidden without permission. As we're writing this at a time that permission is likely to take hours to secure, we're linking.

see/read also :
https://theconversation.com/written-in-stone-the-worlds-first-trigonometry-revealed-in-an-ancient-babylonian-tablet-81472

The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation.

..all those long dull afternoons in class for nought eh !

Brilliant!

Speak for yourself reds8n.

One of the best days of my life was when I realised that trigonometry could be used to accurately plot cannon fire during games of Warhammer or historics

Of course, this was back in the day when you couldn't just measure anything on the table top battlefield. Much more realistic.

But yeah, good article.

 Do_I_Not_Like_That wrote:
Speak for yourself reds8n.

One of the best days of my life was when I realised that trigonometry could be used to accurately plot cannon fire during games of Warhammer or historics

Of course, this was back in the day when you couldn't just measure anything on the table top battlefield. Much more realistic.

But yeah, good article.

For me, it was from working with an optics bench... but it was only useful in Battlefleet Gothic for some reason....

The Auld Grump - base ten kind of sucks - but I favor base twelve over base sixty, for easy of notation.

I also knew it was the Babylonian's fault. Never trust a Marduk worshipper. They're all crazy. Base 60 math. Pft.

 LordofHats wrote:
I also knew it was the Babylonian's fault. Never trust a Marduk worshipper. They're all crazy. Base 60 math. Pft.

Yeah, who would use base 60 for anything nowadays? I'm so glad we all moved onto the decimal system for our timekeeping

Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

 LordofHats wrote:
Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

This will probably look very ugly (and I'm not used to writing in english about mathematics).

On a fundamental level it's just another way displaying stuff and that's it. The binary system, for example, is useful for computing because it maps quite nicely and allows you to do stuff like bitwise operations/bit shifting which are simpler on that hardware than other operations (that need more individual steps).

Base ten just means that a number like 5471 is just a quick way of writing (5 *1000) + (4*100) + (7*10) + (1*1) = 5471
And it looks "orderly" because we have the enough normal numeral to fit in all the spaces. And we have enough finger to easily count in this system and not get confused.

In base 12 the same number 5471 would looks like this (5471) but be this in decimal: (5*1728) + (4*144) + (7*12) + (1*1) = 9301 and you would need to find a way to depict the two extra numerals (what we know as 10 and 11 in decimal, 12 would push you one further to the left). You could write something like 54(11)1 but that would get confusing, in a hexadecimal system we use A (10), B (11), C (12), D (14), E (14), and F (15) so you could use A (10) and B (11) in a duodecimal system and have numbers like 4A2 for (4*144) + (10*12) + (2*1) = 698 in decimal

The smallest bit in the base (far right) is just an exponentiation of 12 (base of the system) with the exponent of 0 (for the first bit) = 1 (the same as for base 10 and all other base systems), the next one is 12^1 = 12 (instead of 10 in base 10) so you have "space for 12 whole numbers instead of only 10" before you push to the next higher "size" category, then 12^2 which is 144 (instead of 100 in base 10), after that you got 12 raised to the power of 3 (12*12*12 = 1728). You can pack more stuff in each box.

If you use an octal system with a base of 8 you would be able "to put fewer numbers in each box", meaning 1, 2, 3, 4, 5, 5, 6, 7, and 8. A 9 (base 10) would be a 11 (base 8 can't display a 9 in one digit because there's not enough space in the first digit box, so to speak), and a 10 (base 10) would be 12 (base 8) because it's (1*8) + (2*1). In a war game where your stats or your movement work in multiples of 8 using base 8 might be useful in some twisted way.

In languages the latin alphabet allows us to work nicely with languages that evolved from latin but it's worse for Japanese or languages that were build on other systems, and even then one could say there are different base systems. English is very minimal latin system with a base of 26 (which we use to build words instead of numbers) while other languages have fun additions like ä, à, á, â, č, ß, š and whatever else they need to make the system more useful/efficient for their own language. You could also formalise a system that uses the ten digits we use regularly, half the alphabet, and some emojis if you really wanted or needed it for some reason. For an exercise we had to convert some numbers from decimal into like a dozen different made up and existing systems and then create one with 30 "spaces" (base 30) per digit and convert the numbers into that (you just had to find enough system to make it readable).

These systems are arbitrary, we just make them up as needed to order numbers in a sequence. Base 10 works well for us, you just need to look at your hands while typing to understand why (if you are not missing fingers). We use other systems for computers because they don't use fingers for counting and the base 60 system probably works really nice for whatever calculations they did. And it's just easier for us to do arithmetic in base 10 than it's with roman numerals (V + M / IV = ?, they needed another, additional system for fractions).

Ultimately, the base of whatever counting system is arbitrary. You can do any and all math with any base system. However certain math will be easier or more intuitive with certain systems. And converting between different systems is a nightmare because you'd have completely different mathematical equations, and might even have processes which don't exist at all due to the quirks of one system(like this Babylonian thing which didn't involve angles at all).

Base 10 makes the most sense for us humans simply because we can relate to it nicely. However there have been some societies which picked other base numbers. It is after all completely arbitrary.

Another thing to consider with alternate math systems is weather that system has the concept of Zero. Many early systems did not, which is a rather limiting thing as some concepts just don't work without it. Its also why Roman numerals can sometimes be difficult to understand at first since we are so familiar with Zero that it's absence is an anathema.

So its worth noting that, while the Babylonian system is clearly superior at calculation Trig, it would probably be hideously clunky for other things. Trig might be a small price to pay for making your basic every day addition and multiplication more intuitive.

I'm confused. Aren't sin cos and tan ratios as well?

I mean, I use them almost daily in my work, and always find that using them as ratios makes things so much easier to figure out.

 LordofHats wrote:
Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

Steven Fry mentioned this in a QI episode. Something about the number of factors was mentioned, as in 10 can only be divided by 1 & 2, while 12 can be divided by 1,2,3,4,6. There is apparently something behind this-I never bothered to look into it more, because the 12-based system is however many thousand years dead.

Long live metric!

 Farseer Anath'lan wrote:

 LordofHats wrote:
Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

Steven Fry mentioned this in a QI episode. Something about the number of factors was mentioned, as in 10 can only be divided by 1 & 2, while 12 can be divided by 1,2,3,4,6. There is apparently something behind this-I never bothered to look into it more, because the 12-based system is however many thousand years dead.

Long live metric!

I still count eggs by the dozen, cousin!

 LordofHats wrote:
Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

it really doesn't make much difference these days with calculators. Before calculators base 6,12,24, etc is convenient because those numbers are divisible by more whole numbers, but refardless of which base you use you're going to cone across irrational numbers because the universe doesn't fit nicely in to any numbering system (see pi, e and dozens of physical constants engineers and scientists have to deal with).

Base 10 seems natural because we have 10 fingers/thumbs and since (from memory) the Indians introduced the place holder system counting to 10 became quite natural. By placeholder I mean the zero, so we count to 9 then restart at 10, count to 99 then restart at 100. That means our puny minds only need to comprehend 10 symbols and we can count as large or as small as we like. A base 60 system with placeholders would also work fine but you'd either need to remember 60 symbols or have a separate set of symbols that it in to a base 60 system (like how we read time, it's a base 60 system but we count to 60 using a base 10 system, because of that even though time is technically a base 60 system it's an unnecessary pain to actually do calculations in base 60).

If you take the example of a circle, we divide it in to 360 degrees, but in modern times when doing circular calculations in science and engineering you tend to use radians rather than degrees anyway, which is a non-dimensional numbering system based on the radius of the circle, so in some ways using pi is our way of transcending the limitations of a base 10 or base 60 or base whatever numbering system.

 Grey Templar wrote:


Base 10 makes the most sense for us humans simply because we can relate to it nicely. However there have been some societies which picked other base numbers. It is after all completely arbitrary.

Yes, it is completely cultural. Base 10 only makes sense to us because the ancestors of the European peoples once upon a time started counting their fingers (and we have 10 fingers). Other peoples on the other hand didn't count their fingers, they counted their finger bones instead (there is three for each finger, so 12 for a hand), giving rise to base 12 systems. The Babylonians counted their finger bones and then used the 5 fingers of their other hand to keep track of how many hands they counted (and 5 times 12 is 60) giving rise to their base 60 system. For the Babylonians, base 60 felt as natural as base 10 feels to us.

 Iron_Captain wrote:

Yes, it is completely cultural. Base 10 only makes sense to us because the ancestors of the European peoples once upon a time started counting their fingers (and we have 10 fingers).

I don't believe that's true (in that you give Europeans too much credit). The modern counting system is based off the Hindu-Aramaic system, which long ago was based off 9 symbols between one and nine and a placeholder for tens, hundreds, thousands, millions, billions, etc. At some stage a smart chap (I believe also in India) came up with the idea of a universal placeholder (ie. the 0 in 10, 100, 1000, etc) and someone also came up with the idea for zero; something that represents nothing, and negative numbers.

So probably best to blame those damned Indians if you're going to blame anyone.

To be honest I think base 10 is fine, as I said earlier if you're doing any real science or engineering in the modern world you're going to come across awkward fractions and irrational numbers no matter what base system you use. We overcome it by non-dimensionalising and using constants.

Hard to take the article seriously when it contains such a fundamental error as assuming Pythagoras' Theorem is named after him because he discovered it. The theorem was well known long before Pythagoras' time and tables of Pythagorean triples (3,4,5 5,12,13 etc) have been found in ancient Mesopotamia, Egypt and ancient China.

If I remember correctly, the theorem was only named after Pythagoras centuries after he lived, because the Pythagorean philosophers claimed that Pythagoras had been the first to produce a (rather elegant) proof of the theorem.

Math-speak???

AllSeeingSkink wrote:

 Iron_Captain wrote:

Yes, it is completely cultural. Base 10 only makes sense to us because the ancestors of the European peoples once upon a time started counting their fingers (and we have 10 fingers).

I don't believe that's true (in that you give Europeans too much credit). The modern counting system is based off the Hindu-Aramaic system, which long ago was based off 9 symbols between one and nine and a placeholder for tens, hundreds, thousands, millions, billions, etc. At some stage a smart chap (I believe also in India) came up with the idea of a universal placeholder (ie. the 0 in 10, 100, 1000, etc) and someone also came up with the idea for zero; something that represents nothing, and negative numbers.

So probably best to blame those damned Indians if you're going to blame anyone.

To be honest I think base 10 is fine, as I said earlier if you're doing any real science or engineering in the modern world you're going to come across awkward fractions and irrational numbers no matter what base system you use. We overcome it by non-dimensionalising and using constants.

People in Europe (and several other parts of the world) were already counting 1-10 on their fingers millennia before that chap in India "invented" zero. People in Europe did not need anyone to teach them how to count, they were already doing that long before they even came to Europe. Counting wasn't invented anywhere, humans have been doing it since pretty much forever. Also, the Indian and European systems of counting (finger counting, base 10) are the same since most European and Indian populations share a very distant common origin in Central Asia (as evidenced by the Indo-European language group).

 Iron_Captain wrote:

People in Europe (and several other parts of the world) were already counting 1-10 on their fingers millennia before that chap in India "invented" zero. People in Europe did not need anyone to teach them how to count, they were already doing that long before they even came to Europe. Counting wasn't invented anywhere, humans have been doing it since pretty much forever. Also, the Indian and European systems of counting (finger counting, base 10) are the same since most European and Indian populations share a very distant common origin in Central Asia (as evidenced by the Indo-European language group).

I don't think you give human beings enough credit. I may not know math well but I know my history. Europeans were using the Roman numeral system until after the Crusading era, and the roman numeral system is a cool by completely convoluted way of representing numerical values. While Roman mathematics understood zero as a value "nulla" or "nihil" had no concise visual representation and only one existent math text shows anyone working in that system giving it one in the later Middle Ages (some guy and his buddies started using 'N' to represent zero, but they didn't use it positionally and it didn't seem to catch on with the rest of Europe). We take the word "algorithim" from the Persian-Muslim scholar Algoritmi (real name Muhammad ibn Musa al-Khwarizmi, "alkhwarizmi" became 'algoritmi' in Latin) and while he didn't invent zero he did create the first explanation of 'zero' to reach Europeans in the 12 century, and the same text included a broad summary of Greek and Hindu mathmatics which had largely become lost or made overly-complicated by Middle Age Europeans operating in the Roman numeral system. While Europeans had been exposed to Hindu-Arabic math earlier in Spain by the Moores, it didn't really catch on until after the widespread publication of Algoritmi's Algoritmi de numero Indorum (the original Arabic text is now lost).

The adoption of the Hindu-Arabic numeral system was a huge stepping stone culturally because it made visual representation of math less cryptic and easier to understand and equations easier to write out by replacing the archaic Roman system with a positional system. The rapid advancement of literacy and science in the Western World actually might not have happened without it making the basics of math magnificently simple. In fact Europeans did not need to be taught to count, everyone indeed counts, but they did benefit massively from being taught how to count better

 LordofHats wrote:
Perhaps some math peep can explain this to me cause I've never really grasped it. I'm aware that there are number systems like base 60, base 12, and base 10 (which is the one we generally use). How exactly do these system differ from one another, and what are the advantages of using base 12 instead of base 10? The article says base 60 makes it easier to do angles which I guess makes sense cause it occurs to me that angles are calculated in values easily divided by 60 but I'm not really clear on exactly what that looks like. If I learn base 8 math will it be easier to create a device able to connect a stable wormhole between two points in space?

I've done math in base 60 and it's almost exactly the same as base 10 except for how you write numbers. For example, 62 would be written as 1,02. 73 would be 1,13. Typically, we don't use commas for any base system less than 10, although I have seen letters used to represent numbers so that one could write in base 16, such as 30 would be 1E instead of 1,14.

Outside of certain number theory niches, the base number doesn't really mean a thing. Binary is useful because on and off are such easy states to differentiate between.

Binary is also sort of the default system for a basic computer because it's how electrical current works. It's either On or Off. And then the computer uses that in a series to transmit information.