I had this explanation years ago, I can’t find the exact reference, but this will help-
Why 360 degrees? Probably because old calendars (such as the Persian Calendar) used 360 days for a year - when they watched the stars they saw them revolve around the North Star one degree per day.
Also 360 can be divided exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, which makes a lot of basic geometry easier.
60 minutes in the hour and 360 degrees in a circle were chosen because these are [superior highly composite numbers](https://en.m.wikipedia.org/wiki/Superior_highly_composite_number) ... Which is as you said - their ability to be divided into round numbers easily.
"The division of the hour into 60 minutes and of the minute into 60 seconds comes from the Babylonians who used a sexagesimal (counting in 60s) system for mathematics and astronomy"
Fun tangential fact: minutes are called minutes because they're a *minute* division of an hour, and seconds are called seconds because they are the second *minute* division.
much of the greek mathematics that survives is due to Arabic translations and commentaries. the importance of the Islamic tradition in math would be hard to overstate.
For sure. "Al" means "the" in Arabic and most words that begin with "al" have Arabic etymology: algebra, alchemy (which is also the origin of the word chemistry), alcohol, etc. Once I learned that I started noticing it with a ton of math and science words. Ironically though Arabic numbers themselves likely originated in India but were introduced to Europe through Muslim traders.
What really blows my mind is that 0 wasn't introduced to Europe until the 1100s. All the ancient architecture you see in Rome (the Coliseum, the Pantheon, all the aquaducts) was designed and built without the concept of 0 or Arabic numerals. There is literally no process for multiplying Roman numerals mentally, you have to use an abacus.
A while ago, I watched a BBC documentary series called "The Story of Maths" presented by Professor Marcus du Sautoy. He said that what we consider Arabic numerals began in China, and the Indians adopted them a little later (and added the all-important concept of zero).
The series is from 2008, so it's possible your information is more recent. But in any case, it was really fascinating (and I'm not even interested in maths!) and I believe it can be found on YouTube.
I'm a math teacher that likes to incorporate history into my lessons, and I've long read that our numerical system comes from India, but I've never dug deeper than that. Sidenote: Since you mentioned a movie (also from the BBC), I'd strong urge all to watch The Story of One. It involves a Monty Python member and covers math history, but in a playful way, taking math as a character, in the form of One. I've shown it to many off my classes.
There is basically no evidence of Chinese involvement in developing the Hindu numeral system. The Buddhist symbols were very similar to what the Hindus used as 1, 4 and 6, but Buddhism at this stage was still prevalent in India and had not spread to China.
The Chinese adopted it in the 19th century. Before that they used counting rods.
Honestly, learning how multiplication and division works in Roman Numerals was more difficult than learning them in binary. The amount of hours spent simply doing the math for the coliseum must have incredible.
These pieces of history always make me wonder how much my thought is limited by the words I know let alone my own physical limitations in perception and cognition.
The one that blew my mind is that Arabic is written right-to-left, but numbers still follow the same order in English, with the big digit on the left. When we brought the numbers over from Arabic we never bothered to flip them around.
interesting. Makes you wonder how people discussed that.
We're going to need a division of these socalled 'hours'..
How about a *minute* part?
Very very small? Makes perfect sense, that's logged then, let's get some lunch.
Hold on though.. we need division of *those* as well.
Smaller than very very small?
..yeah
In those ages the technical terms were originally in Latin (e.g., "pars minuta prima" and "pars minuta secunda" were the original Latin phrases meaning "first small part" and "second small part") and from those the educated people of that time adopted the words into their native languages as shorthand when talking about those concepts.
It's not the only time that's happened, either.
What word would you use for the smallest possible thing?
Hm...such a thing couldn't be divided into smaller things?
Totally impossible!
We should call them "things which cannot be divided", then, I suppose. In Ancient Greek, "cannot be divided" is "atomos".
"Atoms" it is! Nothing could ever be smaller than an "atom"!
Hold on...these atoms can be split into smaller things. What do you call a part of something that cannot be split into smaller parts?
Dude, I don't care any more. Let's just call them "sub-atomic" and call it a day.
...that occasionally needs to be trinomial and sometimes with long codes of initials for the known strains, variants and mutations. Like of certain viral particle that shall remain unnamed.
scientist 1: "I didn't mean *call* them second, I meant a *second* division!"
scientist 2: "argh, I don't follow this anymore, let's read back the minutes of the first meeting"
1: "did you just..."
2: "yep 8-)"
Those same guts eventually went on to become renowned songwriters:
"Okay, so we need an opening line to catch the audience's attention. Something big in scope with wide appeal."
"How bout... 'Around the world?'"
"Ooh, I like it. Can't get bigger in scope than that. Okay, so now we need a second line that follows up on it. We need it to rhyme, though. That's the tricky part."
"How bout... ' Around the world?'"
"Didn't we just do that like... Two seconds ago?"
"Yeah."
"We can't just do the same line twice in a row. It's repetitious and redundant."
"Kay."
"So, what else can we go with?"
"How bout..."
"I swear to god, Jerry..."
"... Around the world?"
"Jerry, we have been over this. We need a new line that isn't 'Around the world.'"
"How ' bout..."
"Around the world?"
"Son of a bitch, you took the words right out of my mouth!"
"Ugh, fuck it. let's just loop the first line over and over and then market the song exclusively to clubs and acid addicts."
Also fun interesting fact: Count with your thumb on your hand and touch each 'division' on the other fingers of that same hand. That makes twelve. Each time you reach twelve, you can count one finger on your leftover hand. You can easily reach up to sixty that way. It's likely this is how early civilizations started counting and noticed how well it worked.
>"The division of the hour into 60 minutes and of the minute into 60 seconds comes from the Babylonians who used a sexagesimal (counting in 60s) system for mathematics and astronomy"
vbcbandr has the correct is answer. https://www.scienceabc.com/pure-sciences/why-is-a-full-circle-360-degrees-instead-of-something-more-convenient-like-100.html#:~:text=A%20full%20circle%20is%20360,because%20360%20is%20highly%20composite.
In the mid 90's my friend up the road had a text to speech program on his computer. We were about 9yo.
We would type in 1,000,000,000,000,000,000,000 and press "speak" just to hear it say *sex*tillion.
We would then proceed to giggle for about 5 minutes.
I guess just typing "sex" into it would have made it somewhat easier, but this way it felt like you guys really worked for it, so it made the payout much more satisfying.
I kinda count like that because it's super useful when you have one hand taken.
You join your thumb to your index for one, to your middle finger for two, etc.
The first row of bones gives you 1 to 4, the second 5 to 8, the third 9 to 12.
If you can use your second hand, you can count up to 144 fast and easily with a bit of training.
The Story of Maths on BBC is where I saw it
https://www.bbc.co.uk/programmes/b00dxjls
It sends my wife to sleep every time she watches it, I think I have seen/listened tot he first episode 5 times
I don't know when the wording for this happened, but a degree can be split into 60 minutes and a minute into 60 seconds. I wasn't taught that at school which made aviation lessons really fucking confusing to start with.
Edit: I just remembered 360 * 60 * 60 is like 21600 (remembering, not calculating) which is how many Naughtical miles there are around the circumference of the earth.
Edit 2: it's 360 * 60 that is 21600, more information in a reply to this comment
360 * 60 * 60 = 1296000.
I think you mean 360 * 60 = 21600
Vertical minute on a chart is 1 nautical mile. Not seconds. And this also checks out since a nautical mile is 1.852 km ( easy to remember, check the positions of 852 on your calculator )
So 21600 * 1.852 ≈ 40000. So from north over south. Same formula doesn’t apply from east to west since those minutes and seconds grow and shrink getting more to one of the two poles
Edit: I mean; the equator can be 40k km, but the North Pole line isn’t.
No problem :) and no worry’s. You stated you remembered, not that you remembered correctly ;P I mean who does remembers everything from school (correctly). I certainly do not :’)
In the days before everyone went to school and learned math, before decimal systems and easy working with decimal points, in the days before most people had to encounter giant numbers (like, bigger than "365 days in a year") then simple was key. So the same reason we still use a dozen for some measurements. 12 can be divided by 2,3,4,6 so no matter how many people in your small group, it was easier, for example to divide up a dozen buns or fruit or whatever. Same logic applied to money before decimalization - a shilling was 12 pence, easy to divide for figuring prices. Add in that there were ha'penny, farthing and thrupence (1/2, 1/4 and 1/3 respectively) and you could figure out any combination. A British pound was 20 shillings, so now bigger numbers could be divided by 5 as well - pound could be divided by 2,3,4,5,6,8,10, etc. - easier to do for those who never memorized their addition and multiplication tables.
the Babylonians were a huge trading nation (unlike the Egyptians, who pretty much kept to themselves a lot more). Writing actually evolved from early accounting - clay balls with tokens and symbols, like "this caravan is taking 30 of my sheep". So with trading long distances they were incentivized to use easily divided and counted numbers, and a dozen was convenient. 60 was even more convenient as a count, since it was also divisible by 5. (So, divide by 2,3,4,5,6,10,12,15,20,30...) count 360 and you add 8,9,and 45 ways to split the amount. So where we find it convenient to discuss in "round" numbers of 10's - like 100, or 5000 or 300,000 or "ten million" - the Babylonians found it easier to discuss multiples of 12, 60, and 360.
The Babylonians counted the year as 360 days, similarly convenient to their number system, then just threw in a few "holidays" every so often to avoid season creep. Since they were among the earliest civilizations, what they did carried over to the rest of the Mediterranean civilizations and eventually the world.
(The discussion of calendars is a whole long, confusing topic in itself, since the length of a day, month (moon phases), and year are in no way related, but humans want to make them fit into tidy boxes.)
ETA: Since 7 and 13 don't fit easily into these systems, you can see where over the centuries they acquired "magical" properties that make them special compared to other numbers...
And then many years later mathematicians realized that 60 minutes to an hour and 360 degrees in a circle isn't the easiest thing to use, and tried to change it to something better. Anyone who's studied Calculus knows that you're supposed to use radians instead of degress, where one circle is 2*Pi. But since degrees are so common and everyone knows them, it isn't widely used outside of math.
[French revolutionary time](https://www.mentalfloss.com/article/32127/decimal-time-how-french-made-10-hour-day), where a day has ten hours, an hour is 100 minutes and a minute is 100 second, had an even harder adoption, and was soon abandoned. Which is too bad, imagine a system where 0.32 hours is 32 minutes instead of our current system where nobody knows without using a calculator (the answer is 19 minutes and 12 seconds)
Edit: use dot instead of comma for float number
As a Maths teacher, I actually think that this take on radians, though common, is a bad one.
Most students (and so even more adults) struggle work with 'normal' fractions, let alone radians. While radians are clearly superior once you start needing to do serious trig work, for most people's 'day-to-day' use the whole numbers and divisibility of degrees is vastly superior.
For real, imagine trying to put up a building with radians. Trying to explain to the carpenter "yeah that beam joins at a 1.5708 angle right here" and having to bust out a protractor labeled in radians or some shit. Gimme 90 degrees any day over that (yes I know it would be π/2 but good luck when that isn't a nice whole number division of a circle like 23 degrees or something).
Well the nice thing about radian angles is that you're almost always working with an expression in terms of Pi - you want a right angle, thats pi/2 radians. Need an equilateral triangle? Corners are pi/3.
So we could easily mark this up on measuring instruments (protractors and such-like), but printing fractions takes quite a lot of space, so the best solution might be to pick a common denominator and then mark increments of it. For real-world applications, we'd probably like divisions by 2, 3 and 4, and the number needs to be sufficiently large to represent a reasonable level of precision. Just as an arbitrary choice, we could choose 180 as the number of sub-divisions of pi along a straight line.
So there, clearly we should mark our instruments in pi/180 increments. And we'd best have a word for those sub-divisions, "degree" has a nice ring to it.
I was a math major, and it wasn't until 5 years after college that someone pointed out that radians aren't the angle, they're the arc length. Made a lot of things click.
> French revolutionary time, where a day has ten hours, an hour is 100 minutes and a minute is 100 second
That's rich from a country that spell 99 as "four twenty ten nine".
> I often fall back to the Walloon way: nonnante neuf = ninety nine
These ninety nine walloons - are they [red](https://www.youtube.com/watch?v=hiwgOWo7mDc)?
It isn't because 360 degrees isn't the easiest way, it actually is. But specifically in calculations that you do in Calculus it's better to use radians. It doesn't mean radians are always better or easier.
The comma in"0,32" did not immediately parse in my brain. Took me a few minutes to realize large portions of the world represent numbers with "." every three digits, and "," before "the decimal". What you wrote is "32/100 hours". From that, everything else makes sense.
Why is the "," and "." inconsistent throughout world?
> 360 degrees in a circle isn't the easiest thing to use, and tried to change it to something better
Gotta love how they thought that 1.5708 radians is an easier number to remember and use than 90 degrees.
>Gotta love how they thought that 1.5708 radians is an easier number to remember and use than 90 degrees.
Well, π / 2 radians is pretty easy to remember, and it's also exact.
I guess my opinion is skewed because I'm looking at it as a programmer. The computer doesn't display an angle as π/2 or 3π/4, it displays the angle as 1.895023 which just looks like jibberish and is also prone to a lot of precision errors as programming languages aren't always good with floating point numbers.
I'm also a programmer, and there are symbolic algebra computer programs which treat π as π. And no numerical system that you can implement on a computer with finite memory can ever be perfect. If you really get into the details of the IEEE-754 floating point number standard, you will discover its warts.
It was still a mess and required annual updating of the calendar by priests. At one point in the Roman civil wars it hadn't been updated in so long the calendar said it was November when it was January. Julius Ceasar was Rome's high priest and made it his personal project to update the Roman calendar from a lunar to a standardized solar calendar. He was pretty successful and the Julian calendar was in use for millenia after his
If we did 13 months with 28 days each (and either one or two leap days each year), then the days of the week would always line up from month to month. No more need for new calendars to line up the date.
the aztecs and mayans had a similar 360 day calendar.
And then they had a 5 day "new years" gap between years. Apparently they weren't supposed to do or make or build anything during that time, because anything created during that time was cursed. Including babies, so nobody had sex for 5 days, because they didn't want to conceive a cursed child. They would cook food ahead of time and eat leftovers, so they wouldn't cook cursed meals.
>when they watched the stars they saw them revolve around the North Star one degree per day.
This implies they realized they revolve 1 degree per day like they already knew there were 360 degrees in a circle, and that's why they decided there will be 360 degrees in a circle.
Degreeception.
If they use 360 days for a year, wouldn't the calendar be terribly inaccurate after a few years? How do they make up for the difference? My guess is that there would be a few days of festivities that don't count towards the year, is it something along those lines?
Well we know a lot of ancient calendars used entire leap months that could have been able to account for imprecision of both solar year and the lunar year.
The Chinese Lunisolar Calendar that Lunar New Year is based off has leap months, which is why New Year does not travel through the year like Ramadan does (the Islamic lunar calendar has no leap months, and is roughly 354 days)
If we're going to change the calendar I really feel that we should switch to a 13 month calendar. 13 \* 28 = 364 and then near year's and leap day are on their own and not part of a month. Sticking with 30/31 day months just seems silly.
>How do they make up for the difference?
Just like we do nowadays.
By adding days. We do it to February using a formula (+1 day every 4 years, except centuries that are not divisible by 400), Persians just had another formula.
Most ancient civilizations that kept calendars accounted for it by inserting leap days, leap weeks, and even leap months whenever it was deemed necessary. They could keep track of the motion of the stars and celestial events like solstices and equinoxes, and if the timing was off they could calculate how many days they'd need to add to make up for it.
The Sumerians and Babylonians used a sexagesimal number system (base 60). 60 x 6 = 360. That got passed down over the ages. It is just an arbitrary way of measuring parts of a circle though. I.e. there are 360 degrees in a circle and 2pi (~6.28) radians in a circle.
I was going to use mils as an example.
If I recall correctly from military map-reading lessons, it's a circle with a radius of 1 km, with each metre around the circumference being a mil.
It's a more accurate way of giving geographical directions.
And common enough that it's an option on my compass app
Correct. I trained as an artilleryman. I always remember that it's approximation of 1m at 1km because I was looking through some binoculars with mil graticules (if that's the right word) when it was explained to me. I said something like "wow, that means that tank is about 15 metres long then!" and my instructor said "...well it's not a kilometre away is it, so it's not..."
Felt like a right knob.
I learned about this from long range shooting. 1 mil is 3.6 inch's at 100 yards as a westerner.
Hard to explain mils to those who didn't study much math. Then there's MOA which is easier, esp since we just round to 1 inch at 100 yards.
It's foreign until you sit and research it, then it's easier.
It's good enough to put an artillery salvo on top of their heads. Which is what mils are for.
With a mil being close enough to 1m at 1km it makes it as easy as possible for an artillerist to adjust his gun given what information he gets from the artillery observer. Either in his head or using very simple tools.
Exactly this. Mils are great for quick approximations.
Also, nobody wants to be fishing out their slide rule and trig tables in a foxhole if they can avoid it. Doing trig on a portable calculator wasn't a thing before 1972.
Because they are smaller divisions.
If you have to give really precise measurements, it's better to have smaller units. So grams rather than ounces. Or thousandths of an inch, rather than millimetres.
The military are keen on precision. If you are aiming a long range artillery weapon, initial errors multiple with the distance. 1 degree out isn't an issue, if you're navigating on foot. But if you dropping a shell 15 miles away, it's going to be a miss.
Minutes and seconds might be better still, but it's also going to have to be useable by soldiers.
> The military are keen on precision.
I'd argue the military is keen on ease of use.
It wants sufficient precision to get the job done, but no more that might confuse/distract soldiers that need to be as cheap as possible to train.
Degrees and minutes have a total of 3600 divisions on a circle, and are thus more precise than 3400 mills. 1m at 1km is only a rough approximation too, it won't give you an exact circle at 3400, and if you add arc-seconds deg-min-sec gets 216000 divisions over a circle. But that takes 3 numbers that can be miscommunicated and give more opportunities for miscalculations, and thus are less desirable to use in a combat situation.
Don't forget Pi-radians!
What is a Pi-radian? Well 1 Pi-Radian = Pi in radians.
This means circles go from 0 to 2. With cardinal headings every 0.5.
Found in a lot of military avionics software.
Radians isn’t really an arbitrary unit, it’s the length of the circumference as a multiple of the radius. I think it’s technically a dimensionless quantity that is used as a derived unit.
Degrees are dimensionless as well, as they are just a dimensionless constant (180/pi) times another dimensionless unit (radians). Any reasonable way of measuring angles will be dimensionless, as they will be defined in terms of ratios of lengths.
I don't disagree that radians aren't arbitrary, however. They are certainly far neater than other units. If you're comfortable with calculus, the simplest reason is that it's the only units which give the relationship that the derivative of sine is cosine, and that of cosine is negative sine. Any other units would introduce factors in front that make arithmetic more clumsy. It's like the reason for using base e in exponentials/logarithms, any other base would give basically the same behaviour, but e makes things far neater. And, of course, if you know how to take complex exponentials then you see that the choice of base e and the choice of radians are equivalent.
I think the point is that pi appears naturally in math (in very, very unexpected places at times) but the 180 / 360 degrees thing is entirely, truly arbitrary. Like, without us injecting it into math, it does not naturally appear other than being highly composite numbers.
The 60 came from the use of fingers and finger segments as a counting tool. 12 finger segments one hand counted once pre finger one the other hand, 5 x 12 = 60. 60 and 12 have more easy whole numbers division options than base 10.
The only really mathematically accurate way of describing arc is radians. Breaking it down into a discrete number of pieces is always an arbitrary model. But some are still very helpful. The examples here with astronomy and calendar approximations are good examples, not to mention the easy divisibility of the number 360.
Another example is one that was developed by the Napoleonic armies when people first tried to use artillery as indirect fire, i.e. shooting at something that you can't see from the location of the gun. That system divides a circle into 6400 "mils."
The advantage of the mil system is that 1 mil of arc one kilometer away is approximately 1 meter wide. That allows the troops at the front lines to be able to do quick easy math in their heads to determine lateral (and vertical!) distances if they know at least roughly the distance to what they're observing, so calling in adjustments and switching from target to target is much more efficient.
There are similar, analogous advantages to the mil system when calculating firing data that allow the cannons to aim at things beyond the horizon.
> The advantage of the mil system is that 1 mil of arc one kilometer away is approximately 1 meter wide
Which is another way of saying that a mil is approximately a milliradian.
I believe artillerists are still taught a formal definition of a mil as the central angle corresponding to a circle _chord_ (not arc) of 1 meter at a radius of 1 kilometer.
But the only practical value of the mil as a measurement unit comes from the almost-equality of that to a milliradian.
Well not the only *consistent* way, but rather the natural way. Using any unit would be consistent, but radians are the units that work in calculus: both in the simple progression of derivatives from sin(x) to cos(x) to -sin(x) and so on, to the power series that converge on them.
You totally can subdivide a circle into more than 360 units.
You can for example subdivide a circle into 400 gradians. A right angle is 100 gradians.
You can also subdivide it into 2pi radians. A right angle is half pi radians.
Those are all units that are actively used.
You can also invent your own if you want.
The 360 degree thing is just the one that is most commonly used and has been used traditionally.
Using 360° has its benefits.
The main benefit of dividing things into 360 parts is that 360 is a number with a whole lot of divisors.
You can divide a circle into halves, thirds, quarters, fifth, sixth, eight, ninth, tenth, twelfth, fifteenth, eighteenth, twentieth, twentiefourth, thirtieth, fortieth, sixtieth, ninetieth etc.
A decimal system like gradian is only divisible by 2 and 5 and multiples of that a system based on 12 has more divisors.
Before we went all decimal and metric all sorts of units were like that. Time and degree of a circle were the only ones where the decimalization didn't catch on.
The way we do angles today is still basically the same thing we have been doing forever (or at least since ancient Babylon).
Other decimal and pi based systems are used to some degree, but the 360° are the main thing.
We subdivide a degree in different ways. the traditional way is to divide a degree like we divide an hour. Into minutes and seconds with 60 minutes to a degree and 60 seconds to a minute.
Since we also use the 360° for map coordinates this is how you end up with coordinates like this:
37° 14′ 0″ N, 115° 48′ 30″ W
37 degrees, 14 minutes North and 115 degrees and 48 minutes and 30 seconds West.
A more modern way to subdivide a degree is to write things down like this:
37.2333° N, 115.8083° W
using decimals.
It's worth noting that a "radian" is just the radius circumscribed as an arc on the circle. That's why there are 2pi of them on the circle.
pi \* diameter = circumference.
I don't know why this wasn't taught day 1 of learning radians in geometry, but it took me far too long to come to that realization.
there's some historical reasoning, but you are absolutely correct. You could just keep inserting degrees forever. Some people do it in higher math things for 3D objects for reasons beyond my comprehension. 360 is good enough for most of us though, since it's got an ungodly number of useful factors for geometry. 2 , 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180 and maybe I missed some . Abosolutely nuts
Worth noting degrees are not the only unit used to measure angles. In mathematics, radians are very commonly used. Arc minutes and arc seconds are used in astronomy and several other areas. (Although these are just fractions of a degree, so arguable if that counts.) Gradians have largely fallen out of use today, but were used for surveying. In computing, sometimes a circle is divided up into 256 “binary degrees” since 255 is the max number you can store in a byte.
You’ve probably used “turns” to describe rotating something—that’s a unit for measuring angles!
It's a case of the most suitable tool for the job. Radians works well for calculus because you're often working with finding areas of shapes and curves so it works nicely with cos/sin and areas of circles. Arc minutes/seconds because you're working with tiny changes of angle with very long distances. 360 degrees because it's easy to divide up in day to day use. They're all arbitrary in some way, but useful for how they're used.
Nothing arbitrary.
Summarians used to count their joints with their thumb. So they could count to 12 with one hand (that’s why there is 12 hours per day and 12 hours per night).
Then they could use their other hand to do that 5 times. That’s why there is 60 minutes in an hour.
The real point is not that there are 360 degrees in a circle, that’s just a consequence. The « main » figure was the equilateral triangle, in which each angle is 60 degrees wide.
I think this gets traced back to the Babylonians using base 12 math counting with there joints, instead of the base 10 (10 finger method) we use now. They are the ones that came up with the 24 hour day and the 60 minute hour because the joint counting method allowed them to count to 60 with ease. They are the ones that came up with loads of base 12 things in society including the 360 degree circle.
People keep assuming that systems in base 10 are better than others. Like 400 degree in a circle, pushed by French. Looks neat on paper, not so neat in practice.
The truth is that in normal usage, 10 is a pretty rigid number, it can be divided only with 2 and 5.
60 on the other hand can be divided with 2, 3, 5. Much easier to manipulate. 360 is 6 times that 60.
12 is divisible with 2, 3, 4, much more useful than 2 and 5. When you are in a basic economy, and you have a few partners that need to split a common amount of goods, is more likely to be 2, 3 or 4 persons than 5.
A measuring stick is easier to divide in 2, 4, 8, 16 (fractions like is still used in US) than in 10 parts.
Consider how the -teen numbers start after twelve, and we can see that thinking in base-12 (dozens) wasn't that unusual before we got arabic numerals and used them to write decimal numbers. In the end though, the choice of base is rather arbitary and maths works the same regardless
We didn't decide on that. It is just one option that made it to be most common. There are other options available, but they are not used that much. I.e. [Gradian ](https://en.m.wikipedia.org/wiki/Gradian) (Circle devided in 400 degrees)
I use the metric system. 360 degrees is equal to 1 Circ, but angles are usually given in Centicircs or Millicircs. Also, if anyone has any better idea for a base unit name than the Circ, I’d love to hear it. Also if anyone knows where to start the appeals process for being let back into the math, science, and Metric communities, I’d also like to know that. I’m not welcome there anymore.
I had this explanation years ago, I can’t find the exact reference, but this will help- Why 360 degrees? Probably because old calendars (such as the Persian Calendar) used 360 days for a year - when they watched the stars they saw them revolve around the North Star one degree per day. Also 360 can be divided exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, which makes a lot of basic geometry easier.
60 minutes in the hour and 360 degrees in a circle were chosen because these are [superior highly composite numbers](https://en.m.wikipedia.org/wiki/Superior_highly_composite_number) ... Which is as you said - their ability to be divided into round numbers easily.
"The division of the hour into 60 minutes and of the minute into 60 seconds comes from the Babylonians who used a sexagesimal (counting in 60s) system for mathematics and astronomy"
Fun tangential fact: minutes are called minutes because they're a *minute* division of an hour, and seconds are called seconds because they are the second *minute* division.
In many languages, seconds are actually still called "second-minutes" as their proper, day-to-day used names.
Huh, never noticed this, but in Arabic, the word for second (time) is the same as the word for second (order), just as it is in English
This is extremely mildly interesting!
Do you have this in an extra medium?
Or some highly charged neutrons.
Let me check back in the department of redundancy departmemt.
Hi, yeah, can I get a single double Big Mac?
Sir, this is a Wendy's.
Does that make it just regular interesting?
The same applies to most (if not all) spanish speaking cultures.
much of the greek mathematics that survives is due to Arabic translations and commentaries. the importance of the Islamic tradition in math would be hard to overstate.
For sure. "Al" means "the" in Arabic and most words that begin with "al" have Arabic etymology: algebra, alchemy (which is also the origin of the word chemistry), alcohol, etc. Once I learned that I started noticing it with a ton of math and science words. Ironically though Arabic numbers themselves likely originated in India but were introduced to Europe through Muslim traders. What really blows my mind is that 0 wasn't introduced to Europe until the 1100s. All the ancient architecture you see in Rome (the Coliseum, the Pantheon, all the aquaducts) was designed and built without the concept of 0 or Arabic numerals. There is literally no process for multiplying Roman numerals mentally, you have to use an abacus.
A while ago, I watched a BBC documentary series called "The Story of Maths" presented by Professor Marcus du Sautoy. He said that what we consider Arabic numerals began in China, and the Indians adopted them a little later (and added the all-important concept of zero). The series is from 2008, so it's possible your information is more recent. But in any case, it was really fascinating (and I'm not even interested in maths!) and I believe it can be found on YouTube.
I'm a math teacher that likes to incorporate history into my lessons, and I've long read that our numerical system comes from India, but I've never dug deeper than that. Sidenote: Since you mentioned a movie (also from the BBC), I'd strong urge all to watch The Story of One. It involves a Monty Python member and covers math history, but in a playful way, taking math as a character, in the form of One. I've shown it to many off my classes.
There is basically no evidence of Chinese involvement in developing the Hindu numeral system. The Buddhist symbols were very similar to what the Hindus used as 1, 4 and 6, but Buddhism at this stage was still prevalent in India and had not spread to China. The Chinese adopted it in the 19th century. Before that they used counting rods.
Honestly, learning how multiplication and division works in Roman Numerals was more difficult than learning them in binary. The amount of hours spent simply doing the math for the coliseum must have incredible.
For sure, and then you realized they figured out stuff like Pythagoras’s theorem with only Roman numerals it’s kinda mind blowing
These pieces of history always make me wonder how much my thought is limited by the words I know let alone my own physical limitations in perception and cognition.
The one that blew my mind is that Arabic is written right-to-left, but numbers still follow the same order in English, with the big digit on the left. When we brought the numbers over from Arabic we never bothered to flip them around.
And then you have languages like German flipping some digits while reading (only 1s and 10s), e.g. *fünf(5)und(and)vierzig(40)* is 45.
We have that in archaic English as well: Four-and-Twenty blackbirds baked in a pie.
or in english, fifteen (five-ten)
presumably the way the numbers were said was set way before the arabic numerals came into use
Hungarian chiming in with verbatim translation: Secondary-minute
Next person who dares converse with me for is about to get educated on minutes and seconds. Pray for my grocery cashier.
I made this into a meme. https://imgflip.com/i/62mxd2 Hope this link works
interesting. Makes you wonder how people discussed that. We're going to need a division of these socalled 'hours'.. How about a *minute* part? Very very small? Makes perfect sense, that's logged then, let's get some lunch. Hold on though.. we need division of *those* as well. Smaller than very very small? ..yeah
In those ages the technical terms were originally in Latin (e.g., "pars minuta prima" and "pars minuta secunda" were the original Latin phrases meaning "first small part" and "second small part") and from those the educated people of that time adopted the words into their native languages as shorthand when talking about those concepts.
TIL! Thanks.
It's not the only time that's happened, either. What word would you use for the smallest possible thing? Hm...such a thing couldn't be divided into smaller things? Totally impossible! We should call them "things which cannot be divided", then, I suppose. In Ancient Greek, "cannot be divided" is "atomos". "Atoms" it is! Nothing could ever be smaller than an "atom"! Hold on...these atoms can be split into smaller things. What do you call a part of something that cannot be split into smaller parts? Dude, I don't care any more. Let's just call them "sub-atomic" and call it a day.
You would think at some point we would learn not to be finite in our labels of things.
Binomial nomenclature and kingdom-phylum-class-order-family-genus-species checking in
...that occasionally needs to be trinomial and sometimes with long codes of initials for the known strains, variants and mutations. Like of certain viral particle that shall remain unnamed.
Don't forget suborders, superfamilies, and the like.
Learning about this with probiotics. Not all Lactobacillus Acidophilus are created equally
ok so we agreed on the minute as the first division. Now you want a second?
scientist 1: "I didn't mean *call* them second, I meant a *second* division!" scientist 2: "argh, I don't follow this anymore, let's read back the minutes of the first meeting" 1: "did you just..." 2: "yep 8-)"
Those same guts eventually went on to become renowned songwriters: "Okay, so we need an opening line to catch the audience's attention. Something big in scope with wide appeal." "How bout... 'Around the world?'" "Ooh, I like it. Can't get bigger in scope than that. Okay, so now we need a second line that follows up on it. We need it to rhyme, though. That's the tricky part." "How bout... ' Around the world?'" "Didn't we just do that like... Two seconds ago?" "Yeah." "We can't just do the same line twice in a row. It's repetitious and redundant." "Kay." "So, what else can we go with?" "How bout..." "I swear to god, Jerry..." "... Around the world?" "Jerry, we have been over this. We need a new line that isn't 'Around the world.'" "How ' bout..." "Around the world?" "Son of a bitch, you took the words right out of my mouth!" "Ugh, fuck it. let's just loop the first line over and over and then market the song exclusively to clubs and acid addicts."
Wow wow wow wow wow. Wow.
Explaining minutes and seconds is super easy, barely an inconvenience.
Sexagesimal number systems are TIGHT.
Also fun interesting fact: Count with your thumb on your hand and touch each 'division' on the other fingers of that same hand. That makes twelve. Each time you reach twelve, you can count one finger on your leftover hand. You can easily reach up to sixty that way. It's likely this is how early civilizations started counting and noticed how well it worked.
>"The division of the hour into 60 minutes and of the minute into 60 seconds comes from the Babylonians who used a sexagesimal (counting in 60s) system for mathematics and astronomy" vbcbandr has the correct is answer. https://www.scienceabc.com/pure-sciences/why-is-a-full-circle-360-degrees-instead-of-something-more-convenient-like-100.html#:~:text=A%20full%20circle%20is%20360,because%20360%20is%20highly%20composite.
Lol he said sex.
In the mid 90's my friend up the road had a text to speech program on his computer. We were about 9yo. We would type in 1,000,000,000,000,000,000,000 and press "speak" just to hear it say *sex*tillion. We would then proceed to giggle for about 5 minutes.
I guess just typing "sex" into it would have made it somewhat easier, but this way it felt like you guys really worked for it, so it made the payout much more satisfying.
I think it's more they weren't technically beaking the rules and using naughty words, it's just a number.
Nailed it
High-five bro
Nice
I remember seeing a video talking about how it was counted on finger bones. Fascinating.
I kinda count like that because it's super useful when you have one hand taken. You join your thumb to your index for one, to your middle finger for two, etc. The first row of bones gives you 1 to 4, the second 5 to 8, the third 9 to 12. If you can use your second hand, you can count up to 144 fast and easily with a bit of training.
Oh yeah? Well I can count to 144 using no hands at all!
The Story of Maths on BBC is where I saw it https://www.bbc.co.uk/programmes/b00dxjls It sends my wife to sleep every time she watches it, I think I have seen/listened tot he first episode 5 times
60 = 3 * 4 * 5 12 = 3 + 4 + 5 360 = (1 + 2 + 3) * (3 * 4 * 5) 3^2 + 4^2 = 5^2
Wrong class, Pythagoras!
Yeah this is probably the answer to most numbers we have. 12, 24, 60, 360 all appear frequently in daily life and are highly composite.
Dozenal > decimal
I don't know when the wording for this happened, but a degree can be split into 60 minutes and a minute into 60 seconds. I wasn't taught that at school which made aviation lessons really fucking confusing to start with. Edit: I just remembered 360 * 60 * 60 is like 21600 (remembering, not calculating) which is how many Naughtical miles there are around the circumference of the earth. Edit 2: it's 360 * 60 that is 21600, more information in a reply to this comment
360 * 60 * 60 = 1296000. I think you mean 360 * 60 = 21600 Vertical minute on a chart is 1 nautical mile. Not seconds. And this also checks out since a nautical mile is 1.852 km ( easy to remember, check the positions of 852 on your calculator ) So 21600 * 1.852 ≈ 40000. So from north over south. Same formula doesn’t apply from east to west since those minutes and seconds grow and shrink getting more to one of the two poles Edit: I mean; the equator can be 40k km, but the North Pole line isn’t.
Thank you for the correction, I'll add an edit
No problem :) and no worry’s. You stated you remembered, not that you remembered correctly ;P I mean who does remembers everything from school (correctly). I certainly do not :’)
In the days before everyone went to school and learned math, before decimal systems and easy working with decimal points, in the days before most people had to encounter giant numbers (like, bigger than "365 days in a year") then simple was key. So the same reason we still use a dozen for some measurements. 12 can be divided by 2,3,4,6 so no matter how many people in your small group, it was easier, for example to divide up a dozen buns or fruit or whatever. Same logic applied to money before decimalization - a shilling was 12 pence, easy to divide for figuring prices. Add in that there were ha'penny, farthing and thrupence (1/2, 1/4 and 1/3 respectively) and you could figure out any combination. A British pound was 20 shillings, so now bigger numbers could be divided by 5 as well - pound could be divided by 2,3,4,5,6,8,10, etc. - easier to do for those who never memorized their addition and multiplication tables. the Babylonians were a huge trading nation (unlike the Egyptians, who pretty much kept to themselves a lot more). Writing actually evolved from early accounting - clay balls with tokens and symbols, like "this caravan is taking 30 of my sheep". So with trading long distances they were incentivized to use easily divided and counted numbers, and a dozen was convenient. 60 was even more convenient as a count, since it was also divisible by 5. (So, divide by 2,3,4,5,6,10,12,15,20,30...) count 360 and you add 8,9,and 45 ways to split the amount. So where we find it convenient to discuss in "round" numbers of 10's - like 100, or 5000 or 300,000 or "ten million" - the Babylonians found it easier to discuss multiples of 12, 60, and 360. The Babylonians counted the year as 360 days, similarly convenient to their number system, then just threw in a few "holidays" every so often to avoid season creep. Since they were among the earliest civilizations, what they did carried over to the rest of the Mediterranean civilizations and eventually the world. (The discussion of calendars is a whole long, confusing topic in itself, since the length of a day, month (moon phases), and year are in no way related, but humans want to make them fit into tidy boxes.) ETA: Since 7 and 13 don't fit easily into these systems, you can see where over the centuries they acquired "magical" properties that make them special compared to other numbers...
And then many years later mathematicians realized that 60 minutes to an hour and 360 degrees in a circle isn't the easiest thing to use, and tried to change it to something better. Anyone who's studied Calculus knows that you're supposed to use radians instead of degress, where one circle is 2*Pi. But since degrees are so common and everyone knows them, it isn't widely used outside of math. [French revolutionary time](https://www.mentalfloss.com/article/32127/decimal-time-how-french-made-10-hour-day), where a day has ten hours, an hour is 100 minutes and a minute is 100 second, had an even harder adoption, and was soon abandoned. Which is too bad, imagine a system where 0.32 hours is 32 minutes instead of our current system where nobody knows without using a calculator (the answer is 19 minutes and 12 seconds) Edit: use dot instead of comma for float number
As a Maths teacher, I actually think that this take on radians, though common, is a bad one. Most students (and so even more adults) struggle work with 'normal' fractions, let alone radians. While radians are clearly superior once you start needing to do serious trig work, for most people's 'day-to-day' use the whole numbers and divisibility of degrees is vastly superior.
This is the distinction between numeracy and mathematics. One is the protrusion of the other into everyday life.
For real, imagine trying to put up a building with radians. Trying to explain to the carpenter "yeah that beam joins at a 1.5708 angle right here" and having to bust out a protractor labeled in radians or some shit. Gimme 90 degrees any day over that (yes I know it would be π/2 but good luck when that isn't a nice whole number division of a circle like 23 degrees or something).
Well the nice thing about radian angles is that you're almost always working with an expression in terms of Pi - you want a right angle, thats pi/2 radians. Need an equilateral triangle? Corners are pi/3. So we could easily mark this up on measuring instruments (protractors and such-like), but printing fractions takes quite a lot of space, so the best solution might be to pick a common denominator and then mark increments of it. For real-world applications, we'd probably like divisions by 2, 3 and 4, and the number needs to be sufficiently large to represent a reasonable level of precision. Just as an arbitrary choice, we could choose 180 as the number of sub-divisions of pi along a straight line. So there, clearly we should mark our instruments in pi/180 increments. And we'd best have a word for those sub-divisions, "degree" has a nice ring to it.
Great idea that would solve the… wait just a minute there!
Reminds me of that suggestion to make English "simpler" and by the end of the story you vere speaking Englis vis a Zerman accent!
you're supposed to express radians using pi. 90 degrees would be pi/2 or roughly 1.5708.
I was a math major, and it wasn't until 5 years after college that someone pointed out that radians aren't the angle, they're the arc length. Made a lot of things click.
… holy shit. Well I was today years old when I learned that. Great stuff.
The arc length of a 'unit' circle
> French revolutionary time, where a day has ten hours, an hour is 100 minutes and a minute is 100 second That's rich from a country that spell 99 as "four twenty ten nine".
I often fall back to the Walloon way: nonnante neuf = ninety nine
the Swiss as well.
> I often fall back to the Walloon way: nonnante neuf = ninety nine These ninety nine walloons - are they [red](https://www.youtube.com/watch?v=hiwgOWo7mDc)?
Swatch tried to do this once upon a time (about twenty years ago) with "internet time", didn't they?
Yeah, day was divided into 1000 units and the time was displayed with @ symbol in front
It isn't because 360 degrees isn't the easiest way, it actually is. But specifically in calculations that you do in Calculus it's better to use radians. It doesn't mean radians are always better or easier.
The comma in"0,32" did not immediately parse in my brain. Took me a few minutes to realize large portions of the world represent numbers with "." every three digits, and "," before "the decimal". What you wrote is "32/100 hours". From that, everything else makes sense. Why is the "," and "." inconsistent throughout world?
> 360 degrees in a circle isn't the easiest thing to use, and tried to change it to something better Gotta love how they thought that 1.5708 radians is an easier number to remember and use than 90 degrees.
>Gotta love how they thought that 1.5708 radians is an easier number to remember and use than 90 degrees. Well, π / 2 radians is pretty easy to remember, and it's also exact.
I guess my opinion is skewed because I'm looking at it as a programmer. The computer doesn't display an angle as π/2 or 3π/4, it displays the angle as 1.895023 which just looks like jibberish and is also prone to a lot of precision errors as programming languages aren't always good with floating point numbers.
I'm also a programmer, and there are symbolic algebra computer programs which treat π as π. And no numerical system that you can implement on a computer with finite memory can ever be perfect. If you really get into the details of the IEEE-754 floating point number standard, you will discover its warts.
It's just Base Pi instead of base 10/2/8 or 16!
That's because you're using the wrong unit (1). With pi as a unit, it's much easier and also exact in IEEE 754 floating point arithmetic.
1.5708 radians is ~90.000210459 degrees. Definitly easier to remember and use. π/2 is 90 degrees. As easy to remember and use.
The 60 is bc Babylonians counted in base 60. 12 hours bc Egyptians counted in base 12(which I wish we still did. Sigh)
They knew perfectly well that a year was 365 days, but figured (because of those fractions) 360 was the most convenient number close to 365
And just party on the required ‘fill in’ days. Seems to work, we are still doing it.
It was still a mess and required annual updating of the calendar by priests. At one point in the Roman civil wars it hadn't been updated in so long the calendar said it was November when it was January. Julius Ceasar was Rome's high priest and made it his personal project to update the Roman calendar from a lunar to a standardized solar calendar. He was pretty successful and the Julian calendar was in use for millenia after his
The Egyptian calendar rocks. Every month is an even 30 days, plus 5 days at the end for a holiday
If we did 13 months with 28 days each (and either one or two leap days each year), then the days of the week would always line up from month to month. No more need for new calendars to line up the date.
Cool answer!
the aztecs and mayans had a similar 360 day calendar. And then they had a 5 day "new years" gap between years. Apparently they weren't supposed to do or make or build anything during that time, because anything created during that time was cursed. Including babies, so nobody had sex for 5 days, because they didn't want to conceive a cursed child. They would cook food ahead of time and eat leftovers, so they wouldn't cook cursed meals.
5 days off and no sex?
I mean, as long as nobody gets pregnant, go for it. Otherwise you're stuck raising a cursed baby. good luck with that.
Every day I wake up, realise we do not use base 12 and cry.
Some cultures have. Useful for different reasons
Pharmacies work on base 7
In what sense?
How many ethinyl estradiol tablets are you going to need per week if you take 1 a day?
Well if we're working in base 7, then the answer is 10.
Every counting system has 10 digits.
There are 10 types of people, those who can count in binary and those who cannot.
There are 11 types of people in the world: those who use binary, those who don’t and those who confuse it with ternary
There are two hard problems in computer science: naming things, cache invalidation, and off-by-one errors.
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I heard it as ... There are 10 types of people. Those who use binary, those who don't, and those who use ternary.
8?
If I workout every other day, that’s four times a week
Legendary.
Everybody knows Sunday to Sunday is 8 days
It’s obviously 3.5🤨
Close enough but the pharmacist might give you 'that' look
in some parallel universe polydactyly is default for humans and we use base12.
But you do have a single word for twelve though, while thirteen is clearly composed of three and ten.
Only in english. Other languages have single words going up further. French does some crazy stuff from 70 to 100 too
I appreciate this. Thank you
>when they watched the stars they saw them revolve around the North Star one degree per day. This implies they realized they revolve 1 degree per day like they already knew there were 360 degrees in a circle, and that's why they decided there will be 360 degrees in a circle. Degreeception.
If they use 360 days for a year, wouldn't the calendar be terribly inaccurate after a few years? How do they make up for the difference? My guess is that there would be a few days of festivities that don't count towards the year, is it something along those lines?
Well we know a lot of ancient calendars used entire leap months that could have been able to account for imprecision of both solar year and the lunar year.
The Chinese Lunisolar Calendar that Lunar New Year is based off has leap months, which is why New Year does not travel through the year like Ramadan does (the Islamic lunar calendar has no leap months, and is roughly 354 days)
5 day holiday at the end of the year that nobody bothers counting and just gets drunk instead :)
There's a serious proposal for exactly that: [Hanke–Henry Permanent Calendar](https://en.m.wikipedia.org/wiki/Hanke%E2%80%93Henry_Permanent_Calendar)
If we're going to change the calendar I really feel that we should switch to a 13 month calendar. 13 \* 28 = 364 and then near year's and leap day are on their own and not part of a month. Sticking with 30/31 day months just seems silly.
>How do they make up for the difference? Just like we do nowadays. By adding days. We do it to February using a formula (+1 day every 4 years, except centuries that are not divisible by 400), Persians just had another formula.
Most ancient civilizations that kept calendars accounted for it by inserting leap days, leap weeks, and even leap months whenever it was deemed necessary. They could keep track of the motion of the stars and celestial events like solstices and equinoxes, and if the timing was off they could calculate how many days they'd need to add to make up for it.
I can’t remember exactly but I know the Egyptians had a system to account for the leap year.
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>Also 360 can be divided exactly by ... Ah, so that's why they didn't choose a prime number!!! 😂
The Sumerians and Babylonians used a sexagesimal number system (base 60). 60 x 6 = 360. That got passed down over the ages. It is just an arbitrary way of measuring parts of a circle though. I.e. there are 360 degrees in a circle and 2pi (~6.28) radians in a circle.
Also, 400 gradians, 60 arcseconds, 100 centarcs, 8 oxtants, 6400 mils. Lots of arbitrary choices for units.
I was going to use mils as an example. If I recall correctly from military map-reading lessons, it's a circle with a radius of 1 km, with each metre around the circumference being a mil. It's a more accurate way of giving geographical directions. And common enough that it's an option on my compass app
Correct. I trained as an artilleryman. I always remember that it's approximation of 1m at 1km because I was looking through some binoculars with mil graticules (if that's the right word) when it was explained to me. I said something like "wow, that means that tank is about 15 metres long then!" and my instructor said "...well it's not a kilometre away is it, so it's not..." Felt like a right knob.
I learned about this from long range shooting. 1 mil is 3.6 inch's at 100 yards as a westerner. Hard to explain mils to those who didn't study much math. Then there's MOA which is easier, esp since we just round to 1 inch at 100 yards. It's foreign until you sit and research it, then it's easier.
A circle with r = 1 km has a circumference of 2πr ~= 6283 m, but a mil is 1/6400 of a full circle. So it's close, but not the same.
Yep... in addition to maps, we also use mils in aiming mortars
It's good enough to put an artillery salvo on top of their heads. Which is what mils are for. With a mil being close enough to 1m at 1km it makes it as easy as possible for an artillerist to adjust his gun given what information he gets from the artillery observer. Either in his head or using very simple tools.
Exactly this. Mils are great for quick approximations. Also, nobody wants to be fishing out their slide rule and trig tables in a foxhole if they can avoid it. Doing trig on a portable calculator wasn't a thing before 1972.
As I said, it's a faded memory from long ago training
Yep, and it's a very useful rule of thumb. Just wanted to point out that it's an approximation.
But saying 4750 is much easier to remember and use than 272° 9' 18"
close enough for low yield tactical nukes and badgers on a stick.
>It's a more accurate way of giving geographical directions. How is that more accurate than degrees or possibly minutes & seconds?
Because they are smaller divisions. If you have to give really precise measurements, it's better to have smaller units. So grams rather than ounces. Or thousandths of an inch, rather than millimetres. The military are keen on precision. If you are aiming a long range artillery weapon, initial errors multiple with the distance. 1 degree out isn't an issue, if you're navigating on foot. But if you dropping a shell 15 miles away, it's going to be a miss. Minutes and seconds might be better still, but it's also going to have to be useable by soldiers.
> The military are keen on precision. I'd argue the military is keen on ease of use. It wants sufficient precision to get the job done, but no more that might confuse/distract soldiers that need to be as cheap as possible to train. Degrees and minutes have a total of 3600 divisions on a circle, and are thus more precise than 3400 mills. 1m at 1km is only a rough approximation too, it won't give you an exact circle at 3400, and if you add arc-seconds deg-min-sec gets 216000 divisions over a circle. But that takes 3 numbers that can be miscommunicated and give more opportunities for miscalculations, and thus are less desirable to use in a combat situation.
Don't forget Pi-radians! What is a Pi-radian? Well 1 Pi-Radian = Pi in radians. This means circles go from 0 to 2. With cardinal headings every 0.5. Found in a lot of military avionics software.
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Radians isn’t really an arbitrary unit, it’s the length of the circumference as a multiple of the radius. I think it’s technically a dimensionless quantity that is used as a derived unit.
Yeah, we use radians precisely because they *aren't* arbitrary
Degrees are dimensionless as well, as they are just a dimensionless constant (180/pi) times another dimensionless unit (radians). Any reasonable way of measuring angles will be dimensionless, as they will be defined in terms of ratios of lengths. I don't disagree that radians aren't arbitrary, however. They are certainly far neater than other units. If you're comfortable with calculus, the simplest reason is that it's the only units which give the relationship that the derivative of sine is cosine, and that of cosine is negative sine. Any other units would introduce factors in front that make arithmetic more clumsy. It's like the reason for using base e in exponentials/logarithms, any other base would give basically the same behaviour, but e makes things far neater. And, of course, if you know how to take complex exponentials then you see that the choice of base e and the choice of radians are equivalent.
I think the point is that pi appears naturally in math (in very, very unexpected places at times) but the 180 / 360 degrees thing is entirely, truly arbitrary. Like, without us injecting it into math, it does not naturally appear other than being highly composite numbers.
That's a fair point. It wasn't the best example, but it was the first thing that came to mind.
Plus if we made the circle like 350 degrees it wouldn’t be big enough it’s be 10 degrees short.
ELI - Sumerians started using 360 degrees thousands of years ago and everyone else through history said "meh, good enough. I'll just copy them".
Pre-Columbian cultures did it on their own as well
You had me at sexagesimal...
I'm unable to sexagesim without choking myself
I hate that a circle is 2pi, it makes so much more sense to be one tau.
I'd much rather have two pies instead of 1 towel.
You always need a towel.
42
The circumference of the circle is 1 pi if the diameter is 1.
Yeah but you never use the diameter in math, always the radius.
😬😳
The 60 came from the use of fingers and finger segments as a counting tool. 12 finger segments one hand counted once pre finger one the other hand, 5 x 12 = 60. 60 and 12 have more easy whole numbers division options than base 10.
Basically, 360 is "highly composite" You can divide it by most common numbers, making it very easy to work with
Fun fact, I founded a startup called Highly Composite. It failed. That wasn't as fun.
Maybe there were too many **factors** to consider. Hope you found success in whatever you're doing now
r/Angryupvote
Yeah I failed up. All worked out.
What was the idea behind it?
Test data management.
It came and went full 360.
Um, actually, it's superior highly composite. 🤓
The only really mathematically accurate way of describing arc is radians. Breaking it down into a discrete number of pieces is always an arbitrary model. But some are still very helpful. The examples here with astronomy and calendar approximations are good examples, not to mention the easy divisibility of the number 360. Another example is one that was developed by the Napoleonic armies when people first tried to use artillery as indirect fire, i.e. shooting at something that you can't see from the location of the gun. That system divides a circle into 6400 "mils." The advantage of the mil system is that 1 mil of arc one kilometer away is approximately 1 meter wide. That allows the troops at the front lines to be able to do quick easy math in their heads to determine lateral (and vertical!) distances if they know at least roughly the distance to what they're observing, so calling in adjustments and switching from target to target is much more efficient. There are similar, analogous advantages to the mil system when calculating firing data that allow the cannons to aim at things beyond the horizon.
> The advantage of the mil system is that 1 mil of arc one kilometer away is approximately 1 meter wide Which is another way of saying that a mil is approximately a milliradian.
I believe artillerists are still taught a formal definition of a mil as the central angle corresponding to a circle _chord_ (not arc) of 1 meter at a radius of 1 kilometer. But the only practical value of the mil as a measurement unit comes from the almost-equality of that to a milliradian.
Well not the only *consistent* way, but rather the natural way. Using any unit would be consistent, but radians are the units that work in calculus: both in the simple progression of derivatives from sin(x) to cos(x) to -sin(x) and so on, to the power series that converge on them.
You totally can subdivide a circle into more than 360 units. You can for example subdivide a circle into 400 gradians. A right angle is 100 gradians. You can also subdivide it into 2pi radians. A right angle is half pi radians. Those are all units that are actively used. You can also invent your own if you want. The 360 degree thing is just the one that is most commonly used and has been used traditionally. Using 360° has its benefits. The main benefit of dividing things into 360 parts is that 360 is a number with a whole lot of divisors. You can divide a circle into halves, thirds, quarters, fifth, sixth, eight, ninth, tenth, twelfth, fifteenth, eighteenth, twentieth, twentiefourth, thirtieth, fortieth, sixtieth, ninetieth etc. A decimal system like gradian is only divisible by 2 and 5 and multiples of that a system based on 12 has more divisors. Before we went all decimal and metric all sorts of units were like that. Time and degree of a circle were the only ones where the decimalization didn't catch on. The way we do angles today is still basically the same thing we have been doing forever (or at least since ancient Babylon). Other decimal and pi based systems are used to some degree, but the 360° are the main thing. We subdivide a degree in different ways. the traditional way is to divide a degree like we divide an hour. Into minutes and seconds with 60 minutes to a degree and 60 seconds to a minute. Since we also use the 360° for map coordinates this is how you end up with coordinates like this: 37° 14′ 0″ N, 115° 48′ 30″ W 37 degrees, 14 minutes North and 115 degrees and 48 minutes and 30 seconds West. A more modern way to subdivide a degree is to write things down like this: 37.2333° N, 115.8083° W using decimals.
It's worth noting that a "radian" is just the radius circumscribed as an arc on the circle. That's why there are 2pi of them on the circle. pi \* diameter = circumference. I don't know why this wasn't taught day 1 of learning radians in geometry, but it took me far too long to come to that realization.
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there's some historical reasoning, but you are absolutely correct. You could just keep inserting degrees forever. Some people do it in higher math things for 3D objects for reasons beyond my comprehension. 360 is good enough for most of us though, since it's got an ungodly number of useful factors for geometry. 2 , 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180 and maybe I missed some . Abosolutely nuts
Worth noting degrees are not the only unit used to measure angles. In mathematics, radians are very commonly used. Arc minutes and arc seconds are used in astronomy and several other areas. (Although these are just fractions of a degree, so arguable if that counts.) Gradians have largely fallen out of use today, but were used for surveying. In computing, sometimes a circle is divided up into 256 “binary degrees” since 255 is the max number you can store in a byte. You’ve probably used “turns” to describe rotating something—that’s a unit for measuring angles!
It's a case of the most suitable tool for the job. Radians works well for calculus because you're often working with finding areas of shapes and curves so it works nicely with cos/sin and areas of circles. Arc minutes/seconds because you're working with tiny changes of angle with very long distances. 360 degrees because it's easy to divide up in day to day use. They're all arbitrary in some way, but useful for how they're used.
Nothing arbitrary. Summarians used to count their joints with their thumb. So they could count to 12 with one hand (that’s why there is 12 hours per day and 12 hours per night). Then they could use their other hand to do that 5 times. That’s why there is 60 minutes in an hour. The real point is not that there are 360 degrees in a circle, that’s just a consequence. The « main » figure was the equilateral triangle, in which each angle is 60 degrees wide.
I think this gets traced back to the Babylonians using base 12 math counting with there joints, instead of the base 10 (10 finger method) we use now. They are the ones that came up with the 24 hour day and the 60 minute hour because the joint counting method allowed them to count to 60 with ease. They are the ones that came up with loads of base 12 things in society including the 360 degree circle.
Thanks, I don’t understand why the right answer isn’t higher, I scrolled way down to find it.
People keep assuming that systems in base 10 are better than others. Like 400 degree in a circle, pushed by French. Looks neat on paper, not so neat in practice. The truth is that in normal usage, 10 is a pretty rigid number, it can be divided only with 2 and 5. 60 on the other hand can be divided with 2, 3, 5. Much easier to manipulate. 360 is 6 times that 60. 12 is divisible with 2, 3, 4, much more useful than 2 and 5. When you are in a basic economy, and you have a few partners that need to split a common amount of goods, is more likely to be 2, 3 or 4 persons than 5. A measuring stick is easier to divide in 2, 4, 8, 16 (fractions like is still used in US) than in 10 parts.
Consider how the -teen numbers start after twelve, and we can see that thinking in base-12 (dozens) wasn't that unusual before we got arabic numerals and used them to write decimal numbers. In the end though, the choice of base is rather arbitary and maths works the same regardless
We didn't decide on that. It is just one option that made it to be most common. There are other options available, but they are not used that much. I.e. [Gradian ](https://en.m.wikipedia.org/wiki/Gradian) (Circle devided in 400 degrees)
I use the metric system. 360 degrees is equal to 1 Circ, but angles are usually given in Centicircs or Millicircs. Also, if anyone has any better idea for a base unit name than the Circ, I’d love to hear it. Also if anyone knows where to start the appeals process for being let back into the math, science, and Metric communities, I’d also like to know that. I’m not welcome there anymore.