Why 360-degrees?

[Turtle]

___I listened to the author of that book on the radio a while back. Loved him! He talked some about that book, but I think he has a new one out.

___So if I may sum up, we agree that calling the ancient Babylonian or Sumerian systems "base 60" is technically wrong (not equivalent) in regard to what we understand our base 10 system? :Alien:

[niviene]

Why not, say, 400-degrees to describe circumference of the circle?

 

I was actually discussing this with my math prof awhile back, and she told me that it was simply because it was divisible by so many numbers... she also said she would use 360 for the N (times compounded) in interest equations (A=P(1+r/n)^nt) because of the same reason and that it was considered acceptable and even normal to use 360 rather than 365... but when I tested that out myself, it made a huge difference in some cases, so she may just have been yanking my chain. Somehow I feel I've been duped! :Alien:

[C1ay]

So if I may sum up, we agree that calling the ancient Babylonian or Sumerian systems "base 60" is technically wrong (not equivalent) in regard to what we understand our base 10 system? :Alien:

It is not what we technically call a base 60 system today. All historical references I can find refer to the system used by the babylonians as a base 60 system though. Even MathWorld uses this terminology. Can they all be wrong?

[Turtle]

___Yes, they can all be wrong. Everyone in history who said man can't fly was wrong!? It is a point I have long noticed & found extremely misleading. This is math afterall, the surest of logical systems we rely on. You may say what the diff , base 60, based on 60, but that is akin to saying what the diff, six minus three or three minus six.

___It's not just the initial wording either, but the explanations that follow often go to great lengths to get around the error. Not only can they be wrong in this, they are.

[C1ay]

___Yes, they can all be wrong. Everyone in history who said man can't fly was wrong!? It is a point I have long noticed & found extremely misleading. This is math afterall, the surest of logical systems we rely on. You may say what the diff , base 60, based on 60, but that is akin to saying what the diff, six minus three or three minus six.

___It's not just the initial wording either, but the explanations that follow often go to great lengths to get around the error. Not only can they be wrong in this, they are.

Their system did group by increasing exponents of 60 though. How is that different than base 60? The angular system of degrees, minutes and seconds that we use today is credited to them and is based on their number system. It is referred to by mathematicians as a base 60 system since 1 degree is 60 minutes and 1 minute is 60 seconds. It is a system where each succesive grouping is the next increment of an exponent of the base 60. What else does such a system need for you to consider it base 60?

[Turtle]

___It needs zero, & it needs a positional arrangement which uses succesive powers of some given value. That requires a single unique symbol for every value up to one less than the value chosen as base.

[C1ay]

___It needs zero, & it needs a positional arrangement which uses succesive powers of some given value. That requires a single unique symbol for every value up to one less than the value chosen as base.

So you are saying base 10 systems of the time were not base 10 either since they also failed to have a zero placeholder?

[Robust]

So you are saying base 10 systems of the time were not base 10 either since they also failed to have a zero placeholder?

The Babylonian Base 10 system - as I interpret from the Pythgorean perfect ratios - does have a zero placeholder - the cipher.

[C1ay]

The Babylonian Base 10 system - as I interpret from the Pythgorean perfect ratios - does have a zero placeholder - the cipher.

Dated when? There was a point in time when the concept of zero was invented. Prior to that there was no zero. Does that mean that there was no base n for lack of a placeholder?

 

Most sources I can find say that a space between numbers was replaced by two slanted wedges in Babylonian math between 400 and 300 BCE. What bases did they use prior to this?

[Turtle]

____That's right; not really base ten. In this regard, one ubiquitously finds refernce to base ten in regard to our having ten fingers. This is entirely a misconception of base & manifest in the current deabate. The zero in the context of base is a placeholder only, & not a number; zero as a number is an algebraic concept.

___If our number system were based on our number of fingers, we would use base eleven. Further, any system of enumeration is an attempt at symbolically representing amount, and the concept of base is nothing more than a method of efficiency in handling amounts. One chooses an appropriate base depending on conditions as we must use base 2 in computers in deference to the nature of transistors. :Alien:

[Robust]

Dated when? There was a point in time when the concept of zero was invented. Prior to that there was no zero. Does that mean that there was no base n for lack of a placeholder?

 

Most sources I can find say that a space between numbers was replaced by two slanted wedges in Babylonian math between 400 and 300 BCE. What bases did they use prior to this?

My archeological study dates to 5,000 BC, indicating (at least to me) via Pythagoras that Base 10 was then the Babylonian mode. As before mentioned, I find from Pythagoras that the entire Base 10 concept was predicated upon the cipher as representing Base 10....and have shown the proof of its working in my initial posting on this forum;i.e., Base 10 Anomaly.

[C1ay]

My archeological study dates to 5,000 BC, indicating (at least to me) via Pythagoras that Base 10 was then the Babylonian mode. As before mentioned, I find from Pythagoras that the entire Base 10 concept was predicated upon the cipher as representing Base 10....and have shown the proof of its working in my initial posting on this forum;i.e., Base 10 Anomaly.

So then, You claim to have evidence of the use of zero prior to the current record? Have you written a paper on this to claim your fame?

[Qfwfq]

___Acknowledged C1ay. I still have a nit pick with what they used being called Base 60. One of the articles from your link made the point that the Babylonian system needed a context to begin, that is if there was no value for a 'place' in a number the 'place' was left out. That is equivalent to us just writing a 1 for 100 & expecting you to know what is meant.

___It is a matter of semantics I admit, but I still contend theirs was a system based on 60, not a base 60 system.

Turtle, the symbolism one uses for a zero placeholder isn't essential to saying which is the base. What counts is the base of which powers are taken and consequently the number of possible values for each "digit". You can even choose to use mixed base numbering. Base b essentially means that:

 

123456 = 6*1 + 5*b^1 + 4*b^2 + 3*b^3 + 2*b^4 + 1*b^5

 

But what happens if one of the digits is zero?

 

120456 = 6*1 + 5*b^1 + 4*b^2 + 2*b^4 + 1*b^5

 

In the explicit power summation there's no need for a digit 0!!!

 

Whether you use a notation 120456 or 1;2;;4;5;6 is irrelevant.

 

How do you mark the digits greater than 9? For hex, programmer's use ABCDEF and this could be easily extended up to 36. One could use any larger base "on top of" base ten by writing 48;6;13;21;37 and the likes, mixed base you might call it.

 

One could even choose to write a number in base a/b/c/d... so that the digit left of the unit figure is to be multiplied by a, the next left one by a*b etc. In this manner base 60 "on top of" base ten could be written like 590007 equivalent to 59;;7 and it is clearly also base 10/6/10/6/10/6...

[Turtle]

___Thanks Q! That is all very similar explanation to those I found in other sources, & I don't contend it. That you & others have to go through some extra work to make the point, is in a sense my point. Their system differs enough from ours that IMO it doesn't merit the same accord.

___Your last paragraph looks like an encryption scheme & it puts in mind if anyone has evidence of Babylonians or Sumerians using encryption. I recall a story of the Roman 'code' sticks but nothing earlier.

____Regardless of the Base discussion, it seems we have agreement on why 360 is used, ie. because of a large number of proper divisors. No matter the base, it is amount that matters & changing base or notation doesn't change the amount.

[Robust]

So then, You claim to have evidence of the use of zero prior to the current record? Have you written a paper on this to claim your fame?

The best evidence we have IMO is (as mentioned) the perfect ratios of Pyhtagoras 0:1:1:2:3:4 , where the cipher represents Base 10. Thus,as given, pi/40 giving the least possible distance between each angular degree on circumference of the circle - or, as also given, radius/radian. It all coincides with a circumference of 360 degrees.