What numerical system did they use in ancient times?

Question

it seems like it used to be 1-6, or possibly 1-12? we have 360 degrees of a compass 60 min=1 hr 12 ft to a fathom, & also is the modern 1-10 system flawless?

Answer 1

The ancient Babylonians had a base-60 system... 60 is evenly divisible by 1, 2, 3, 4, 5 and 6, which explains part of it.

Also if you imagine using your thumb to point to segments on your fingers you see that you have 3 segments on each of the 4 fingers. That explains 12 hours in a day (and 12 more hours for night). If you count 5 groups of 12 using the fingers of your left hand you get to 60, e.g. 60 minutes and 60 seconds.

So Base-60 was a combination of Base-5 and Base-12. Base-12 explains 12" in a foot, and terms like "dozen" and "gross" (12² = 144).

As you noted, the Babylonians also divided the circle into 360 degrees, with 60 minutes per degree and 60 seconds per minute of angle.

So basically the Babylonians used Base-12 for some things, Base-5 for others and Base-60 for lots of other things.

As for Base-10 that we use today, it is flawed when it comes to fractions. For example, in base 10, 1/3 is 0.33333.... but in another base you wouldn't have an infinitely repeating decimal. I'm sure there are other things that are imperfect with a Base-10 system... for example logic and computers love to think in on-off (0, 1). It takes effort to convert from the language of computers back to Base-10.

Answer 2

Don't know what you mean by "flawless", but it's definitely easier to deal with measures that are in the same powers as our number system.

The 360 degrees in a circle and 60 minutes in an hour is a relic of Babylon. See below.

As for 6 (not 12) feet to a fathom...another source I found says that it was chosen that way because a man's arms can stretch about that far. See below.

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From http://www.newton.dep.anl.gov/askasci/ast99/ast99606.htm:

In the Old Babylonian period, there was a unit of time corresponding to the "barleycorn." In Sumerian, one barleycorn (called se) was 1/180 of a shekel. In late Babylonian astronomical texts, it was 1/6 of a finger. Since 1 finger was 1/12 or a degree, we have 1 degree = 72 barleycorns and 15 degrees = 1080 barleycorns. So why are we worried about 15 degrees? That is the amount of arc length the sun travels in the sky in one hour, where one hour is 1/24 of a day. It was convenient to divide the day into 12 daylight and 12 night time parts. At 15 degrees per hour, and 24 hours per day, this would give 360 degrees of rotation per day. From this, it is an easy jump to 360 degrees in a circle.

If you look into Biblical texts,you will find that hours were broken up into 1080 parts (we now divide our hours into 3600 seconds) corresponding to these barleycorns. The relationship of 1 finger = 6 barleycorns is well known in Arabic, Syriac, and Sanskrit astronomy.

Some scholars claim that 1080 (and 360) were used because these numbers are easily divisible by so many numbers (notably 7 excepted). There does not seem to be great evidence for this.

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From http://www.unc.edu/~rowlett/units/dictF.html:

a traditional unit of distance equal to 2 yards or 6 feet (approximately 1.829 meters). The word comes from the Old English fæthm, meaning "outstretched arms", because a fathom is the distance between a man's outstretched fingertips. This is a generic unit that has been used in many cultures since ancient times. Other versions include the Spanish braza, the French toise, the German klafter, the Danish favn (6.18 feet or 1.88 meters), the Swedish famn (5.84 feet or 1.78 meters) and the Japanese ken. In England, the fathom was a common unit during Saxon times, and it continued to be used for many purposes through the medieval era. In fact, the length of the foot may have been defined, early in the twelfth century, specifically to assure that 1 foot = exactly 1/6 fathom. Today the fathom is used almost exclusively at sea, measuring water depth, the length of ships' cables, etc.

Answer 3

Most societies probably used base 5 or base 10. The Babylonians, or Chaldeans, however decided to adopt a different base, either base12 or base60. Usually it is presented as 60. The reasoning behind base 12 is that it is the LCM of the 1st 4 integers, while base60 is the LCM of the first 5 integers. Note that the way we use base 60 (and presumably the way the Babylonians used it) is actually as a modified base10. We count to 59 in base 10, then count multiples of 60 in base 10 to 59 again, THEN we add in a modified base12.
A true base12 system has 12 characters, and a true base 60 would have 60 distinct characters.

Answer 4

I think your question seems to be whether or not we always used a base-10 system for counting.

The 360 degrees in a circle isn't so much related to counting, it's more derived from the fact that early mathematicians used hexagons to approximate circles when trying to approximate pi.

There are actually six feet in a fathom, not twelve, and this measurement comes from the fact that an average man's arm span was about six feet.

As far as numerical systems goes, my understanding is that base-10 has for the most part always been standard. You are attempting to compare numerical systems to systems of measurements, and this is like comparing apples to oranges in my opinion.

For what it's worth, computers use binary counting, since computer chips can only be on or off, thus either a zero or one.

Answer 5

No numerical system is inherently flawed. It is simply a choice. For example, computers use base 2. It never caught on until there was a need for it. For similar reasons, octal and hexadecimal were used because of their easy convertability into binary.

By the way, the Maya used a system of counting based on 20.

Answer 6

Arabic.

Answer 7

http://jhunix.hcf.jhu.edu/~blee27/essays/ancient_mathematics.htm

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