Babylonian
Speculative phase:
(1) Think for yourself why 60 might be a
convenient, significant or especially useful number to use as the base for a
number notational system. What is special about the number 60? How is it
different from 10?
60 is much bigger than 10, but maybe it is
easier for the Babylonians when dividing/using fractions. 60 is divisible by
2,3,5,6,10,12... It contains many factors. Hence, base 60 might be easier to
use than base 10 in ancient times when calculations are not as convenient as nowadays.
(2) Then think for yourself how we still
use 60s in our own daily lives, in Canada, and across cultures if you have
knowledge of other systems (like the Chinese zodiac and time-telling system,
for example.) Why is 60 significant in so many situations involving time and/or
space?
In China, one day is separated into 12 sections. One year is separated into 12 months. 60 is a multiple of 12. In western cultures, one day is separated into 24 hours. One hour is separated into 60 minutes. One minute is separated into 60 seconds. I think 60 is convenient because when measuring time because it is large enough to use and it is divisible by many factors. if we use 10 as the base, it might be too small a number to use. Also it might get too complicated. If we get too many hours in a day, it is easy to lose track.
12 is a good number because each 2-hour section represents a different type of weather/ environment in one day. Consider 9-11 and 11-13, one may be morning with mild weather and one may be the noon with shining sun. if we divide one day into 10 hours, it might not be as accurate when representing each situation. Also, this number is consistent with the number of months in a year. Consistency is an important factor of Chinese culture.
The Chinese 60-year calendar cycle is based
on the combinations of a cycle of ten heavenly stems and twelve earthly
branches. Each year is named by a pair of one stem and one branch. The Year of
Jia Zi (Jia from the heavenly stems and Zi from the earthly branches) is the
beginning of the sexagenary cycle. The next Jia Zi Year will come 60 years
later.
The same reason might apply to this way of
counting, for it is a large enough number to use, so that we don’t confuse the name
of recent years. 60 years is already a long time in ancient times and one generation
may already has come to an end. Also, the base of 12 earthly branches is
consistent with the number of months in a year. It is also consistent with the
number of Chinese hours in a day.
Research phase:
(3) Finally, do a bit of research via the
internet and/or the library to find out what others have learned about the
significance of 60 in Babylonian numeration systems, in our contemporary world,
and possibly across cultures.
1.
There is this conjecture that :
“The most commonly accepted theory holds that two earlier peoples
merged and formed the Sumerians,” USA Today reported. “Supposedly, one group
based their number system on 5 and the other on 12. When the two groups traded
together, they evolved a system based on 60 so both could understand it.”
That’s because five multiplied by 12 equals
60. The base 5 system likely originated from ancient peoples using the digits
on one hand to count. The base 12 system likely originated from other groups
using their thumb as a pointer and counting by using the three parts on four
fingers, as three multiplied by four equals 12.
Also, “The Babylonian mathematics system
may not be as popular as it once was, but it has advantages over the base 10
system because the number 60 “has more divisors than any smaller positive
integer,” the Times pointed out.”
2. The number 60, a superior highly
composite number, has twelve factors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20,
30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many
fractions involving sexagesimal numbers are simplified. For example, one hour
can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes,
10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1
minute. 60 is the smallest number that is divisible by every number from 1 to
6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6.
Source: https://en.wikipedia.org/wiki/Sexagesimal
3. Another author also wrote that:
“To be clear, base 60 has a big advantage
over base 10: 60 is divisible by 3, and 10 isn’t. It’s easy to write the
fractions 1/2, 1/4, and 1/5 in base 10: they’re 0.5, 0.25, and 0.2,
respectively. But 1/3 is 0.3333…. Its decimal representation doesn’t terminate.
That really isn’t too much of a problem for us because we are comfortable
representing numbers as either decimals or fractions. But the Babylonian number
system did not represent fractions in terms of numerators and denominators the
way we do. They only used the sexagesimal form, which would be like us only
using decimals instead of writing numbers as fractions. In sexagesimal, 1/3 has
an easy representation as. It’s 20/60, which could be written as .20 in a
sexagesimal system. “
4. 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. They derived their number system from the Sumerians who were using it as early as 3500 BC. The use of 12 subdivisions for day and night, with 60 for hours and minutes, turns out to be much more useful than (say) 10 and 100 if you want to avoid having to use complicated notations for parts of a day. Twelve is divisible by two, three, four, six and 12 itself - whereas 10 has only three divisers - whole numbers that divide it a whole number of times. Sixty has 12 divisers and because 60 = 5 x 12 it combines the advantages of both 10 and 12. In fact both 12 and 60 share the property that they have more divisers than any number smaller than themselves.
Great post May! I find your description of the Chinese sexagenary cycle of years really interesting. There are so many different influences for this structuring of time. I hope that you have a chance to share some of this knowledge with the rest of the class.
回复删除P.S. Please make sure that you complete your posts for the first reading "Why teach math history?" and for the "Crest of the Peacock" first chapter.