Fact: 12, 24, 60, these are the big numbers of your everyday life. These are surely different from the nickel, dime, quarter, and dollar. Just why is that?
You may have questioned why do we, users of a base 10 system of numbers, seem so dedicated to the dozen (12), the 24 hour clock ( = 2 x 12), the minute (= 60 seconds), and the hour (= 60 minutes). The answer lies in divisibility.
Clearly, a dozen (12) is divisible by 2, 3, 4, and 6. This means you can ask for a quarter dozen, third of a dozen, half a dozen, etc. The math is simple. If the basic multiple were ten (10 = 2 x 5) we could only do a half or a fifth. That would be it.
For the day (24 hours), we can divide the number of hours by 2, 3, 4, 6, 8, and 12. This allows a half day, a third of a day (typical shift), a sixth of a day (= 4 hours - the morning shift), and more.
For an hour (60 minutes) we can divide by 2, 3, 4, 6, 12, 20, 30. This allows a half hour, a quarter hour, a third of an hour, and more. The divisibility makes the calculation convenient and easy. Imagine an hour of 100 minutes. Then we could divide it into only half, fourth, fifth, tenth, twentieth, and a couple of others. There is less flexibility.
The ancient Babylonians used a base 60 system of enumeration for good reason. Why? It's all in the divisibility. Also, it made dealing with fractions a whole lot easier. BTW, handling fractions were a problem for all the ancient civilizations. Naturally, the multiplication tables were horrific. But they had tricks to make calculations easy (well, easier than memorizing all products x times y for x and y in the range from 1 to 59). Isn't it interesting that in the schools, the multiplication tables taught are for all products x times y for x and y in the range from 1 to 12. There is that number, 12, once again.
The year of 365 1/4 days, we can do nothing about. It is what it is. But note, we have conveniently divided the year into 12 months. Sure, the numbers of days differ a bit, but calculations are easy in this context. On the other hand, imagine a year with ten months. Divisibility is only possible into halves and fifths. This is the same as if the basic multiple unit consisted of ten items, just like the discussion about the dozen.
There are 360 degrees in a circle. This is 30 x 12. Thus a circle consists of 12, 30 degree arcs. So, twelve occurs again. And 360 has the ultimate number of divisors. We use them all: quarter circle = 90 degrees, half circle = 180 degrees, eighth circle = 45 degrees, and so on. All the directions, N, S, E, and W have multiple of 90 degrees; all the half directions, NW, SW, NE, and NW, likewise. All are on the compass with whole number degrees. Convenient, huh?
Fingers and toes. These are the basis of our monetary and (some of our) measurement systems. This means tens and twenties. It is an apparent explanation of one system in contradistinction to the more facile base 12, 24, and 60 systems of everyday counting. For Americans, don't forget the foot (12 inches) and the yard (36 inches). The mile is 1760 yards. Don't forget the usual divisions of the inch into 2, 4, 8, and 16 parts. Yet, your car’s odometer measures distance in miles and tenths of miles. Go figure. No wonder kids are confused. Geez, now even I'm confused.
In an unrelated note, today's date is 12/12/12. Lots of twelves. Won't happen again in your lifetime.
You may have questioned why do we, users of a base 10 system of numbers, seem so dedicated to the dozen (12), the 24 hour clock ( = 2 x 12), the minute (= 60 seconds), and the hour (= 60 minutes). The answer lies in divisibility.
Clearly, a dozen (12) is divisible by 2, 3, 4, and 6. This means you can ask for a quarter dozen, third of a dozen, half a dozen, etc. The math is simple. If the basic multiple were ten (10 = 2 x 5) we could only do a half or a fifth. That would be it.
For the day (24 hours), we can divide the number of hours by 2, 3, 4, 6, 8, and 12. This allows a half day, a third of a day (typical shift), a sixth of a day (= 4 hours - the morning shift), and more.
For an hour (60 minutes) we can divide by 2, 3, 4, 6, 12, 20, 30. This allows a half hour, a quarter hour, a third of an hour, and more. The divisibility makes the calculation convenient and easy. Imagine an hour of 100 minutes. Then we could divide it into only half, fourth, fifth, tenth, twentieth, and a couple of others. There is less flexibility.
The ancient Babylonians used a base 60 system of enumeration for good reason. Why? It's all in the divisibility. Also, it made dealing with fractions a whole lot easier. BTW, handling fractions were a problem for all the ancient civilizations. Naturally, the multiplication tables were horrific. But they had tricks to make calculations easy (well, easier than memorizing all products x times y for x and y in the range from 1 to 59). Isn't it interesting that in the schools, the multiplication tables taught are for all products x times y for x and y in the range from 1 to 12. There is that number, 12, once again.
The year of 365 1/4 days, we can do nothing about. It is what it is. But note, we have conveniently divided the year into 12 months. Sure, the numbers of days differ a bit, but calculations are easy in this context. On the other hand, imagine a year with ten months. Divisibility is only possible into halves and fifths. This is the same as if the basic multiple unit consisted of ten items, just like the discussion about the dozen.
There are 360 degrees in a circle. This is 30 x 12. Thus a circle consists of 12, 30 degree arcs. So, twelve occurs again. And 360 has the ultimate number of divisors. We use them all: quarter circle = 90 degrees, half circle = 180 degrees, eighth circle = 45 degrees, and so on. All the directions, N, S, E, and W have multiple of 90 degrees; all the half directions, NW, SW, NE, and NW, likewise. All are on the compass with whole number degrees. Convenient, huh?
Fingers and toes. These are the basis of our monetary and (some of our) measurement systems. This means tens and twenties. It is an apparent explanation of one system in contradistinction to the more facile base 12, 24, and 60 systems of everyday counting. For Americans, don't forget the foot (12 inches) and the yard (36 inches). The mile is 1760 yards. Don't forget the usual divisions of the inch into 2, 4, 8, and 16 parts. Yet, your car’s odometer measures distance in miles and tenths of miles. Go figure. No wonder kids are confused. Geez, now even I'm confused.
In an unrelated note, today's date is 12/12/12. Lots of twelves. Won't happen again in your lifetime.
And today 12/12/12 is Craig's 68th Birthday. Divisible by 2, 4, 17, and 34!
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