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Title: the long hundred


Ebbe - April 13, 2007 12:46 AM (GMT)
I am currently reading an interesting book about historical number and number writing systems, and how the can be infered from the various languages.

There I came across the curious old scandinavian use of the long hundred: During the Viking Age, the Old Norse word "hundraş" usually meant 120, not 100. So the year had approximately "tre hundraş" days. There was also (less common) a long thousand, meaning 1200.

The long hundred was also in use in Britain for cloth, nails and herrings.

The author gives two reasons for the use of the factor twelve: The well-known divisibility, but also the custom of the allowance, like 12=10+extra 2, or 120 = 100+extra 20.

So, while their number system neglected the dozen (the 120 was counted as twelve tens, not ten dozen, which would also rather be called "small gross"), its "big brother" saw widespread use.
There should be also noted the more German counting measure "Schock" (translated "shock", but here meaning 60), 5 dozen.

I want to add that 120 is also a nice number base for divisibility, although it requires a sub-base. The smallest integers up to 18 all have "nice" reciprocals (here 10=ten, T0=short hundred, E0="eleventy"):
1/2 = 0.60
1/3 = 0.40
1/4 = 0.30
1/5 = 0.24
1/6 = 0.20
1/7 = 0.17 recurring
1/8 = 0.15
1/9 = 0.13.40
1/10 = 0.12
1/11 = 0.10:T9 recurring
1/12 = 0.10
1/13 = 0.09:27:83 recurring
1/14 = 0.08:68 (68 recurring)
1/15 = 0.08
1/16 = 0.07:60
1/17 = 0.07 recurring
1/18 = 0.06:80
1/19 = 0.06:37:T7:44:25:31:69:56:T1 recurring
1/20 = 0.06
That is due to the fact that 120 is not only divisible by 2, 3, 4, 5, 6, 8, but also its neighbours by the next primes 7, 11 and also 17 (119 = 7*17, 121 = 11*11).

In base 120, Pi = 3.16:E8:E2:12:77..., which can well be approximated by 3.17. In fact, that's Ptolemy's approximation, accurate to almost 4 decimals; he used, of course, the sexagesimal variant 3.08:30.

icarus - April 13, 2007 02:00 PM (GMT)
120 is part of the sequence:

2 2
6
12
60
120 <-----
360
2520
5040
55440
720720...

which seems to yield numbers with the highest number of divisors compared to the integer's size. It is one of the optimum bases. In addition to its happiness with 7, 11, and 17, it is the marriage of decimal and dozenal, and thus could serve as a mixed radix, where every other digit could be intercalated decimal and dozenal.

Dan - July 22, 2007 08:32 PM (GMT)
QUOTE (icarus @ Apr 13 2007, 08:00 AM)
120 is part of the sequence:

2            2
6
12
60
120 <-----
360
2520
5040
55440
720720...

What sequence is that? I recognize the numbers 2, 6, 12, 60, 360, 2520 as Lauritzen's "peak numbers" (an integer n such that no positive integer less than 2n has more factors than n), but the others aren't.

uaxuctum - July 22, 2007 10:22 PM (GMT)
It's the sequence of superior highly composites, of course.

The Mighty Dozen - July 22, 2007 11:26 PM (GMT)
If you're interested, this 120 long hundred existed into Old English, too. You'ld say "tenty", "eleventy" (probably the origin of Tolkien's "eleventy-first", by the way, given he was a scholar of Old English), and so on.

icarus - September 17, 2007 02:05 AM (GMT)
in response to dan the online encyclopedia of integer sequences a002201 lists the superabundant numbers, if i am correct, but a004490 is the colossally abundant; i think this sequence is the better, but these are identical for the first several. Sorry for overlong reply, we had been dealing with DAJ.




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