From: <philtill@aol.com>

Date: Tue Oct 09 2007 - 15:22:12 EDT

Date: Tue Oct 09 2007 - 15:22:12 EDT

Phil,

All these speculations about base-7 arithmetic are fascinating, but is there really any evidence that any past civilisations used base-7 (apart from it making the patriarchal ages work).? A quick perusal of Wikipedia (not the found of all knowledge, I admit) shows that the commonly used bases are 2,5,8,10,12,16, 20 and 60.? There is an entry on "septenary" arithmetic that gives no historical connections (apart from use in a computer game!).? The big problem with it is that few fractions in "decimal" ("septimal") notation can be expressed other than by infinitely recurring digits ( e.g. 1/2 has a problem). Only when the demoninator is a power of seven is this possible.

Generally, number bases with lots of factors are preferable (which is why 60 was used by the ancient Babylonians).

Best wishes,

Iain

The strongest evidence for base-7, IMO, is census data in Numbers.? It is a 1 out of about 500,000 probability that only 7 contiguous numerals would be used in that many digits if it were by chance.

The factoring issue is a good observation but were the Hebrews or other ancients really that logical about it?? The most common Mesopotamian system was evolved and selected out from many divergent?systems that existed in the various city-states, and so it was probably not so intentionally designed as much as evolved.? If the Hebrews did design to use a base-7 (to set themselves apart from other nations), then it may have inherently been less useful than a designed system, since they were not so adept at arithmetic as we are and would not have foreseen the possible problems.? If they discovered that it didn't work very usefully in subsequent years, then that could explain why it was dropped before the census of David occurred.? They may have eventually adopted the base-10 we see in the Davidic census because that was the Egyptian system and they had close contact with Egypt.

Even our (and Egypt's) base-10 fails the factoring test.? It only factors by 2 and 5, so it is marginally better than a base-5 system which is no better than?base-7.? 10 and 5 feel like important numbers to us only because we are used to them in our number system.? They have no real advantages.? Why did the Egyptians?pick 10?? Maybe from the number of fingers, but that neglects the?zero numeral.? Our hands are actually a base-11 system if you include zero, since zero = "no-fingers" plus the 10 fingers you can put up makes a total of 11 numerals and that is base-11, not base-10.

One hand alone represents base-6, not base-5,?for the same reason.? If your second hand represents the next higher digit, then you can count up to 35 (6^2-1) with two hands.? You can use tokens to represent the next higher digit (units of 36), and Earle in his book speculated that this is how the Mesopotamian system with its use of 6's evolved.? It is a more natural finger-counting base than 5 or 10.

Phil

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Received on Tue Oct 9 15:37:43 2007

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