# Number Base Comparisons

CRGreathouse
Homework Helper

Factors and Prime Factors: octal has two factors, one prime; decimal and two factors, both prime; dozenal has four factors, two prime.

We must also consider that binary multiplication and division is the simplest and most easily attempted form of all; easier than tenery or decimal or any other kind.
We've also to consider that arrangement by six is actually the most efficient method;
That sixty-four - the square of eight- is the first cubic and square number;
That twelve is the first abundant number;
And a host of other things (hopefully this has gotten the ball rolling...)
Base 12 has an easy test for divisibility by 2, 3, 6, and so forth: just look at the last (few) digit(s). Base 10 likewise has an easy test for divisibility by 2, 4, 5, etc. Bases 8 and 16 give only rules for 2, 4, 8, and so on, more restricted than both the preceding.

Base 12 has an easy test for divisibility by 11: sum the digits and test for divisibility by 11. Base 10 has a similar rule for divisibility by 3 and 9. Base 8's rule is for 7. Base 16 gets 3 and 5.

Base 12 has an easy rule for divisibility by 13: sum alternating digits and test for divisibility by 13. (12A -> A - 2 + 1 = 9, so 12A is not divisible by 13.) Base 10's rule is for 11, and base 8 gets 9 (and so also 3). Base 16 gets 17, which is less useful (but overlooked, so maybe good).

Base 1000 (and so base 10 by extension) has a great rule for divisibility by 7, 11, and 13: sum alternating digits (since 11*13*7 = 1001). I don't know of a similar rule for bases 8, 12, or 16 -- but this might be worth exploring.

Base 10 has a good binary conversion: 2^10 ~= 10^3. Hexadecimal has a similar rule, but not quite as nice: 16^5 ~= 10^6. The others don't have anything obvious.

What bothers me about the British sozenal system, IS THAT'S ABOUT WHAT IT WAS WHEN I WAS IN ENGLAND. They were on the 12 pence to a shilling system.

WE ARE ON THE METRIC SYSTEM and 12 pence to the shilling makes accounting more complicated.
Not in base 12 it isn't and that is what was proposed, not accounting with 12 pence to a shilling in a base 10 number system.

Is it so difficult to understand that the proposed system would have a numeric radix of 12 and a system of weights and measures that use the same radix for ALL magnitudes in the same way the metric and S.I. systems use 10. That the existing system works for the scientific community is NOT in dispute, but it does NOT satisfy the requirements of the general user. The preponderance of 12 as a basis for many non metric measures that evolved from actual use and its applicability to the purposes to which it was applied, is prima facia evidence for the superiority of 12 over 10 as a practical unit.

To put it simply, the non-scientist user likes a simple means to divide things and twelve does it better than 10! If it didn't, the use of 12 as a grouping value would have fallen out of use with the introduction of the Hindu-Arabic decimal place value system.

As for accounting, with a 12 month year, an annual rate is easier to convert to a monthly rate in base 12. Furthermore, even with the existing partially duodecimal subdivision of time, calculation of wages based on halves, quarters, thirds, sixths and even eighths and ninths of and hour, minute or second results in far fewer occasions for rounding due to non-terminating non integer values.

In conclusion, I do not see why the use a base 12 metric system would be such a problem for technologist and scientists. Not only are all the existing facilities of a decimal system duplicated, but they are extended by using a radix of 12. The previously noted greater divisibility is supplemented by a finer graduation in measures and a reduction in the number of digits needed to denote values. There are even more advantages of base 12 vs 10 but this is not the place to present the complete picture.

CRGreathouse
Homework Helper
As for accounting, with a 12 month year, an annual rate is easier to convert to a monthly rate in base 12.
That's not a valid argument, unless you're saying that the cost of converting to a 10-month year is too great.* If you're going to assume that weights and measures will be redone to accommodate base 12, you need to make the same assumption for base 10.

* You don't want to argue this, because the cost of converting other measures to base 12 would be greater than the cost of changing the number of months. You should stick to arguing the long-term benefits of both.

That's not a valid argument, unless you're saying that the cost of converting to a 10-month year is too great.* If you're going to assume that weights and measures will be redone to accommodate base 12, you need to make the same assumption for base 10.

* You don't want to argue this, because the cost of converting other measures to base 12 would be greater than the cost of changing the number of months. You should stick to arguing the long-term benefits of both.
Not at all. I just don't see the point of changing the calendar to a decimal format. The whole idea of changing to base 12 is to simplify arithmetic. Changing the calendar to decimal just makes things worse. This is not to say I don't underestimate the cost of changing to a base 12 system, or even that it would ever be practical. This is a discussion (or so I thought) on the theoretical benefits of a completely duodecimal basis of numeration, weights and measures, calendar and time. The practical issues are a completely separate topic, but all I ever hear in return is evasion and obfuscation.

CRGreathouse
Homework Helper
Not at all. I just don't see the point of changing the calendar to a decimal format.
A claim was made that base 12 makes working with months easier. Assuming for the sake of argument that this was true, I pointed out that a decimal calendar is easier in a decimal system and that adopting such a calendar would be easier than adopting base 12 everywhere base 10 is now used.

NThis is a discussion (or so I thought) on the theoretical benefits of a completely duodecimal basis of numeration, weights and measures, calendar and time. The practical issues are a completely separate topic, but all I ever hear in return is evasion and obfuscation.
I thought I addressed the benefits of bases 8, 10, 12, and 16 reasonably. But where on this thread are you hearing evasion and obfuscation?

As far as conversions between bases go: is there any advantage representing a number as a vector in an n-th dimensional vector space?

So, say I wish to represent the number 73 in base 10, it'd be $$73=7\times \vec{v}_1+3\times\vec{v}_0$$ where $$\vec{v}_n=10^n$$, so its representation is on $$\mathbb{Z}^2$$ lattice.

The equivalent representation in base 2 is 1001001, so as a vector, it'd be $$73=1\times \vec{w}_6+1\times\vec{w}_3+1\times\vec{w}_0$$ where $$\vec{w}_n=2^n$$, so its representation is a point on a $$\mathbb{Z}^6$$ lattice.

Is there an obvious relationship between the vector in both spaces? I would be very interested in some feedback!

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I don't think the vector space paradigm applies. A basis of a vector space is made of independent vectors; while, in your binary example, $w_n = 2 w_{n-1}$.

Edit: oh, that was possibly a stupid remark of mine. You mean a vector space over the trivial field {0,1}. Then the question is, over which fields you mean, when using a base greater than 2? Since, for non-prime bases, I think, modular arithmetic does not form a field.

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I don't think the vector space paradigm applies. A basis of a vector space is made of independent vectors; while, in your binary example, $w_n = 2 w_{n-1}$.

Edit: oh, that was possibly a stupid remark of mine. You mean a vector space over the trivial field {0,1}. Then the question is, over which fields you mean, when using a base greater than 2? Since, for non-prime bases, I think, modular arithmetic does not form a field.
That has cleared the picture up considerably - I was aware of the connection between the "unit vectors" and was more concerned with $v_n v_m= v_{n+m}$ idea.

With regards the underlying field in each case; it seems a little much to use a different field for each conversion and expect it to go smoothly.

Just for fun...

Convert the number CAT36 to base 35

What do you get?

HallsofIvy