which the author, Icarus, answers:9. How if any would having a different system affect higher mathematics, above the fundamentals?

That question and its answer was in great part my reason for doubting whether much good would be served by getting mathematicians interested in dozenalism. Since most mathematicians are not interested in such basic things as elementary arithmetic or even numbers for the sake of numbers, said I, then why would they consider dozenalism (or any alternative base, for that matter) worthy of their time? There are pointful answers to that objection of mine, but I've been questioning my initial assumption: that dozenal has no appeal beyond the basics. Here I will survey the various number sets and arithmetic operations and the way dozenal does or does not bring benefits to their use.Using a different number base would have a limited effect on higher mathematics. Arithmetic is different, but operates using the same rules. Constants would need conversion. Calculus would not be any different.

**Natural Numbers and Integers**

The natural numbers begin from 0 (or 1, according to preference; I prefer to go by Peano's axioms) in a succession: 0, 1, 2, 3,... They satisfy addition but not subtraction, which may yield numbers out of their range. The integers include the negative numbers as well, thus bringing both addition and subtraction into closure.

As far as I can tell, the positional bases are equal in their utility for integer arithmetic just so long as they're a fair size, i.e. not too small like binary nor too large like base 50

_{z}using a discrete symbol per numeral as in argam. That is to say, if one is limited to addition and subtraction, there is no great pressure to prefer dozenal over decimal; the operations are neither eased nor hampered by the choice of the base. Also, whatever the base, subtraction of

*any*number from another can

*always*be performed by taking the base-complement and adding. This is easily done by substituting every digit of the subtractor with its Ï‰-complement (the number to be added to it in order to get one less than the base), for example the nines' complement in decimal or the elevens' complement in dozenal, then add 1 to the result. Thus:

633

_{d}âˆ’ 289

_{d}â†’ 633

_{d}+ 711

_{d}= 1344

_{d}â†’ 344

_{d}

Multiplication too is defined for both natural numbers and integers. Although it could be executed as repeated addition, various algorithms, most of them relying on a memorised multiplication table, make easier work of this process. Here the base most certainly does matter, for the multiplication table changes according to the base. Those of large bases are unusable, while too small a table fails to help on account of its triviality; the best bases for multiplication are those that have compact and

*patterned*multiplication tables. The decimal multiplication is compact; the dozenal, only slightly less so, and this is offset by its rich set of patterns for all but two rows (those of 5 and 7).

**Rational Numbers**

Division is not defined for all integers; it is under rational numbers that it is closed.

Common (or vulgar) fractions, though considered less advanced, are actually the mathematically pure way of representing rational numbers - numbers expressed as ratios of two integers. A half is 1/2, a third is 1/3, two thirds is 2/3, and two (an integer rational) is 2/1. Only a denominator of 0 is undefined. In contrast, base fractions (decimals, dozenals and so on) are

*integralisations*of rational fractions, a form in which they can be handled using the laws of integer arithmetic.

To add 1/2 and 1/3 directly is not straightforward: the fractions must be brought to a common denominator before their numerators can be added. Thus 3/6 and 2/6 are added, giving 5/6. As well, comparison can become difficult. Base fractions do away with the difficulties of rational arithmetic, at the cost of losing the ability to represent most fractions accurately - all those whose denominator is not a prime factor of the base. In like manner, division can be achieved by multiplying the reciprocal and shifting the product; which numbers one can divide by depends on the prime factorisation of the base.

I contend that rational arithmetic has always been the reason for choosing an alternative to decimal. The irrepresentability of thirds (of any fraction with the prime 3 in the factorisation of the denominator, in fact) in decimal is the reason why the Babylonians clung to base 2Â²Â·3Â·5, and why the Romans used base 2Â²Â·3 for representing the fraction-parts of numbers while keeping to decimal in the integer-parts of the same.

Those two choices, and others such as base twelve-on-ten and base 6, all come at their prices: the Babylonian usage made tables a constant necessity; the Roman numeration introduced a discrepancy between the integralisation of fractions to the handling of the integers; twelve-on-ten requires getting to term with alternating-radix arithmetic, dealing with two bases in one; base 6 is far too small; and bases 20

_{z}and 30

_{z}are too big.

It could be said that dozenalists prefer dozenal to decimal for the same reasons that the ancient Babylonian sexagesimals did theirs, but with the additional requirement of not having any complications in arithmetic beyond those they are already familiar with in decimal. At its size, dozenal preserves the familiar post-Stevin arithmetic of decimal intact. The only price incurred is the shunting of the prime 5 to the wayside. Given the lesser importance of 5 in contrast to 3, and the fact that some ways to alleviate the loss exist (such as a workable divisibility test and fraction approximations using the set {25, 4X, 73, 98}), this is probably not too bitter a pill to swallow.

With dozenal, the integralisation of rational numbers brings all the most important fractions under its wing: binary powers in a representation more compact than that of decimal (one digit per each 2â†‘2n), thirds and the multiples of the two groups (sixths and so forth). Now 1/2 and 1/3 can be translated to 0.6

_{z}and 0.4

_{z}respectively, their sum easily given as 0.X

_{z}, their average as 0.5

_{z}(5/12

_{d}as decimalists have to make do with).

The reciprocals of the base also form a relatively dense set: {2,6}, {3,4}. Doubling and halving can make for multiplication and division by other 3-smooth numbers: for example, halve a number twice to multiply it by 3, or double a number four times to divide it by 9 (14

_{z}is the dozenal reciprocal of 9).

The rational properties of dozenal, including but not limited to fractional representation, reciprocals and snap-in intervals, make dozenal the best rational base. It is its excellence in integralising rational numbers that make it such a good choice. It seems to me safe to say that most advantages of dozenal a dozenalist can think of are related to rational arithmetic.

But can we go beyond that?

**Real Numbers**

While the set of rational numbers encompasses all the others as shells within shells, getting to the real numbers is not so simple. The irrational numbers, whether algebraic (roots of polynomials) or transcendental, are hard to numerate; ever since ancient Greek times, geometry rather than arithmetic has more often been used in the attempt to construct them. Continued fractions, Taylor series, Dedekind cuts and Cauchy sequences are just a few of the methods used in defining them; the first two are numeric, the last two are geometric, but all have in common that they are endless, approximations to which the irrational number in question is the limit.

Of the extand methods,

*continued fraction expansions*are the successive rational approximations of the irrational numbers. Thus Ï€ can be approximated as the rational numbers 3/1, 1X/7

_{z}, 257/95

_{z}, 50221/171X6

_{z}and so on. If we want integralised approximations, it is best to skip the intermediaries and choose the successive base fractions: 3, 3.2

_{z}, 3.18

_{z}, 3.185

_{z}, 3.1848

_{z}... 3.18480949

_{z}, 3.184809494

_{z}. No matter the base, approximation is all we can hope for.

Powers and roots, being algebraic, seem to be the closest we can stay to straightforward arithmetic. It was here that I first tried to find out how dozenal might be of help; I ended up with a few sobering insights.

While

*n*Â² and

*n*Â³ do really mean 'multiplied by itself two times' and 'multiplied by itself three times' respectively, the feel for the numbers 2 and 3 here is technical, not intimately tied to the base as it is in rational arithmetic. There is no hint that the 'twoness' or 'threeness' of the exponent matters when doing numeric calculations. Similarly, dozenal lets you denote cube roots as

*n*â†‘0.4

_{z}rather than having to use the radical symbol, but beyond that, I get no impression that the number 0.4

_{z}is of any significance. For, while you can always subtracting by adding (the base-complement), and for a subset of numbers you can divide by multiplying (the reciprocal), there is no equivalent way of taking the root through exponentiation. (I shall discuss logarithms later.)

Irrational roots can be pried by various methods, such as Newton's method (based on calculus, though known already to the Babylonians for the case of square roots) and binomial completion. The latter method involves algebra, namely the use of the binomial expansions for (10

*a*+

*b*)Â² and (10

*a*+

*b*)Â³ to extract digit by digit of the square and cube root, respectively. I've given this method in my post on dozenal quarter-squares, with a few links for more elaboration; this is an opportunity to add two more interesting links: 1, 2.

As the post on dozenal quarter squares shows, I

*have*found a dozenal application for this area of irrational arithmetic: by using the reciprocal relationship {3,4}, the extraction of cube roots can be made tolerable when one looks up the term 300

*a*Â² easily in the quarter/triple squares table. However, once the euphoria had worn off, I needed to be honest with myself: how frequently would the application be put to use? Not much, I'd have to admit; certainly nothing near the rational arithmetic utility of dozenal. Even I would extract a square or cube root manually only when actually studying roots, not when needing to calculate a root as part of another procedure. An application it is, but too esoteric to matter.

Another method of extracting roots is by using logarithms. I have found that a cube root can be painlessly found on a dozenal logarithm table by taking the logarithm of the fourth power of the number, then dividing the log by 10

_{z}and finding its antilog. The logarithm of 2â†‘4 is 1.14808

_{z}on my table (note that I've had to add the characteristic; the entry in the table is for 1.4

_{z}, and is comprised only of the mantissa, 14808

_{z}), then divide by 10

_{z}, giving about 11481

_{z}, the closest antilogarithm of which is 1.315

_{z}. This procedure uses the fact that 1/3 is 4/10

_{z}, which is particular to dozenal and cannot be employed in decimal. But again,

*how much*is this of use? People don't use logarithm tables today; scratch that, then, as a useful application in dozenal for real numbers.

Then I mused further: so logarithm tables are no longer in use, but logarithms in general still are. In fact, there has been among the alternatives proposed to floating point in computers the use of (binary) logarithms: LNS or logarithmic number systems. I wondered to what extent the base influenced the efficiency of such numeric representation. Here is a vertical number line mapping linear numbers to their binary logarithm equivalents (all the numbers are given in dozenal):

Code: Select all

```
+âˆž â†‘ +âˆž
â”‚
â”‚
54 â”¼ 6
â”‚
â”‚
â”‚
âˆš1228 â”¼ 5.6
â”‚
â”‚
â”‚
28 â”¼ 5
â”‚
â”‚
â”‚
âˆš368 â”¼ 4.6
â”‚
â”‚
â”‚
14 â”¼ 4
â”‚
â”‚
â”‚
âˆšX8 â”¼ 3.6
â”‚
â”‚
â”‚
8 â”¼ 3
â”‚
â”‚
â”‚
âˆš28 â”¼ 2.6
â”‚
â”‚
â”‚
4 â”¼ 2
â”‚
â”‚
â”‚
âˆš8 â”¼ 1.6
â”‚
â”‚
â”‚
2 â”¼ 1
â”‚
â”‚
â”‚
âˆš2 â”¼ 0.6
â”‚
âˆšâˆš2 â”¼ 0.3
â”‚
1 â”¼ 0
â”‚
â”‚
â”‚
âˆš0.6 â”¼ âˆ’0.6
â”‚
â”‚
â”‚
0.6 â”¼ âˆ’1
â”‚
â”‚
â”‚
âˆš0.16 â”¼ âˆ’1.6
â”‚
â”‚
â”‚
0.3 â”¼ âˆ’2
â”‚
â”‚
0 â†“ âˆ’âˆž
```

_{z}are accurate, but 0.9

_{z}(3/4) is not, because 3 is not a rational power of the base. Even more devastating to the attempt to connect a particular base to LNS is that prime factorisation is inextricable in an exponential space: that is, the powers of 2, 3 and 6 are all disjunct, even though 6 is the product of 2 and 3. In exponentiation, the prime factors of the base are tightly bound together like protons in an atomic nucleus, and acquiring or removing factors, the analogue of nuclear fusion or fission, if possible at all, will not likely yield the desired result. Dozenal logarithms, in and of themselves, give no advantage, for they do not have rational representation for numbers such as 3 and 4, only for the rational powers of dozen.

I'm quite sure my investigation of real number application has been far from complete, but even this brief foray gives me the sobering hint that real arithmetic is base-agnostic. I'd be glad to be proved wrong about this. It stands to reason that exponential operations cannot leverage the number base, as the number to be raised to a certain power

*is*the base. As for transcendental numbers and functions (trigonometric functions and such), I reasoned that the utility of bases for them could lie only in the divisions of the circle, which brings us back to rational arithmetic.

Or does it?

**Complex Numbers**

Algebraic closure was finally achieved by conceiving the imaginary unit,

*i*, whose square (

*i*Â²) is âˆ’1. All equations have a solutions once complex numbers are used, even those that have none when using reals. A complex number consists of a real part plus an imaginary part, such as 2+3

*i*. In the geometric interpretation, if a real number is a point on a horizontal line, then a complex number is a point on a plane whose x-axis shows the real part and y-axis the imaginary part.

It is in the unification of arithmetic and geometry where complex numbers have enabled one of the greatest mathematical syntheses. Euler and De Moivre bridged the gap that had lain between arithmetic and geometry since ancient Greek times. By mapping the complex plane between rectangular and polar coordinates, trigonometry could be used to perform arithmetic. With the following identities:

*a*+

*bi*=

*r*(cos

*Î¸*+

*i*sin

*Î¸*)

and

*e*â†‘(

*iÎ¸*) = cos

*Î¸*+

*i*sin

*Î¸*

complex analysis could be carried out using trigonometric functions. For example, the cube roots of 1 (unity) could be found by converting the thirds of a unit circle to rectangular coordinates. A scientific calculator will convert the polar (1,120Â°) to rectangular (âˆ’1/2,âˆš3/2), meaning that the cube of âˆ’0.6+0.X485XX

*i*

_{z}is 1. Such computations seem far-fetched to the general public, but they are used for analysis in the sciences all the time.

At this point the discussion requires expert opinion, meaning not mine. I can, however, venture to say that the rational advantage of dozenal could extend to complex arithmetic if exact divisions of the circle are needed. The dichotomy between sexagesimal degrees and rational fractions of Ï€ (or of Ï„ = 2Ï€) could be ended by using dozenal. The twin facts are sexagesimal degrees have held on and mathematicians use rational fractions of Ï€ because of the single reason that fractions of the circle with the prime 3 in the denominator are so very important. If so, dozenal might be as useful for higher mathematics in the field of complex numbers as it is in elementary arithmetic in the field of rational numbers.

**Conclusion**

The survey first laid out the mathematical rationale (no pun intended) for the importance of dozenal that the man in the street can relate to: rational arithmetic, the arithmetic of division. After that, I investigated whether dozenal might be of such good use in the higher fields. My conclusion is that real arithmetic gains no practical advantage from dozenal (or any particular base), while complex arithmetic might do so through the dozenal representation of circle divisions in the use of trigonometry. My answer cannot be final, of course, as I lack the full information to judge this. Feedback is therefore welcome. It would make me happy to agree with Icarus that there is a solid case for taking dozenal to mathematics beyond the elementary level.