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 Is Dozenal Only A Rational Optimum?, examining higher mathematical apps
Treisaran
Posted: Feb 7 2015, 07:51 PM


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It is asked in the DSA's Dozenal FAQs:

QUOTE
9. How if any would having a different system affect higher mathematics, above the fundamentals?


which the author, Icarus, answers:

QUOTE
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.


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.

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 50z 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 ω-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:

633d − 289d → 633d + 711d = 1344d → 344d

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 2²·3·5, and why the Romans used base 2²·3 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 20z and 30z 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 2↑2n), thirds and the multiples of the two groups (sixths and so forth). Now 1/2 and 1/3 can be translated to 0.6z and 0.4z respectively, their sum easily given as 0.Xz, their average as 0.5z (5/12d 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 (14z 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 π can be approximated as the rational numbers 3/1, 1X/7z, 257/95z, 50221/171X6z and so on. If we want integralised approximations, it is best to skip the intermediaries and choose the successive base fractions: 3, 3.2z, 3.18z, 3.185z, 3.1848z ... 3.18480949z, 3.184809494z. 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 n² and n³ 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↑0.4z rather than having to use the radical symbol, but beyond that, I get no impression that the number 0.4z 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 (10a+b)² and (10a+b)³ 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 300a² 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 10z and finding its antilog. The logarithm of 2↑4 is 1.14808z on my table (note that I've had to add the characteristic; the entry in the table is for 1.4z, and is comprised only of the mantissa, 14808z), then divide by 10z, giving about 11481z, the closest antilogarithm of which is 1.315z. This procedure uses the fact that 1/3 is 4/10z, 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

   +∞ ↑ +∞
      │
      │
   54 ┼ 6
      │
      │
      │
√1228 ┼ 5.6
      │
      │
      │
   28 ┼ 5
      │
      │
      │
 √368 ┼ 4.6
      │
      │
      │
   14 ┼ 4
      │
      │
      │
  √X8 ┼ 3.6
      │
      │
      │
    8 ┼ 3
      │
      │
      │
  √28 ┼ 2.6
      │
      │
      │
    4 ┼ 2
      │
      │
      │
   √8 ┼ 1.6
      │
      │
      │
    2 ┼ 1
      │
      │
      │
   √2 ┼ 0.6
      │
  √√2 ┼ 0.3
      │
    1 ┼ 0
      │
      │
      │
 √0.6 ┼ −0.6
      │
      │
      │
  0.6 ┼ −1
      │
      │
      │
√0.16 ┼ −1.6
      │
      │
      │
  0.3 ┼ −2
      │
      │
    0 ↓ −∞


The use of LNS, like the slide rule of old, is beset with the complexity of addition and subtraction, only without the remedy that the human operator might carry out those phases. Leaving that problem aside, which has nothing to do with the base chosen, this number line shows what is perhaps the greatest drawback of logarithmic scale arithmetic: only the base and its powers and roots are accurate. In binary logarithms, 2 and √2 and 0.6z are accurate, but 0.9z (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 (i²) is −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+3i. 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 θ + i sin θ)

and

e↑() = cos θ + i sin θ

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 (1,120°) to rectangular (−1/2,√3/2), meaning that the cube of −0.6+0.X485XXiz 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 π (or of τ = 2π) could be ended by using dozenal. The twin facts are sexagesimal degrees have held on and mathematicians use rational fractions of π 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.
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dgoodmaniii
Posted: Feb 8 2015, 04:37 AM


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This is some deep juju, Treisaran. Most of this will take some thought to form a real opinion.

Still, I do think dozenal is important even beyond basic math, though Arithmetic certainly contains its strongest advantages. The sciences, particularly the practical sciences like engineering, would benefit from a dozenal metric system especially. Architecture and the like, too; we've already heard icarus talk about how much having a dozenal system (feet and inches) helps him in his trade.
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Treisaran
Posted: Feb 8 2015, 06:32 AM


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QUOTE (dgoodmaniii)
his is some deep juju, Treisaran.


It ought to be. A lot of the time I was away from the board was devoted to getting up to scratch on the stuff I daydreamed through at high school. wink.gif

QUOTE
Most of this will take some thought to form a real opinion.


By all means take your time, you know I do ramble.

QUOTE
The sciences, particularly the practical sciences like engineering, would benefit from a dozenal metric system especially. Architecture and the like, too; we've already heard icarus talk about how much having a dozenal system (feet and inches) helps him in his trade.


But these benefits stem from the rational properties of dozenal, and engineers have been at the forefront of dozenalism from the onset. I was thinking about the pure mathematician this time; the folks who eat and drink and breathe complex numbers and partial derivatives and that sort of stuff.
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wendy.krieger
Posted: Feb 8 2015, 08:32 AM


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The great part of mathematics is rune-shuffling, (algebra, algorithms, etc) which does not resolve to numbers. The arts of arithmetic are now middle-school matter.

Fractions of various kinds do occur, and in my search for polytopes, i greatly eliminated square roots to make these calculations. The typical distances that one gets for a polytope of some shape is N/n, where N is the natural numbers, and n some fixed number, eg N/7. You get lots of interesting numbers from finding various radii^2, which tend to be N/n-N/m. When n and m are typically small, then base 120 serves this end well. This is the main attraction to me.

The trig functions are best served outside the radian, by the cycle, and then suppose that the angle is a pure fraction of occupied space. The polygloss has an entry advocating exactly this approach, since the prismatic edge of a p-gon prism is the same angle as the p-gon itself. In terms of known functions, the conversion of a number to a angle is fract(N) = N -int(N) eg 17.32 -> .32

The rationals do occasionally occur with irrationals, and here well-divised bases do better here. This is mostly because the irrational is from a cyclotomic number set (like the span of chords), and the rational comes from some polytope, like the simplex. So something like 1.61803398875, you can add 1/2 or 1/3, the former can be directly seen as 2,11803398875, the latter is 1.951367322. The same done in dozenal, gives 1.74EE67728, 2.14EE67728 and 1.E4EE6728.

The trouble to make this work is that you need large numbers of digits in any case. The numbers 2.5320888806 and 1.73205080757 both arise in the enneagon, but the difference very near 0.2 is disceiving. It really is not 1/5, because that number can not occur in this context.

One should not suppose that the optimum system exists with the particular fraction system in use. The sumerians made use of tables, however there is no direct record of alternating arithmetic. Fibonacci used added fractions, but makes no use of invoice-arithmetic, but relies heavily on reduction. Coupled with the right arithmetic, both alterating bases and added fractions are fine examples of how to greatly simplify the arithmetic.

Modern mathematicians speak of bases. Sumerian sixtywise, etc. But before bases, there was count-number, and division number. The different parts of the mind services these, leading to a greater variety of division numbers.

The sumerian sixty-mal is not a count-base, but a division-base, this means that the more digits do not make larger numbers but extended fractions, thus 1C is not 90 (1,30) but 1½ (1;30). The romans used a decimal count-base and a dozenalish division-base, the actual form is weights in the pound. The mayans and the greeks used mite-fractions, that is, eg 1944 parts, where 2000 make the foot, etc.
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Kodegadulo
Posted: Feb 8 2015, 09:44 AM


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1. The sum of the set of all natural numbers \(N\) is minus uncia, i.e., the negation (additive inverse) of the reciprocal (multiplicative inverse) of unqua:
$$\sum_{n = 1}^{\infty} n = -\frac{1}{10_z} = -0.1_z$$
If that seems inexplicable, understand that this finding appears to have applications in quantum mechanics and string theory, which are also inexplicable. Those are realms of physics where infinities threaten to blow up equations, yet inexplicably refrain from doing so, with the help of inexplicable results such as this. The universe is stranger than you think. smile.gif

2. The unqual Fibonacci number is biqua:
$$F(10_z) = 100_z = 10_z^2$$
It's also the only Fibonacci number (other than 1) to be a square.

3. Triqua appears in the formula for Klein's j-invariant for elliptical curves:
$$j\left(e^{\frac{1}{3}\tau\ i}\right) = 0,$$$$j\left(i\right) = 1000_z$$(where \(\tau = 2\pi\))
Theorems regarding elliptical curves were intermediate steps in Andrew Wile's proof of Fermat's Last Theorem. \(e^{\frac{1}{3}\tau\ i}\) is one of the cube-roots of unity. Triqua is of course the cube of unqua.

4. Triqua-et-uni (\(1001_z\)) is the Hardy-Ramanujan Number, or the smallest Taxicab Number. It is the smallest number that can be expressed as the sum of two cubes in two different ways:
$$1001_z = 1^3 + 10_z^3 = 9^3 + \operatorname{Ӿ}^3$$(where \(\operatorname{Ӿ} = 10_d\))
It's also the first Fermat Near-Miss Number, i.e., the first solution for the Diophantine equation \(x^3 + y^3 = z^3 + 1\). Analysis of Diophantine equations was also instrumental for Andrew Wile's proof.

Conclusion: There's something interesting going on with the number unqua and its powers. Perhaps mathematicians would catch tidbits like these more readily if they habitually saw numbers in unqual base. Does anybody know of any more similar tidbits?
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Treisaran
Posted: Feb 8 2015, 05:36 PM


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QUOTE (wendy.krieger)
The great part of mathematics is rune-shuffling, (algebra, algorithms, etc) which does not resolve to numbers. The arts of arithmetic are now middle-school matter.


Arithmetic may be learnt and finished learning in middle school, but it's applied by people of various professions, from shopkeepers to engineers. Apropos of engineers, it may be said in rough generalisation that (pure) mathematicians think up the equations and their solutions while engineers plug the numbers in those very equations.

QUOTE (wendy.krieger)
The trig functions are best served outside the radian, by the cycle, and then suppose that the angle is a pure fraction of occupied space.


I agree. While once I refrained from weighing on the pi vs. tau debate, saying it did not hold my interest, now it does - the more so as I learn about complex number trigonometry. I think π and τ both have their place. Whichever produces more clarity and elegance in the particular context is the best.

Regardless of that choice, the thing that slays me about complex numbers is this synthesis of arithmetic and geometry. Two complex numbers can be added or multiplied using trig functions. The relevance to dozenal is that thirds, sixths etc. of the circle are of inescapable importance, and here dozenal can play a decisive part. Imagine the unification of the circle's practical divisions (currently: sexagesimal degrees) and pure-mathematical ones (currently: divisions of pi radians written as vulgar fractions) under the banner of dozenal fractions everywhere. This bridge that the complex numbers lay between arithmetic and geometry would then be a seamless extension of high-school geometry!

QUOTE (wendy.krieger)
So something like 1.61803398875, you can add 1/2 or 1/3, the former can be directly seen as 2,11803398875, the latter is 1.951367322. The same done in dozenal, gives 1.74EE67728, 2.14EE67728 and 1.E4EE6728.

The trouble to make this work is that you need large numbers of digits in any case. The numbers 2.5320888806 and 1.73205080757 both arise in the enneagon, but the difference very near 0.2 is disceiving. It really is not 1/5, because that number can not occur in this context.


Irrational numbers have never been tamed the way fractions have been. You can find out that tan(0.06zτ) equals 2−√3 through equation solving or that it equals about 0.327024z through numeric analysis, but you can never get to the closed-form solution from the numeric one. This is in contrast to the interchangeability of vulgar fractions and base fractions (to a great part; even 0.2497z ought to be recognisable as 1/5).

Another predicament is that approximations are chancy: having rejoiced at √2 being so near 1.5z, and more so for φ being nearly 1.75z (this because, as Kodegadulo mentions, 100z is a Fibonacci number), e gives a disappointment: the rhythmic and memorable decimal 2.718281828d has to be matched in dozenal by the approximations 2.875236z or 2.875z. All depends on the factors in the continued fraction expansions, which are the same chance-and-necessity affair as the distribution of prime numbers.

Those two observations taken together explain why I don't think alternative number bases are of much (if any) consequence with the set of real numbers. Therefore, I think, if there's a venue for dozenal to attract interest in higher mathematics, I think it lies with the complex numbers, specifically in their intersection of arithmetic and geometry.

QUOTE (wendy.krieger)
The sumerian sixty-mal is not a count-base, but a division-base, this means that the more digits do not make larger numbers but extended fractions, thus 1C is not 90 (1,30) but 1½ (1;30).


Not quite. As Donald Knuth put it in The Art of Computer Programming, the Babylonians employed a floating-point notation with the exponents omitted - slide rule notation without a slide rule. Sexagesimal 1:30 could be interpreted as 60d+30d or as 1+30/60d as the context required. I actually like this notation, as I do the slide rule, but modern notation and technology have passed them by.

QUOTE (Kodegadulo)
2. The unqual Fibonacci number is biqua


IIRC, all the dozenal-power entries of the Fibonacci sequence have trailing zeroes in dozenal.

QUOTE (Kodegadulo)
Conclusion: There's something interesting going on with the number unqua and its powers. Perhaps mathematicians would catch tidbits like these more readily if they habitually saw numbers in unqual base.


It would be marvellous to find a systematic dozenal relationship in some areas of higher mathematics. As I said above, the existence of a round dozenal denominator in the continued fraction expansions of √2 and 0.6(1+√5)z (φ) is a matter of chance - chance that could be regularised through a sieve (cf. Eratosthenes for primes), but chance nonetheless. In contrast, the fact that dozenal divisions of the circle are important even in the trigonometry of complex numbers is a real boon for making dozenal inroads into higher mathematics.

By the way:

Although logarithms of negative numbers are undefined in the lin-log mapping line in my opening post, they do get a definition as (drum roll, please) complex numbers. For example, ln(−1) = iπ (an example in which π is handier than τ). This brings us to trigonometry once again. I wonder... perhaps a logarithmic number system encoding values in terms of dozenal fractions of the unit circle in the complex plane might give rise to an efficient numeric coding. There's so much to explore.
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Kodegadulo
Posted: Feb 8 2015, 07:09 PM


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QUOTE (Treisaran @ Feb 8 2015, 05:36 PM)
QUOTE (Kodegadulo)
2. The unqual Fibonacci number is biqua


IIRC, all the dozenal-power entries of the Fibonacci sequence have trailing zeroes in dozenal.

Indeed, that is a consequence of F(10z) being 100z. This winds up resetting the last two digits of the Fibonacci number to zero. Over the next 10z numbers, the initial pattern of the final 2 digits repeats in reverse, alternating between exact repetitions and 100z-complements, returning to another multiple of 100z at F(20z). Thereafter the cycle repeats indefinitely every 20z Fibonacci numbers:

[z]

\(n\) \(F(n)\)
0 0
1 1
2 1
3 2
4 3
5 5
6 8
7 11
8 19
9
Ӿ 47
Ɛ 75
10 100
11 175
12 275
13 42Ӿ
14 6Ӿ3
15 Ɛ11
16 15Ɛ4
17 2505
18 3ӾƐ9
19 6402
Ӿ2ƐƐ
14701
\(n\) \(F(n)\)
20 22Ӿ00
21 37501
22 5Ӿ301
23 95802
24 133Ɛ03
25 209705
26 341608
27 54Ɛ111
28 890719
29 121Ɛ82Ӿ
1ӾƐ0347
310ƐƐ75
30 5000300
31 8110275
32 11110575
33 1922082Ӿ
34 2Ӿ3311Ӿ3
35 47551Ӿ11
36 75882ƐƐ4
37 101214Ӿ05
38 176Ӿ979Ɛ9
39 2780Ɛ0802
432Ɛ885ƐƐ
6ӾƐ079201
\(n\) \(F(n)\)
40 Ɛ22045800
...


The fact that 100z is the only perfect-square Fibonacci number (other than 1) means that unqual is the only base in which this pattern of repetition occurs.
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wendy.krieger
Posted: Feb 9 2015, 01:40 AM


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The log-scale thing that Treisaran describes is pretty close to the model i use to describe numbers, weights and measures over history etc.

You suppose that the log scale runs from large left to tiny right. This matches how we read digits, the hundreds-digit is to the left of the tens digits. The line can now be regarded as a sliding rail, where lengths representing 10, 12, etc can be placed on like little rods. So a rod of 12 might represent at one end,a 'foot', and at the other end, an inch, thus |--12--|. Multiples of the foot go to one end, of the inch to the other end.

A measurement is then a series of 'digits' connected to these columns, thus a foot is |3 | | 4 | 8 |, when the vertical bars here are ten-wides, and the alignment is that seven further bars to the right make a meridian quadrant: that is, metric.

Of course, you have a positional place system with no radix: that is a float.

The actual range of any activity is over fewer places, so the english land measure rules might be supposed as |8| 40| (columns of 8 and 40, between mile, furlong, and rod. One can suppose there was some means of calculating in these numbers, since there is a variable column (a space), between the right of these and the foot. A woodland mile was 320 woodland perches, but the size of the perch is different to the statute perch of 16½ ft, it is nearer 24 of the same foot.

The variable space can be used to move the foot too, supposing that the statute perch is 16½ feet, it can be reduced to 15 to give drusian feet, or increased to 20 to give Pauli feet.

A measure might be represnted by a pin, defining a point or range. Some of these pins are attached to the log scale themselves, such as the seam and picul, which represent the loads suitable for a pack horse, or the oriental yoke or pole, on which two pans are carried. Neither of these change with base or system, and so one must suppose these pins are set on the scale itself.

Other pins are set at the little end of the sliding rods, so that if the rod moves, the size the pin represents moves too. Such pins are most likely to arise from the calculating devices, such as the third division of a unit, or the twelfth-deal or uncia. These pins move without reference to any particular size.

When one greatly regularises the scale of slides, such as putting alternating 4's and 5's, or 12's and 10's, or simple 10's or 12s, the means of calculations becomes easier, since the same tools are used for a great variety of units. It also greatly restricts the choice of places that the ends of these rods can come to rest at. For example, the repetition of 10's in metric lead to a series of fixed units, which either work or don't work. Any base has this limitation; the facination of binary divisions and systems, is that there are so many fixed points that finding one close to what is needed is much easier.

If a process (such as the roman weight-fractions) falls out of use, it may 'dump' pins onto the scale. So while in the roman weight-fraction, the dram is 1/96, when the uncia of 8 drachms ceases to be a way of writing fractions,the drachm comes to be a weight of something near 3 grams.

A division base, such as the sumerian, adds successions of units to the little-right, while a count base adds a succession of units to the large-left. So for example, to example with decimal, 125 as a count is X5 dozenally, because the common referent is the unit right. 125 as a division-number is 13 dozenally, that is 1¼, since the unit referrant is the 1 on the left. 1.25 d = 1.3 z

The notion that sumerian numbers represent a 'float' is misleading. The indian money system talks of lakhs (100,00,000), but these are a count of damns, rather as we 'float' dollars and cents to a long count of cents.

Similarly the sumerian numbers do admit a scheme where one might talk of a unit of 60 or 3600 in number, and then a series of divisions, but the division-names come from the calculator, and only the first unit is named. So we could write 125 in this system as 2 shocks, 5 minutes, and 1¼ as 1 and 15 minutes. The same scheme is used today, where we say degrees or hours as the units, and minutes, seconds, thirds, etc, are carried at the end of successive 60-rods.

Fibonacci numbers &c

One can accumulate, for any number, a list of 'interesting features'. There is quite an impressing list for 120, for example.

The fibonacci numbers are simply rep-units for j(-3), most of my algorithms make use of this. They're governed by the same set of rules that ordinary rep-unit and periods are, except that for j-bases, the initial period can be upper (divide p+1) or lower (p-1). The gaussian restriction correctly works on these bases too.

144 is nothing special here. The period of 1/2 is three places, and the orthogonal 2 still applies, so the six-place period is 1/8 and the 12 place period is 1/16. Likewise, 1/3 gives a four-place period, its square 1/9 gives 4*3 = 12 places.

The fibonacci numbers repeat over a fixed cycle for every number, and eventually, for every extra place of the number, that period by the number. For 10, the last digit repeats over 15 places, for 100, 150, for 1000, 750, and thence after for 10 times the previous.

On the other hand, an octagon of edge 13 is a special thing since no octagon number of either series is ever again square. Unlike 144, this comes from a sevenite. 31 is the other small sevenite in j(6).

The base that really does magic with the fibonacci series is 18, for not only being 6^phi, it is phi^6.

Complex numbers

One can easily demonstrate the complex relation that if r cis(a) represents a transformation on the plane, by a dilation by r,and a rotation by a, and this is represented by x+iy, then for all x,y, that i²=-1, without having to resort to taylor series or e^ix. It's actually harder to prove the latter statement than the former one.

One can see from the general proof that cis(a) is a proto-exponential form in that cis(a)+cis(B )=cis(a+B ), and that it is necessarily cyclic, ie there exists c such that cis©=1 and that cis(a+c)=cis(a). But to specifically associate c with e^(2pi) can not be done without using taylor series etc.

The measurement of circles by cycles, then immediately supposes that the c=1 (in some angle system), and that one derives a = fract(A), where A is the supplied number.

For the bulk of my calculations, i use fractions of a circle, or to some part, let C=34 as the need comes.
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Kodegadulo
Posted: Feb 9 2015, 02:07 AM


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Did anyone actually understand anything of what Wendy just wrote? It seems like a lot of disconnected gibberish to me.
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wendy.krieger
Posted: Feb 9 2015, 08:45 AM


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The Dynamics of Measurement

This diagram shows a logrithmic scale left (large) to right (small), and vertically high (human scale) to low (computing). It is a useful diagram to show how unit names etc work, and change over time.


CODE

Part 1

                 field
 HUMAN
                     bus                          hair
                      |                             |
  ----------km----hm----Dm-----m----dm----cm----mm----hµ----Dµ----
 
                                 ft      dgt
                                  |       |
                                  |       |
          mile--fur---rod -(v)-  ft----in----ln----pt
                                  3     3     4    6   = 1 metre    
                                  1     1     1    6   = 1/3 metre
                                  ------------------
                                  4     4     6    -   = 1 1/3 metre
                                  ------------------
          Surveyor's tables.       Builder's tables.
         
 Part 2        

           mile          ch     yd  hand  zoll  line              Named Units.                                    
           
            km    hm     Dm     m    dm    cm    mm    hµ   Dµ    Name power
            m3    m2     m1     m    im   iim  iiim   ivm   vm    Number power
                                |
            km                  m                mm               Comma Unit
                                |                                
                                m                                 One unit    
                                            1                     Number Only
   
 CALCULATOR                            



Part 1

The top end has names for all sorts of things, which people might be familiar with. One then says 'two bus-lengths', to give the reader an approximation of size. I've only shown two units here, but ye can put many more on it.

A unit descends if it is made into a measue, but continues to carry its previous meaning. So foot and digit are specific measures but retain their relation with body parts. Units that have a direct connection do not tend to migrate much in size.

Further down, you have measures that are names of multiples or divisions, that is, column-tables. The inch, line, point do not directly connect to real things like foot and digit, and are there because there is a table of twelfths there. The digit is 1/16 foot, does not map onto the dozenal foot scale.

The surveyors tables are a different set of units, which were established by lugging a rod (pole or perch), aroud and laying these end to end. Eventually a chain displaced this. The length of the rod varied according to the land, the idea being that a number of square rods were supposed to substain a family. This meant that in poorer country the rod offered more feet, but the furlong and mile remained the same.

Part 2

In part 2, we see the sae decade names, where units above each other are the same. The first row shows names dragged from their original place to fill the new role. This actually happened in the netherlands.

The next two rows show named powers, and powers with simple numeric names, like 'third-metre' = third disma of the metre'.

Engineers work similar to most folk, in that they prefer integers to fractions, so you are more likely to see these line up with the comma units.

The next step down is to suppose a single unit (eg the grafut is the tgm unit of length), and then below that, a raw numeric (in cgs units, it is 30,48).

The Dynamics

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Kodegadulo
Posted: Feb 9 2015, 11:06 AM


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Wendy, aren't your last two posts completely off-topic? Treisaran's thread here is considering the question of whether dozenal base is more than just good for everyday arithmetic, whether it offers mathematicians something useful for higher mathematics, and in particular for gaining insights about irrational as well as rational numbers. If you want to give us a dissertation on "the dynamics of measurement" I suggest you move your posts to a thread of your own. Besides, I don't see anything about dozenal in your last post, it seems to be all about decimal and SI prefixes.
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Treisaran
Posted: Feb 9 2015, 05:21 PM


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QUOTE (wendy.krieger)
The log-scale thing that Treisaran describes is pretty close to the model i use to describe numbers, weights and measures over history etc.


It was just one of the trails I tried going on in my search for a dozenal application in the set of real numbers. It didn't work.

QUOTE (Kodegadulo)
Did anyone actually understand anything of what Wendy just wrote?


QUOTE (Kodegadulo)
Wendy, aren't your last two posts completely off-topic? Treisaran's thread here is considering the question of whether dozenal base is more than just good for everyday arithmetic, whether it offers mathematicians something useful for higher mathematics, and in particular for gaining insights about irrational as well as rational numbers.


I do often get the impression reading Wendy's posts of having tuned out to a broadcast on a different wavelength... I've tried my best to understand her ideas and connect them with mine, but it isn't easy.

Anyway, if anyone manages to find some area of real number arithmetic where dozenal comes all into its own, or some other area in complex number arithmetic beside the one I've suggested (dozenal for complex trig), this thread would be just the place to post or at least introduce such ideas on.
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Kodegadulo
Posted: Feb 9 2015, 07:48 PM


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QUOTE (Treisaran @ Feb 9 2015, 05:21 PM)
Anyway, if anyone manages to find some area of real number arithmetic where dozenal comes all into its own, or some other area in complex number arithmetic beside the one I've suggested (dozenal for complex trig), this thread would be just the place to post or at least introduce such ideas on.

I wonder if it's possible to provide a mathematical proof that categorically shows that it is impossible for any rational base to impose a systematic pattern upon irrational or transcendental numbers? Perhaps that is already a corollary of Cantor's theorem, or some similar theorem.

Still, even if we must limit ourselves to the realm of rational numbers, I think there's plenty of fertile ground. The fact that dozenal-base is "smooth" with regard to the first two prime numbers (the only two primes directly adjacent to each other) must provide some advantages for dealing with prime numbers.

Your point about complex numbers is well-taken. Certainly anything that involves geometry of some kind could take advantage of the rational divisibility of dozenal base.

Music and mathematics have always been tied together, so the well-tempered chromatic scale being a dozenal division of binary logarithms might be an interesting avenue to pursue.
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wendy.krieger
Posted: Feb 10 2015, 06:51 AM


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The main place where i have found fractions useful, or a number system that handles these, is in the study of polytopes.

Here one might find (a + b sqrt©)/d where a,b,c,d are integers.

If one knows the simple multiples of sqrt©, it's easier to recognise the consist of the number as above in a base like dozenal than other bases (including twelfty), since b/d is likely to be an integer. An example is 2.894427190999, the last bit is 4 sqrt(5) = 8.94427191. One can see from this that this (as 2/sqrt(5)), is recoginisable because 5 divides 10.

One comes across examples like 1/3 + sqrt(3) etc in hyperbolic geometry, which is much easier to see in dozenal, since 1/3 = 0;4.

As i noted earlier, the real fun when you start to deal with the middle radii of the simplexes in higher dimensions, for which having twelfty by the side is a positive boon.

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Treisaran
Posted: Feb 10 2015, 04:19 PM


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QUOTE (Kodegadulo)
I wonder if it's possible to provide a mathematical proof that categorically shows that it is impossible for any rational base to impose a systematic pattern upon irrational or transcendental numbers?


I'm not good at proofs, only at conjectures, but I suppose there might be something with the fact that arithmetic hyperoperations change or shed features the higher you go (rather similar to the situation with the higher number fields like complex numbers, quaternions and octonions). Beginning with hyperoperation #0, the successor function, here's a table for summary:

Hyperoperation Form Construction Features
Successor function a + 1 N/A A unary operation
Addition a + b a + 1 + 1 ... + 1 for b times A binary operation; inverse (subtraction) can be calculated using the additive complement of the base for any b
Multiplication a·b a + a + a ... + a for b times A binary operation; inverse (division) can be calculated using the multiplicative complement of the base for numbers dividing the base or its powers
Exponentiation ab a·a·a...·a for b times A non-commutative operation; two inverses (logarithm and root), neither of which can be found by a trivial method
Tetration a↑↑b (a↑(a↑(a))...a) for b times A transcendental (non-algebraic) function


In a nutshell, once you go to hyperoperation #3, exponentiation, the relationship of the operation with the number base is lost. As I observed in the opening post: the 2nd and 3rd power and root operations no longer bring the numbers '2' and '3' into play as is the case when multiplying by 2 or 3 (in dozenal: can be carried out by dividing by their reciprocals 6 or 4 respectively).

QUOTE (Kodegadulo)
Still, even if we must limit ourselves to the realm of rational numbers, I think there's plenty of fertile ground.


I know, but I did so want to find something yet unseen, break some new ground...

QUOTE (Kodegadulo)
Your point about complex numbers is well-taken. Certainly anything that involves geometry of some kind could take advantage of the rational divisibility of dozenal base.


I wonder how many patterns could suddenly spring before one's eyes once dozenal is used for complex trig. The computer algebra system I use, Maxima, isn't dozenal, but manually converting to dozenal before and after doing the complex-number maths operations in it would be a good start in investigating it all.

My conclusion from all this business so far is that, although dozenal may not necessarily be only a rational arithmetic optimum, its fruitful use outside the set of rational numbers still depends on being able to carry over its rational-division advantages onto higher number sets. Quod erat demonstrandum the rational divisions of the circle (τ/n) for use in complex-analytic trigonometry. But like all early conclusions (and even some late ones), it's provisional.
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Sennekuyl
Posted: Feb 11 2015, 01:01 AM


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Can superior highly composite numbers form something like an Ulam spiral?

There was an article by a bloke (? William? Word<something>? not sure) that suggested SHCN were random, if I understood the article...
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Dan
Posted: Feb 11 2015, 04:16 AM


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QUOTE (Sennekuyl @ Feb 10 2015, 08:01 PM)
Can superior highly composite numbers form something like an Ulam spiral?

Interesting question. Here's what the first few HCNs (not necessarily "superior") look like on a square spiral.

user posted image
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Sennekuyl
Posted: Feb 12 2015, 05:26 AM


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That kinda does seem to be a spiral... Didn't think HCNs were random, just the superior kind.


Forgot to mention this bloke was arguing that SHCNs were as, if not more so, useful as primes. And due to its SHCN status placement in the human useful range of bases twelve is a better base than decimal. Anyway I'll try answer my own question.

EDIT: Can't seem to form complete sentences. :s
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Sennekuyl
Posted: Feb 15 2015, 04:05 AM


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To fill in the gaps of my vague allusions to a possible use of dozenal beyond basic arithmetic, I've found the article referred to earlier.

The bloke = Bill Lauritzen
The article = http://www.earth360.com/math-versatile.html

And yes, he states that versatile (SHCs) numbers are as random as primes. Therefore they might be useful for a dozenal cryptology?

My original post ran afoul of the x -> y question problem and the question about Ulam spirals was to demonstrate versatile numbers equivalence to primes. Not strictly a benefit of including/converting to a dozenal base but maybe somewhat simpler in dozenal than decimal... ?
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Ruthe
Posted: Jul 10 2015, 12:28 AM


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I understand that the original point of this thread was an investigation of the utility of an uncial system to higher level of mathematics than just simple arithmetic. However, I would make the point that why should we try to justify an uncial system to higher levels when it makes very little difference in such endeavors of higher mathematics.

The French Academy introduced the metric system based on the decimal system simply because it was the incumbent method when in practical terms, the number of people at that time were not highly numerate and would not have difficulty in learning a new system of numeration, and would also probably have found such a system of greater application in view of their usage of existing measures, particularly of length. Only those highly numerate classes would need to unlearn one system and learn a new one.

What was not envisaged by members of the academy was the numbers of people using the system of measure based predominantly on twelve. Twelve denier to a sou, 12 pouce to a pied, and dozens and grosses in France, and the parallel measures in Britain, Germany, Spain, Holland and Portugal to name the main states.

These people were used to dealing in dozens and their uncial parts and multiples. The French academicians were only intent on providing a system of use for their own scientific purposes. They gave no thought for the carpenter, baker, butcher, farmer, banker and other tradesmen and all their vast numbers of customers. People were used to dealing in feet and inches, shillings and pence, all with the disadvantage of a decimal numeral system.

The point I am making is that there is no need to search for additional reasons to convince mathematicians of the superiority of an uncial system, they work in the abstract areas where number is just a result of calculation using whatever immutable laws, methods and algorithms they devise. The billions of working people are the ones that were ignored when the metric system was devised. If you want an example, look no further than the costly experiment made by the California Department of Highways when they spent untold millions converting their systems and all their requirements documents to metric and forced suppliers to tender in metric, only to have to reverse the whole mess when forced by the multitude of suppliers to revert to the customary system of measures.

In that case they only returned to their original state, a non- cohesive system of measure based on a non standard set of measures and with a non- compatible system of decimal numeration. But at least it shows the level of demand by the blue collar segment of Californian workers. Had the rest of the world been given a say on the adoption of the metric system, I doubt it would have been accepted.

So let's not waste effort on trying to change the minds of that segment of the population that couldn't care less.

If you want a segment of people that would be worth convincing, go after the commercial sector. Find compelling reasons to change to an uncial system that will increase profits or reduce costs and it will happen.

Money Talks
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Treisaran
Posted: Jul 10 2015, 02:48 PM


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QUOTE (Ruthe)
I understand that the original point of this thread was an investigation of the utility of an uncial system to higher level of mathematics than just simple arithmetic. However, I would make the point that why should we try to justify an uncial system to higher levels when it makes very little difference in such endeavors of higher mathematics.


It was more a series of thought-experiments than a purposeful attempt at outreach to mathematicians, but now that the experiments are through, I agree with you. I answer the topic title question with an almost certain 'yes', and even in complex trigonometry the benefits of dozenal follow from its being a rational arithmetic optimum.

QUOTE
So let's not waste effort on trying to change the minds of that segment of the population that couldn't care less.

If you want a segment of people that would be worth convincing, go after the commercial sector. Find compelling reasons to change to an uncial system that will increase profits or reduce costs and it will happen.


I concur. Perhaps Icarus might disagree with the observation, but I see little interest in the exploration of alternative number bases on the part of the mathematical community. Most probably because mathematicians consider rational arithmetic (which is where number bases really make a difference, as I've shown in this thread) an elementary topic. I'd be happy to be wrong about this, of course, but I currently can't see a contrary case made.
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dgoodmaniii
Posted: Jul 10 2015, 07:16 PM


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I'm not so sure about this. I think dozenal will be a benefit much more to the engineers and the commoners than it will be to theoretical mathematicians, but I also think that dozenal could well wind up being the key to some long-disputed questions in higher math.

The sum of all positive integers being -0;1 is just one great example which shows that twelve may well plumb the depths of the nature of number much farther than we now realize. We know that six governs the locations of the primes, the nature and properties of which is still a fertile area of mathematical research; we know that the whole factors of the dozen (two, three, and four) are vitally important in plane geometry.

I'm not a theoretical mathematician. But while solving these problems certainly doesn't depend on using a dozenal base, it strikes me that saying dozenal wouldn't make the answers to these questions clearer is a bit premature.
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Ruthe
Posted: Jul 11 2015, 12:34 AM


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QUOTE
I'm not so sure about this.  I think dozenal will be a benefit much more to the engineers and the commoners than it will be to theoretical mathematicians, but I also think that dozenal could well wind up being the key  to some long-disputed questions in higher math.

I am not denying uncials may assist in those areas, but just saying let them find that out for themselves without wasting time trying to convince them. Your comments regarding primes is very relevant and I would support any further investigation that sheds light on them and have gut feeling uncials may help.
QUOTE
I'm not a theoretical mathematician.  But while solving these problems certainly doesn't depend on using a dozenal base, it strikes me that saying dozenal wouldn't make the answers to these questions clearer is a bit premature.

Again the same thing applies. I don't think and didn't intend to imply uncials should be ignored in solving these problems, only that the theoretical mathematicians are quite capable of of deducing the usefulness of uncials for themselves. But we don't have to shove uncials in their faces as it only tends to end in them denigrating the use of uncials and defending their precious decimals and in the process begin to make the debate a personal attack against proponents of uncials.

Let them, we have better things to do with our time, like seeking a commercial advantage of uncials in parallel with a coherent system of uncial weights and measures.

PS. I occurs to me that using the second as currently defined as a major foundation of such a system of measures would cause more problems once the human race begins to populate other bodies in our solar system that have wide variations in their unique rotation lengths. This would lead to a system of times similar to that of the totally chaotic standards of time with the first railways across the UK where each stop along a line would have their own version of local time making timetables impossible. This would also be made worse when considering units of mass and weight from asteroid to moon, planet to satellite and so on. Has anybody got any comments on this and if so, should it form a new thread?

PPS I seem to recall an article in one of the science publications ( New Scientist?), that there is evidence that the universe may be a 4D dodecahedron where if one could look far enough in one direction, they would eventually see the backs of their own heads. Funny where you find 12, 10u.
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Double sharp
Posted: Sep 30 2017, 05:14 PM


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QUOTE (Kodegadulo @ Feb 9 2015, 07:48 PM)
QUOTE (Treisaran @ Feb 9 2015, 05:21 PM)
Anyway, if anyone manages to find some area of real number arithmetic where dozenal comes all into its own, or some other area in complex number arithmetic beside the one I've suggested (dozenal for complex trig), this thread would be just the place to post or at least introduce such ideas on.

I wonder if it's possible to provide a mathematical proof that categorically shows that it is impossible for any rational base to impose a systematic pattern upon irrational or transcendental numbers? Perhaps that is already a corollary of Cantor's theorem, or some similar theorem.

Still, even if we must limit ourselves to the realm of rational numbers, I think there's plenty of fertile ground. The fact that dozenal-base is "smooth" with regard to the first two prime numbers (the only two primes directly adjacent to each other) must provide some advantages for dealing with prime numbers.

Your point about complex numbers is well-taken. Certainly anything that involves geometry of some kind could take advantage of the rational divisibility of dozenal base.

Music and mathematics have always been tied together, so the well-tempered chromatic scale being a dozenal division of binary logarithms might be an interesting avenue to pursue.

Incoming thread necro! laugh.gif

It depends what you mean by a systematic pattern. Liouville's number is defined as L = 0.110001000000000000000001000... in decimal, with 1's in the places that are perfect factorials (1, 2, 6, 24, 120, etc.) and 0's everywhere else. It is transcendental, but it is not hard at all to systematically determine all places of it.

The proof of transcendentality for this number is refreshingly straightforward, so here is one by David Feldman. Suppose p(L) = 0 for some polynomial p, which can be assumed without loss of generality to have integer coefficients. That means that p+(L) = −p(L) where p+ contains only the terms of p with positive coefficients, and p contains only the terms of p with negative coefficients.

Clearly, p+ and p cannot have the same degree. Without loss of generality, assume that p+ has a higher degree than p. (It goes the same way the other way round.) Let the degree of p+ be k, so that the leading term of p+(x) is cxk. Then the term p+(L) gives the places c10kn! for n = 1, 2, 3, .... When n gets large enough, the contributions from p cannot possibly catch up to those of p+ and cancel them out. But this is a contradiction, since we defined them to do just that. So there is no such polynomial p in the first place and L is transcendental.
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