Dozenal Fifths Better than Decimal Thirds

Sevenths
The prime number seven is not a factor of ten. Consequently, the fraction a seventh written in decimal form does not terminate. The digits after the decimal fractional jot repeat after cycles of six digits for sevenths. The maximum number of digits in the representation of a unit fraction 1/p that is the reciprocal of a prime number p by positional notation of some base B before repetition of the cycle is p - 1, which is one less than the prime number. In decimal a seventh has the maximum number of digits that it could have before repetition.

Sevenths rarely need to be encountered. It could be suspected that if the average person on the street were to be surveyed, most would not know the full cycle of digits in decimal positional form of sevenths, without a massive publicity campaign. This would be evidence that sevenths are really not that important, although the fact that sevenths are not expressible succinctly in decimal positional notation could be the very reason for them being avoided. A seventh is probably most commonly encountered as an approximation 0.14 by two significant figures to the fractional part of the constant pi, the ratio of the circumference of a circle to its diameter.

In dozenal, the number of numerical characters before repetition of the cycle in the positional notation of the fraction a seventh is the same as in decimal. Elsewhere, others have supposed that this implies that dozenal is just as bad at representing sevenths as decimal is. However, the absolute error between the dozenal positional representation to two significant figures of a seventh and an exact seventh is, written decimally, 21/144 - 1/7 = 1/336, whereas in decimal positional notation the error is 7/50 - 1/7 = -1/350. Since the error is less in decimal, this means that to two significant figures, dozenal is actually worse than decimal at representing sevenths. From this can be concluded that how well a base represents a fraction is not equivalent to how few digits there are in its cycle before repetition.

To three significant figures, on the other hand, dozenal positional notation is better absolutely at approximating sevenths with a smaller error than decimal. For dozenal, the difference between ①⑧⑦ cubic dozenths and a seventh is the reciprocal of seven cubic dozens, which is less than the decimal error from 143/10^3 - 1/7 = 1/7000. That dozenal is more accurate than decimal here is partly due to the base twelve being larger and hence having a finer resolution than base ten.

However, increasing the size of the base does not generally guarantee improved representation of a particular fraction to a certain number of significant figures, as already seen by the error from the representation positionally of a seventh to two significant figures in dozenal being greater than the error from decimal to the same number of significant digits. Another example is that the error, as 9/8^2 - 1/7 = -1/448, from a seventh to two significant figures in octal is less than in the base nine, decimal, base eleven, or dozenal, despite octal being smaller than those other bases.

For the fraction of any prime
In representation positionally to a number \(n\) of significant figures by base \(B\) of a unit fraction \(\frac{1}{p}\) that is the reciprocal of a prime number \(p\) the absolute error ε is determined as $$ε = \frac{mp - B^{n}}{p B^{n}},$$ where \(mp\) is the nearest multiple of the prime number \(p\) to the \(n\)th power \(B^{n}\) of the base \(B\).

Thus, selection of bases the lowest powers of which are the minimum distance from some multiple of the prime can be investigated for finding those bases that are best for positional representation of the reciprocals of the prime number in question.

Fifths
Bases, such as decimal, that can represent fifths positionally without error are multiples of five. Other bases that are the next best at approximating fifths have powers only one unit away from a multiple of five, and the error decreases as the multiple of five increases. So, consider the multiples of five sequentially starting from the smallest and examine those of the numbers adjacent to the multiples of five that may be realistic as bases.

Table: Errors of Fifths in Bases                           

Number of Fives, m	Base, B	Reciprocal Error, 1/ε = 1/(m/B - 1/5), decimally	In Degrees, 360°ε	Image Pentagram
one	senary	-30	-12
two	nine	45	8
two	eleven	-55	-6.545
three	twice seven	70	5.143
three	four squared	-80	-4.5
five	two dozen	120	3
seven	six squared	-180	-2
ten	seven squared	245	1.470
eleven	four-and-a-half dozen	270	1.333
a dozen plus one	eight squared	320	1.125
four squared	nine squared	-405	0.889
a dozen plus seven	eight dozen	-480	0.75
two dozen	eleven squared	-605	0.595
two dozen plus five	twelve squared	720	0.5
one	quinary	undefined	0

Discussion
The approximations to fifths start to become useful in the double digits or to two significant figures of the bases.

Visually, the error is best shown or easier to detect as degrees deviation from a fifth of a perigon in a stellation graph or pentagram with intersecting chords rather than a convex polygon or pentagon.

An error of three degrees appears for the double dozen, making the base twelve with a further subdivision rounded to the half interval one of the first bases reasonably approximating a fifth for estimations.

For constructions, an error less than one degree approaches a limit of accuracy in constructed drawings on a typical page and for discriminating from regular polygons without overlapping them.

Base six to two significant figures with an error of two degrees still leaves something to be desired, and would be better to three significant figures. However, at that many sectors, any base, such as near the second power of base three times five, with a power in that range would have a similar acceptable level of accuracy but fewer digits. Senary was thus rather disappointing.

Base seven squared is not a realistic proposal as a replacement for decimal, but is shown for comparison.

Base eight to two significant figures is quite a decent base.

Dozenal to two significant figures with an error of only half a degree, a limit in many protractors, is about as good as would be wanted for drawn constructions.

Thirds
In comparison, in decimal a third cannot be unambiguously represented positionally by just one significant figure and distinguished from a quarter, and jot three tends rather to be thought of as three tenths.  It would be necessary to use at least two significant figures to convey a third. However, the error in decimal to a third by two significant figures, as 33/100 - 1/3 = -1/300, is still larger and more noticeable than the error for fifths in dozenal to the same number of significant figures. Therefore, absolutely dozenal is better at representing fifths than decimal is at representing thirds. This enhancement is mainly due to the larger size and resolution of base twelve.

Conclusion
In general, in a worst case scenario, the maximum absolute error from the approximation of the reciprocal of a prime \(p\) by \(n\) significant figures of a base \(B\) would happen if the difference between \(B^{n}\) and the nearest multiple \(mp\) of the prime \(p\) to \(B^{n}\) is half of \(p-1\). Then the formula for the error \(ε\) would become $$|ε| ≤ |\frac{{p-1}}{2 p B^{n}}|.$$ For a large prime \(p\), the approximation \(\frac{p-1}{p}≈1\) may be made, such that in a worse than worst case scenario, the formula for the error \(ε\) becomes $$ε < \frac{1}{2 B^{n}}.$$ Thus, the reciprocal of the error for fractions of all prime numbers would not be lower than a resolution of twice the base raised to the number of significant figures. Hence, if a particular resolution be required, such as that the error would not be more than half an angular degree say, then the power of the base required would be half that resolution, and the \(n\)th root of that half resolution may be computed for a desired number of significant figures. For example, in the case of a half degree of angular resolution, the base would be half that resolution a six factorial part of a perigon, which is 360, and if that accuracy is desired by no more than two significant figures, then the base would be about the square root of 360, which is about base a dozen plus seven. Since a resolution of a half angular degree is adequate for most construction drawings and designs, there would not be any need or benefit in the base being larger than about base twenty or vigesimal. It would be quite pointless to have a base larger than that, because any enhanced resolution would be hardly humanly discernible. Thus, a useful range for bases may be limited to between about bases six and twenty. Base twelve appears about the average of this useful range.

What lies behind all of this is a rather brutal misunderstanding of how various things ought be implemented and measured.  The dozenal is given 13 measures to the decimal 12, and thus it should be ahead by at least 7% per digit.

For example, decimal goes down to (on the metre) to 10 mm, whereas in the dozenal, you have 7 mm.  If you are dividing the circle into parts nearest the degree, you might get 288 degrees dozenally, or 300 decimally.  1/7 is just 1/7 of a division off either of these (41, 43), and the measured accuracy is not all that different.

Any given fraction will, at the limit, be really as measurable to the limits of the gradings.  So if the gradings are to 1:288, the ultimate resolution is a number of 288ths and a fraction of it.  For example, \( \frac 17 = \frac {41}{288} \frac 17 \), or \(\frac {57}{400}\frac 17\).  The accuracy of the seventh depends partly on how well one is going to divide the grade-spacing into sevenths. 

The error will necessarily improve as the gradations get finer, but it's not a significant feature of the base.  Dividing the metre into 1024 mm is not appreciably different to 1000 mm, although the bases are different. 

Dozenal remains worse at fifths, then decimal does at thirds, because the decimal period is shorter, and more readily seen.  So something like 3.14166666 vs 3.1849729, which both represent 377/120, the decimal repeats many periods, all be it with a three-digit header, against dozenal's two-digit period.

A further example, is the fraction-approximation of e, which is 2.71828182846.  Most people would have looked at the four-digit period (1828) with a single-digit header (7), but the thing is better represented by the six-digit semi-symetric period 718 281 718 281 with no header.  718+281 = 999, so it's 2 719/1001, which is one of the convergences.  The fraction on 7r1828  is something in 99990. 

Of course, it is not pointless having a large base.  Bases are not monolithic, but there is a division-chain, for which one can go as many places as needed, omitting the penultimate division. For example, with 120, we get 2.2.3.2.5 - 2.2.3.2.5 etc.  If your precision-limit is 50, then you can use 2.2.3.2.(5).2, by swapping the last 5 with the next 2.  This gives 48.  One would not necessarily divide a unit to thirds, unless sixths were involved  as 2,(2),3.  For any given working precision, different bases take the lead.  For example, 18 as 3.2.3 gives 54 at this point, as 3.2.3 3, but it could go 36 (as 3.2.3 (3).2  (bracketed numbers are omitted divisions)

I have a scale ruler here of metres to 40, but the division of the resulting metre (25 mm), is 2.2.5, which is more suited to metres and twentieths, rather than decimal metres (which would give 2.5.2).

wendy.krieger wrote:a rather brutal misunderstanding of how various things ought be implemented and measured.  The dozenal is given 13 measures to the decimal 12, and thus it should be ahead by at least 7% per digit.

Twelve to the power of twelve or thirteen or onezeen is about ten to the power of thirteen or fourteen or a dozen plus two. Thus, what takes fourteen digits in decimal takes about onezeen dozenal numerical character figures. $$1 - \frac{\ln{10}}{\ln{12}} ≈ 0.0734 ≈ 7.3\%$$
Clearly, knowledge that a reason for dozenal fifths being better than decimal tenths being an effect of scale was demonstrated:

Phaethon wrote:That dozenal is more accurate than decimal here is partly due to the base twelve being larger and hence having a finer resolution than base ten.

Phaethon wrote:This enhancement is mainly due to the larger size and resolution of base twelve.

To correct the reciprocal errors for the size of the base, divide by the power of the base, which is the base raised to the number of significant figures. Doing this adjustment for scale of the base to the error for sevenths by three significant figures, decimal and dozenal are comparable. However, to two significant figures, even after adjustment for scale of the base, the bases are not equally good at representing sevenths. Octal would still be better at representing sevenths than expected for two significant figures compared to bases nine, ten, eleven, and twelve. Despite correction for scale, some bases remain better or worse than others.

Even so, the improvement in dozenal without adjustment for scale should not be dismissed. By analogy, in comparing the number of rooms in buildings, what value is there in declaring that the skyscraper has more rooms because it is taller? What matters is that the skyscraper has more rooms per surface area taken up on the ground, while the density within the structure has less impact on the ecosystem.

wendy.krieger wrote:If you are dividing the circle into parts nearest the degree, you might get 288 degrees dozenally, or 300 decimally.

Three hundred is not a decimal sort of division of angle, because it requires a trisection that does not belong to the decimal scale. A decimal pattern of division near two-and-a-half square dozen degrees would be four hundred, producing gradians.

wendy.krieger wrote:Dozenal remains worse at fifths, then decimal does at thirds, because the decimal period is shorter, and more readily seen.

Fifths in dozenal to an even number of significant figures are not as bad as thirds in decimal. In reality, unless the work is being done with vulgar fractions, decimal thirds have to be rounded to be used for computations, and this rounding leads to computational errors that can multiply and accumulate to produce larger errors in results. The error in the approximation with two significant figures in decimal to thirds of a circle can be readily seen with the naked eye and could be catastrophic in construction drawings, by the axes not lining up in isometric projection, for example.

wendy.krieger wrote:18 as 3.2.3 gives 54 at this point, as 3.2.3 3,

That base nine times six is a multiple of twice nine is a reason that it was included in the table of approximations to fifths.

wendy.krieger wrote:the fraction-approximation of e, which is 2.71828182846.  Most people would have looked at the four-digit period (1828) with a single-digit header (7), but the thing is better represented by the six-digit semi-symetric period 718 281 718 281 with no header.  718+281 = 999, so it's 2 719/1001, which is one of the convergences.  The fraction on 7r1828  is something in 99990.

In July 2010, I determined that rational convergents to the base of the natural logarithm may be devised by a recurrence relation and summation series as follows. Let \(q_{0} = 0\), \(q_{1} = 1\), and \(q_{n} = q_{n-2}+q_{n-1} a_{n}\), where $$a_{n} = \left(\frac{2n}{3}\right)^\left(\frac{4\cos^{2}{\frac{n\pi}{3}} - 1}{3}\right).$$ Then, $$e = 2 + \sum_{n=1}^{n=∞} \frac{(-1)^{n+1}}{q_{n} q_{n+1}}.$$ From this, the first few fractional convergents are $$\frac{8}{3}, \frac{11}{4}, \frac{19}{7}, \frac{87}{2^{5}}, \frac{106}{39}, \frac{193}{71}, \frac{1264}{465}, \frac{1457}{536}, \frac{2721}{1001}, \frac{23225}{8544}, \frac{25946}{9545}, \frac{49171}{18089}, \frac{517656}{190435}.$$ Notable of these is 87/2^5. Since dozenal is good at binary divisions, that fraction is exactly ②;⑧⑦⑥ in base twelve, but ②;⑧⑦⑤ is a more accurate approximation to the base of the natural logarithms. The fraction from the repeating digits 1828 is 271801/99990.

Comparison of Errors to Unit Fractions by Different Bases
How accurately bases can represent the simplest unit fractions, which are reciprocals of the smallest whole numbers, overall can be compared by charting the errors between the exact fractions and the approximations to them by their representations in bases to a small number of significant figures or digits after the fractional point in positional notation.

For "fractionals" that have a finite or terminating number of significant figures after the fractional point, those that have just one significant figure are better than those requiring two before termination, which in turn are better than those that require only three numerals to represent the fraction exactly.

As argued previously in this topic, this does not mean that fractions that have a non-terminating and repeating period of fewer digits are necessarily better at being approximated than those having a longer period of digits in a repeating cycle after the fractional point. For example, in base ten or decimal, the fraction an eleventh has a longer period of two significant figures before repetition than the fraction a third does, yet the accuracy of the decimal form to two significant figures of an eleventh fraction in decimal is much more accurate than the approximation to the same number of significant figures for a third fraction. This means that decimal is much better at representing elevenths than thirds.

There comes a point in the number of significant figures after the fractional point where it takes so long before termination for perfect accuracy that the termination is too late and has little or no advantage over an approximation by fewer significant figures. For example, a fraction such as 401/2000 might be better approximated as 0.2 than perfectly accurately as 0.2005, because it is quicker to do calculations using fewer significant figures.

Likewise, if the period of a non-terminating but repeating fractional part is too long, it might as well be irrational because the degree of precision required to reach so many significant figures in the fractional part would be unattainable. For example, in decimal the positional notation form of the fraction 1/17 has so many digits in the period before repetition that the fraction would have to be approximated in practice by truncation to a finite number of significant figures with rounding.

Nevertheless, fractions with very long periods can be more accurate than those with short periods within the same base or in different bases, just as some irrational numbers are approximated extraordinarily well or better than other irrational numbers in some bases rather than other bases, even though all the irrational numbers have equally infinitely long fractional parts in all of those bases. For example, the irrational Golden Ratio is approximated well to two figures after the fractional point in dozenal because of the square of twelve being a Fibonacci number.

Even if a fraction represented by positional notation in a base is repeating or non-terminating but with a short period or a small number of numerals in the cycle before repetition, it would have to be truncated with rounding or approximated in order to be written or used in some types of calculation.

What matters therefore for how good different bases are at representing fractions positionally is not how many numerals there are in a repeating period or even in a long terminating number, but how soon and how well the number is approximated after a few significant figures.

To compare bases for how good they are at representing fractions, I have drawn up a table of the reciprocal errors of the simplest unit fractions from a half to a twelfth to two significant figures in different bases.

I have colour coded the cells of the table by the magnitude of the error. For fractions that do not terminate in positional notation in given bases, the colours for depicting the errors are determined numerically by a score that is the logarithm to the base two of the ratio of the reciprocal error to the most extreme error expressed reciprocally as the number ninety or ⑦⑥ in the table. That is,
\[\text{Error score} = \log_{2}{\frac{1}{90ε}} \]
Colours were assigned to the resulting bands of scores as follows:

Table of Colour Codes for Error Scores

Error Score = log2(1/90ε)	Colour
∞1	Blue
∞2	Cyan
4-∞3	Green
3-4	Yellow
2-3	Orange
1-2	Red
0-1	Maroon

Numbers after the infinity sign ∞ represent the number of significant figures before termination. A reciprocal error that would be about equivalent to providing accuracy to three significant figures is given the same colour code of green as fractionals terminating at three significant figures for a base about the size of decimal to dozenal. With an error score of at least four, the reciprocal error works out to be 90*(2^4) = 1440 = ⑩⓪⓪, and a base producing this accuracy at three significant figures is the cubic root ⑩⓪⓪^(1/3) which is about base eleven between decimal and dozenal. This is saying that fractions approximated to an accuracy equivalent to about three significant figures by just two significant figures are at about the same rank in the error score as fractions represented perfectly accurately at three significant figures, for the sake of simplicity of the colour scheme for the scores.

Table of Reciprocal Errors to Unit Fractions by Two or Three Significant Figures in Various Bases

Base	6	8	⑩	①⓪	①④	②⓪
Fraction	Reciprocal Error
1/2	∞1	∞1	∞1	∞1	∞1	∞1
1/3	∞1	-①④⓪	-②①⓪	∞1	⑤④⓪	∞1
1/4	∞2	∞1	∞2	∞1	∞1	∞1
1/5	-①③⓪	②②⑧	∞1	⑤⓪⓪	-⑧⑩⑧	-①⑧⓪⓪
1/6	∞1	①④⓪	②①⓪	∞1	⑤④⓪	∞1
1/7	-①⑨⓪	-③①④	-②⑤②	②④⓪	④①⑨⁏④	-①②⓪⓪
1/8	∞3	∞1	∞3	∞2	∞1	∞1
1/9	∞2	-④⓪⓪	-⑥③⓪	∞2	-④⓪⓪	∞2
1/⑩	⑦⑥	-①①④	∞1	-②⑥⓪	④⑤④	⑩⓪⓪
1/⑪	-⑪⓪	②⑤④	-⑦⑦⑧	-⑪⓪⓪	-⑥⑥②⁏⑧	-⑪⓪⓪
1/①⓪	∞2	-①④⓪	-②①⓪	∞1	-⑤④⓪	∞1

Discussion on the Tabulated Results
Base Six

• Base six has many cool coloured cells of blue, cyan, and green, but also has several reds.
  
• The maroon cells for tenths and elevenths may not be of too much concern as those are not very common fractions.
  
• The red cells for fifths and sevenths are more worrying however, and base six is worse at representing fifths and sevenths than any of the other bases examined in the table.
  

Base Eight

• Octal has a large proportion of warm coloured maroon, red, and orange cells. It appears to be the worst base for representing fractions in the table.
  
• The main good point about octal is that it is better at representing sevenths than the bases six, ten, or twelve. This does not seem like enough of an advantage to overcome the overwhelming weaknesses.
  

Base Ten

• Decimal has a lot of red. Need any more be said?
  
• Decimal is better at representing ninths and elevenths than thirds and sixths. Its priorities seem to be in the wrong order.
  

Base Twelve

• Dozenal has mainly cool coloured cells.
  
• Dozenal represents a half with fewer significant figures than a third, thirds more accurately than fifths, and fifths more accurately than sevenths. It has these priorities in just the right order as though it were designed perfectly.
  
• Dozenal is better at representing fifths, sevenths, and elevenths than base six is.
  
• Base twelve is worse at representing sevenths than most of the other bases in the table. The lower accuracy of sevenths advantageously would discourage use of mathematically awkward divisions of seven in metrology, geometry, and trigonometry.
  
• Dozenal is unusually good at representing elevenths. This is a bit out of order, but can be overlooked.
  
• Dozenal has four times the resolution of base six its half.
  

Base Four Squared

• Base twice eight has mostly not cool coloured cells.
  
• It is better at approximating fifths than thirds, which is in the wrong order of priority.
  
• It does not seem to be much better than decimal.
  
• It seems worse than dozenal.
  

Base Twice Twelve

• The double dozen has only cool coloured cells on account of its large size. Being twice as large as twelve, the double dozen has four times the accuracy or resolution.
  

It would look as though base two dozen would be a good choice, but it comes at a hardware cost. Imagine having to have two dozen numerical keys on the keyboard. That is almost as many as for an alphabet! Also, it would be far more difficult to devise the extra numerals and terminology since there is nothing familiar to work on above twelve.

For these reasons, base twelve appears to be the best base for representing fractions in positional notation.

Unit Fraction Errors Normalised Relative to Base Size
The normalised error Ɛ relative to the size of the base B of a fraction to \(n\) significant figures after the fractional point is derived from the inequality in the opening post of this topic as
\[ Ɛ=B\sqrt[n]{2|ε|}\]
where ε is the absolute error between a fraction and its approximation to \(n\) significant figures after the fractional point in positional notation by the base \(B\).

Table of Colour Codes for Normalised Error Scores

Normalised Error Ɛ Relative to Base Size	Colour
⓪ \(\leq\) Ɛ \(<\) ⓪⁏②	Blue
⓪⁏② \(\leq\) Ɛ \(<\) ⓪⁏④	Cyan
⓪⁏④ \(\leq\) Ɛ \(<\) ⓪⁏⑥	Green
⓪⁏⑥ \(\leq\) Ɛ \(<\) ⓪⁏⑧	Yellow
⓪⁏⑧ \(\leq\) Ɛ \(<\) ⓪⁏⑩	Red
⓪⁏⑩ \(\leq\) Ɛ \(<\) 1	Maroon
Ɛ = 1	Purple

Table of Errors Relative to Base Size of Unit Fractions by Two Significant Figures

Base	6	8	⑩	①⓪	①④	②⓪
Fraction	Error Relative to Base Size
1/2	0	0	0	0	0	0
1/3	0	⓪⁏⑨⑨⑦	⓪⁏⑨⑨⑦	0	⓪⁏⑨⑨⑦	0
1/4	0	0	0	0	0	0
1/5	⓪⁏⑦⑦⓪	⓪⁏⑦⑦⓪	0	⓪⁏⑦⑦⓪	⓪⁏⑦⑦⓪	⓪⁏⑦⑦⓪
1/6	0	⓪⁏⑨⑨⑦	⓪⁏⑨⑨⑦	0	⓪⁏⑨⑨⑦	0
1/7	⓪⁏⑥⑤	⓪⁏⑥⑤	⓪⁏⑨⓪⑩	⓪⁏⑪①④	⓪⁏⑪①④	⓪⁏⑨⓪⑩
1/8	1	0	1	0	0	0
1/9	0	⓪⁏⑤⑦⑪	⓪⁏⑤⑦⑪	0	⓪⁏⑪③⑨	0
1/⑩	⓪⁏⑩⑧⑩	⓪⁏⑩⑧⑩	0	⓪⁏⑩⑧⑩	⓪⁏⑩⑧⑩	⓪⁏⑩⑧⑩
1/⑪	⓪⁏⑧⑩④	⓪⁏⑦②⑩	⓪⁏⑤①⑤	⓪⁏⑤①⑤	⓪⁏⑧⑩④	⓪⁏⑩②⑩
1/①⓪	0	⓪⁏⑨⑨⑦	⓪⁏⑨⑨⑦	0	⓪⁏⑨⑨⑦	0

From the results of the preceding table, the rank of merits of the bases for approximating unit fractions from a half to the reciprocal of twelve from best to worst is:

1. six
   
2. twice twelve
   
3. twelve
   
4. ten
   
5. eight
   
6. four squared