**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.

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