Natural Fraction

The natural fractions are 1/2, 1/4, 3/4, 1/3, and 2/3. [1, 2]

The natural fractions are the most commonly used fractions. The half is the fundamental fraction, achieved with a single break (even if not in exact halves). The quarters derive from breaking each of the pieces again, and the thirds from breaking a larger piece but not a smaller. We observe that the smallest primes appear as denominators in the natural fractions, the smallest prime with multiplicity. These correspond to sharing among the smallest groups of individuals.

The natural fractions have been important historically and internationally. Numbers amenable to the natural fractions, such as the dozen, have enjoyed longstanding common use [3, 4].

Other important fractions

We might perceive a need to prioritize other denominators in some order of utility [3]. The odd eighths and the irreducible sixths are perhaps next in importance. Fifths require more concerted attention. Lamadrid has suggested that Euler's Gradus Suavitatis might serve as a handy extrapolation of the sequence of denominators {(1), 2, 4, 3} in the set of natural fractions. Sorted by smallest prime factor, the rows Lamadrid considered appear as follows:

1
2
4	3
8	6
16	12	9	5
32	24	18	10
64	48	27	36	20	15	7
128	96	72	40	54	30	14
256	192	144	81	80	108	60	45	28	25	21
512	384	288	160	216	120	56	162	90	50	42
1024	768	576	320	243	432	240	112	135	324	180	100	84	63	75	35	11
2048	1536	1152	640	864	480	224	648	360	200	168	486	270	150	126	70	22

Thus, fractions with denominators {(1), 2, 4, 3, 8, 6, 16, 12, 9, 5, 32, 24, 18, 10, …} appear arranged in order of importance.

Surrogation of natural fractions

Independent of number base, the natural fractions are inherent in daily activity and necessitate representation. For this reason they might crop up through surrogation in all number bases, where at most two numerals are required to represent them. Decimally, the natural fractions are conveyed by the natural numeral 5, the regular quasi-numerals “25” and “75”, and the coprime quasi-numerals “3_” (repeated 3) and “6_” (repeated 6). In duodecimal, the natural fractions are all regular and are simply conveyed by the natural numerals 6, 3, 9, 4, and 8 respectively. In base 11, all the natural fractions are coprime: “5_”, “28_”, “82_”, “37_”, and “73_”, respectively.

Base	1/2	1/4	3/4	1/3	2/3
2	1	1	11	01_	10_
3	1_	02_	20_	1	2
4	2	1	3	1_	2_
5	2_	1_	3_	13_	31_
6	3	13	43	2	4
7	3_	15_	51_	2_	4_
8	4	2	6	25_	52_
9	4_	2_	6_	3	6
10	5	25	75	3_	6_
11	5_	28_	82_	37_	73_
12	6	3	9	4	8
13	6_	3_	9_	4_	8_
14	7	37	a7	49_	94_
15	7_	3b_	b3_	5	a
16	8	4	c	5_	a_
17	8_	4_	c_	5b_	b5_
18	9	49	d9	6	c
20	a	5	f	6d_	d6_
24	c	6	i	8	g
30	f	7f	mf	a	k
36	i	9	r	c	o
60	30	15	45	20	40
120	60	30	90	40	80

In bases with the flexibility of duodecimal and sexagesimal, we may have further quasi-numerals that signify the odd eighths, fifths, and irreducible sixths, numbers further along Lamadrid's suggested extension according to Gradus Suavitatis.

[1] Wolfram code to generate GS chart:

With[{n = 12},
  Map[Reverse@ SortBy[#, FactorInteger[#][[1, -1]] &] &, Take[#, n]] &@
    Values@ KeySort@ PositionIndex@
      Array[Total[Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] - 1] + 1 &, 2^n]] // TableForm

More efficient code:

f[m_] := Block[{s = {Prime@ PrimePi[m + 1] - 1}},
  KeySort@ Merge[#, Identity] &@ Join[{1 -> {}},
    Reap[Do[
    If[# <= m,
      Sow[# -> s]; AppendTo[s, Last@ s],
      If[Last@ s == 1,
        s = DeleteCases[s, 1];
    If[Length@ s == 0,
      Break[],
      s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1] ],
    s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1] ] ] &@
      Total[s], {i, Infinity}]][[-1, -1]] ] ];
  MapAt[Reverse,
    Map[ReverseSortBy[Times @@ # & /@ #,
      FactorInteger[#][[-1, -1]] &] &, Values@ f[12] + 1], 1] // TableForm

References

[1] Flegg 1989, Chapter 5, “Fractions and Calculation”, Section “Natural and unit fractions”, page 131, specifically: “In spoken English certain simple fractions like ½ or ¼ or ¾ have special names. We do not say ‘one-fourth’ but one quarter, and ½ is pronounced as one half. The French have a special name tiers for ⅓. Fractions of this kind, which serve the purposes of everyday life, may be called natural fractions. In ancient Egypt the situation was similar. The Egyptians had special words for the natural fractions 1/2, 1/3, 2/3, 1/4 and 3/4.”

[2] Neugebauer 1957, Chapter IV, “Egyptian Mathematics and Astronomy”, Section “Natural and unit fractions”, page 74, specifically: “…the separation of all unit fractions into two classes, ‘natural’ fractions and ‘algorithmic’ fractions … As ‘natural’ fractions I consider the small group of fractional parts which are singled out by special signs or special expressions from the very beginning, like [2/3, 1/3, 1/2, and 1/4]. These parts are individual units which are considered basic concepts on an equal level with the integers. They occur in everywhere in daily life, in counting and measuring.”

[3] Dudley 1969, Chapter 14, “Duodecimals”, page 114, specifically: “Any base would serve as well as 12 to give practice, but some parts of arithmetic—notably decimals—are nicer in base 12 than they are in base 10. Besides, there is a good deal of twelveness in everyday life: items are measured by the dozen and gross, there are 12 months in a year, 12 inches in a foot, half a dozen feet in a fathom, two dozen hours in a day, and 30 dozen degrees in a circle. The reason for this abundance of twelves is the easy divisibility of 12 by 3, 4, and 6; we want to make such divisions much more often than we want to divide things by 5. … counting by the dozens is manifestly better.”

[4] Menninger 1957, “Number Sequence and Number Language”, “Our Number Words”, “The Number Twelve as the Basic Unit of the Great Hundred”, pages 156-7, specifically: “But why precisely 12? The importance of this number in the lives of common people, in commercial transactions, and in legal affairs is probably due to its easy divisibility in so many ways… The commonly used fractions of …the shilling could all be expressed in terms of whole numbers of pennies” and the ensuing example.

Menninger 1957, “Number Sequence and Number Language”, “Our Number Words”, “The Roman Duodecimal Fractions”, page 158, specifically: “The Roman fractions were originally based on a system of equivalent weights; 1 as (or pound) = 12 unciæ (ounces). … The underlying tendency of all subdivisions of measures is to avoid fractions and to express them instead as whole numbers of smaller units.”

Menninger 1957, “Number Sequence and Number Language”, “Our Number Words”, “The Roman Duodecimal Fractions”, page 158, specifically: “In their computations the Romans used no fractions other than these duodecimal fractions.”

[5] Glaser 1971, Chapter 4, “The Rest of the 1700s”, Section “Duodecimal versus a Decimal Metric System”, page 69, specifically: “Any advantages of base 12 would be due to its being richer in divisors than 10 and certain common fractions such as ⅓ and ¼ would have simpler equivalents in base 12 notation.”