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SurrogationGlossary |
Surrogation
Surrogation is a general term applied in this work to a single numeral or a group of numerals that connote a base b number other than the value of the numeral or group. An example of such is the decimal numeral 5, which connotes “half” or “halfway”. A group of decimal numerals, “25” conveys “one quarter (of a hundred)”. When we see “25” written in decimal society, we tend to think “one quarter” perhaps even more than we do “twenty five”. For many, the notion that 25 signifies the square of the larger prime factor of the base is lost upon them, but most would rather think about it signifying a quarter. A quasi-numeral (as opposed to a “natural” or plain numeral as normally regarded) is a group of numerals that serve as surrogates that signify other numbers. The lone numeral 5 is not a quasi-numeral as it already is a natural decimal numeral. Instead we might say that 5 is a decimal surrogate for the notion of “half” or “halfway”.
Surrogation is subjective
Surrogation is a subjective effect of human pattern recognition, the imputation of meaning on a group of symbols, here numerals rather than letters, to mean something other than their strict interpretation according to standard positional notation.
Surrogation depends on position or degree of repetition
While natural numerals like 5 always are interpreted as “five”, their connoting “halfway” only occurs when they are the last nonzero digit in the number. The decimal number 55 could mean “halfway between 50 and 60”. This also pertains to the quasi-numerals. The number 25000 could be read as “a quarter (of a hundred grand)”, but in a number such as 25,738, the “25” tends not to be interpreted as a quasi-numeral, instead it is read algorithmically as usual, “twenty five thousand seven hundred thirty eight”. There are other quasi-numerals that feature repeating digits, such as decimal 3_ (the underscore notation used in this work connoting a repeated digit). At what point do we perceive a repeating 3? A single 3 decimally stands as three as normal. Does 33 connote a third? It may weakly suggest a third if we know a related total is 100 (as in percent). The number 333 is more clearly a repeated 3 and its surrogation for the concept of the third is stronger. Certainly 3333 or 33,333 seem to emphatically communicate a third of something.
Surrogation appears in all bases that might serve general human computation, principally because people tend to want handy ways of conveying common fractions.
Decimal surrogation
Decimally, the major surrogates pertain to the natural fractions 1/2, 1/4, 3/4, 1/3, and 2/3. These are the natural numeral 5, the quasi-numerals 25 and 75, and repeated 3 (noted in this work as 3_) and repeated 6 (i.e., 6_). Since surrogation is the province of human cognition, skilled workers and those who deal with vulgar fractions as a matter of course may perceive more surrogation than the average person. Workshop personnel in the US often know the multiples of the sixty-fourths on account of their intensive use of the US Customary system of measure. To them, “125” and “375” would convey 1 and 3 eighths, respectively; “4375” could signify “seven sixteenths”. Furthermore, some may see more complex quasi-numerals like 1_6_ as “one sixth” as 1/6 = .1666…, and 8_3_ as “five sixths” since 5/6 = .8333…. To some, 142857_ would evoke “one seventh”. Despite its brevity, 1_ in decimal would convey “one ninth” but for its rarity in everyday use. For the average person the decimal numerals, both natural and quasi, would include 0, 1, 2, 25, 3, 3_, 4, 5, 6, 6_, 7, 75, 8, and 9.
As a rule in this work, we consider surrogation that involves no more than two numerals as acceptable to the general public, were base b used as a base of general human computation. This does not mean that some people would not perceive more complex surrogation.
Ambiguous surrogates
Surrogation can be ambiguous. Decimally, “625” might more often be taken to mean “five eighths” but also could pertain to one sixteenth, since .625 = 5/8 and .0625 = 1/16 decimally. In base twelve, “9” strongly conveys “three quarters”, while duodecimal .09 = one sixteenth, therefore the surrogate for a sixteenth in that base is neutralized. This said, many multiples of the sixteenth in base twelve would seem to be accessible quasi-numerals (e.g., 3/16 = “23”, 5/16 = “39”, etc.).
Surrogation in other bases
In hexadecimal, the numerals 4, 8 and c might certainly convey “one quarter”, “halfway”, and “three quarters”, respectively. As 16 is a prime power, the numerals 1, 2, 4, and 8 (and the multiples of 2 and 4 less than sixteen, most prominently c, digit-twelve) may tend to prove “more powerful” as they are the only proper regular numbers in that base. These four numerals followed perhaps by zeros represent all numbers that have terminating hexaecimal unit fractions; indeed they are the powers of two written in base sixteen. Along with these natural numerals, we anticipate several other surrogates that are quasi-numerals. In today's hexadecimal RGB color codes, we observe the hexadecimal quasi-numerals that correspond with the fifths, i.e., 3_, 6_, 9_, c_, and f_, which signify 1/5, 2/5, 3/5, 4/5, and full channel, respectively. The underscore notation denotes a repeated digit. Therefore hexadecimally, if we see a repeated 3, we might think “one fifth” in much the same way that we consider decimal repeated threes signifying one third. Along with 3_ signifying “one fifth”, we have 5_ conveying “one third” and a_ (repeated digit-ten) conveying “two thirds” hexadecimally. Thus, the numerals, natural and quasi, in hexadecimal might include 0, 1, 2, 3, 3_, 4, 5, 5_, 6, 6_, 7, 8, 9, 9_, a, a_, b, c, c_, d, e, f.
In flexible bases we may have a plethora of surrogates. In base twelve, the natural fractions 1/2, 1/4, 3/4, 1/3, 2/3 are expanded in single-digit terminating fractions: .6, .3, .9, .4, and .8 respectively. Therefore all the natural fractions have surrogates in the natural numerals 6, 3, 9, 4, and 8 respectively. The duodecimal fifths are rather long and may not be perceived by the average person in a duodecimal culture. The duodecimal eighths, ninths, and sixteenths (one dozen fourths) have two-digit terminating expansions; therefore 16, 46, 76, and a6 might evoke 1/8, 3/8, 5/8, and 7/8, respectively. None of the common dozenal surrogates would involve repeating digits, since these would pertain to the seldom needed elevenths. In base twelve we might have the numerals 0, 1, 14, 16, 2, 23, 28, 3, 39, 4, 46, 5, 53, 54, 6, 68, 69, 7, 76, 8, 83, 9, 94, 99, a, a6, a8, b.
Through extrapolation, a base like sexagesimal might have even more surrogates.
In base 120, not only might we have non-repeating surrogates, but quasi-numerals that pertain to sevenths, elevenths, and seventeenths. In base 120, 1/7 = .17,17,17,… , 1/17 = .7,7,7,… , and 1/11 = .10,109,10,109,…. This may seem complicated with digits expressed in decimal, but using the numeral h = 17, the seventh would be conveyed by the surrogate h_.
Surrogation and natural fractions
It is anticipated that the natural fractions 1/2, 1/4, 3/4, 1/3, and 2/3 appear as surrogates and quasi-numerals in all bases. This is because of the simplicity and commonness of these fractions in everyday life, and because these fractions can be conveyed by one or two nonrepeating or repeating digits.