Co-existing bases

Numbers are in a sense, transcriptions of stones on a stone-board.  Regardless of the direction of writing, the largest columns are shown to the left, and the smallest to the right.  The same order is preserved on calculators and cash registers.  This is because most people are right-handed.   When columns are divided to rows, the higher values are further away from the operator: the cash-registers run their numbers upwards, the stone-boards and abacuses run their rows with the units at the bottom.

The various notations of numeral punctuation serve to split the string of digits into column-groups, columns, and rows, with the radix-point appearing as a kind of column-group.

The punctuation runs from heavy to light, as units, "; : , ." space, half-space abutment and accent.

The first four can be rendered heavier from . by adding a tail to get , and then adding a centrepoint · .  So ; = . + ̧ + ·  .

The order of functions, from heavy to light is:  radix-point, comma, column, row, where a digit occupies a column or row point, and heavier punctuation is used throughout.

For example 5.184,000  the comma is heavier than the point, serves as radix.  The point is a comma-marking, setting off columns, while abbutal is lower down.  The point can be replaced by a space 5 184,000 and 5 184. also serves the same end, but 5,184.2 does not, because the lighter point is a heavier function.

Neugebauer for example discusses sexagesimal using variously units and the closed base form, ie 3° 8' 30" (units) = 3;08,30  (closed form).

Units

Units are when a number is set against a column-name or unit.  For example, writing sexagesimal as degrees, minutes, and seconds, is a unit-form.  The chinese style of writing digit-column numbers, such as 5C 2X 8 for 528, is likewise unit-form.  Mayan glyphs also show a unit-form notation, where the column-holder is written above the glyph representing the column.  This is most likely to arise from abacus-style notations.

Unit-point

The sumerian pattern of giving a sexagesimal unit+fraction after a marker, and the chinese pattern of giving only the first colum are examples.  

For example, sexagesimal might write 192 as s3A2, that is (shocks, 3 12, meaning 3;12 shocks = 3 12;).  The chinese name the first value, the column it falls, and then subsequent places.  So 540 would be 5c4  (five hundreds four).  504 is 5c04  five in the hundreds, then zero, four.  

This notation usually has an medial zero (eg the zero in 504), and prehaps leading zeros (eg 1 second = 0h 0 1) but no trailing zeros.

Simple Digit

The range of digits is suffice to show the full array of what might occupy a column.  This is the basis of what mathematicians say having digits from 0 to b-1.  This leads for people to create a vast array of glyphs to meet this need for large numbers.  Historically, this is not the case, and we see row-forms existing.

The general open-base form simply writes these digits as decimal numbers, and uses a heavier column-marker.

Comma-forms

Long strings of digits might be broken into groups of 3, 4, etc digits, to increase the readability.  The commas often align with unit-names for groups of digits, such as 16777216 written 16,777,216 is 16 million 777 thousand 216,  cf 16 million, 77 myriad 72 hundred and 16, does not align with the given commas.

Row digits

For sufficiently large bases, it is usually given to represent the digits in each row of the column, and use column-separators.  When columns are significant, the style is to use abuttal for the column, and a heavier point (including a space), to separate columns.  The full set of rows is shown for all but the first column, where leading zeros may be suppressed.  So 6-5 (5 minutes past 6), in a packed sexagesimal, should be written as 605, not 6 5.  With the dash, it is supposed that context applies for the units.

Note here that close-abuttal serves for small strings of digits, since it is clear that three digits fit to two columns, and the 6 leads.  In twelfty, one can write the number 1.00.06 as 1 0006, but not as 1.0.6 or 1..6.

Accent

One can include a high row as an accent, for example in base 20, one can write 19 as '9, the ' representing a unit in the high row.  The current year 2019 is un this style written 50'9, being 5 0 19, base 20.

Separate High forms

One can eliminate the need for column-spacing by having an alternate form for high digits.  In the cuneform numbers, the units are marked by vertical pips, the horizontal ones mark the tens.  In the mayan, the low digits are stones, the high digits are bars.  We can represent such high forms by the letters A,B,C,... as far as needed.  The sumerian number for 30 is 'C', and for 33 is 'C3'.  This more faithfully shows what is written, and is what lead to the discovery of the semimedial zero, as in A  7 = 10,07 or 17, but writing the zeros in the latter, obliverates the need for a semi-medial zero.

Coexistance of Several Bases

If several bases exists in the same document, it suffices either by context, or an introduction, to state which combination of symbols apply to a given base.  The Humphreys point (;) suffices to differentiate dozenal from decimal, which uses a lighter point (. or ,).  Likewise, twelfty is normally written with a heavy radix (:), with lighter comma markings.  

If the practice is to do some calculations in a base, and never use a different base, that such calculations are underway, suffices to set the base.  For example, I normally use base 120 only to represent the orders of symmetries of polytopes, and have to convert these to decimal on demand.  The group E8, for example, has an order of 3.43.24.00.00, but i should need a calculator to render this as a decimal.

The method used in Essig's 'douze is votre dix futur', is to box the dozenal number, viz [123] for dec 171.

An alternate marker for bases is to use a general form, with a decimal marker like b12, or a double marker b12b6 (base 12, the vertical base is 6),

Twelfty-numbers are written in even-digit groups, while decimal is written in groups of 3.  There is no confusion on 5,184 = 43.24 for example.

The demands of alternating-arithmetic usually means that alternating bases are set to individually spaced columns, such as 43 24.  The dots (and heavier signs by upgrade) are then inserted as needed.  But the normal packed form is into groups of four.

In the class-2 bases, the notation is usually packed forms (ie abuttal), but it suffices to use unique digits, so q implies J6, f implies J3 and R implies J4.  These are variously, sqrt(2), Ø and sqrt(3)+1, which do not make sense in the other systems.

The span of an Equation

Another base-like system is the span of an equation, that is, the powers of x, where x^n = ....., eg x^4 = 4x²-1.  In each column, it follows the form of an open base, but the columns can contain negative values as well.  The usual base-style multiplication works, but instead of having few rows and many columns, you have few columns and many rows.  The notation is to write, eg "1,0,0,0,0 = 4,0,-1." as the column system.

The order of punctuation from heavy to light is given as
1. Denominational symbol
2. Semicolon
3. Colon
4. Comma
5. Point
6. Space
7. Half-space
8. Abutment, meaning simply the placing of two figures next to or beside each other
9. Accent, which is described as diminution of the numerical figure to its upper portion, or the raising of it above the surrounding characters. To these might be added the placement of a denominational symbol above numerical figures, analogous to the accents above letters in the orthographies foreign to English.

The hierarchy of divisions or "order of functions" from most to least notable was
1. Fractional indicator
2. Grouper of digit places
3. Place for a value or power of the base
4. Subdivision of a power of the base

Use of a point in a number to mean anything other than a marker between units and fractional parts is likely not to be recognised as such.

In writing dozenal numbers, a semicolon for the fractional point is compatible with the principle of being heavier than the separators between groups of numerical characters or figures. The separator between groups of figures can be a comma or the even lighter space, both of which have been used for this purpose in decimal numbers.

What was referred to as "Units" appears to have been a form of notation where a symbol effectively behaving as a denomination is used to express the power of the base to which the adjacent numeral refers as a multiplier. An example is the degree symbol, along with prime and double prime for minutes and seconds still in use. The principle is similarly seen for hours, minutes, and seconds notated by xx hrs xx min xx sec, or more simply xx h xx m xx s, where xx represents the numerals. The consecutive ratios of the adjacent denominations can more readily be irregular. For example, the pre-decimal currency used letters for pounds, shillings, and pence. Normally, however, units are understood to be the count of the zeroth power of the base. This type of notation may be called multiplicative denominational.

The "Unit-point" appears to be a particular kind of denominational marker that also serves in some way to show a separator between whole and fractional parts of a number. If such a symbol were the only denominational mark, the system would become positional. Positional notation was referenced as "closed form", or "packed form" without spaces between the digits.

With use of denominational symbols in contrast to positional notation, absence of a power of the base otherwise indicated by a symbol for zero does not need to be explicitly stated. The omission of zero follows the continuing custom of natural spoken language whereby there are words as denominations for individual powers of the base that are not stated when there are zero of them between larger and lesser powers. Further, leading or trailing zeros can be omitted without loss of comprehension. In other words, denominational systems are capable of being zeroless. It is nevertheless possible to augment a multiplicative denominational system with a symbol for zero. The presence of zero is not a sufficient condition for the notation to be fully positional in the sense of portraying the power of the base by no other means than the position of the figures in relation to each other.

Of the "simple digit" modes of notation, there are those which supplement the Indo-Arabic decimal digits by additional characters for figures, sometimes by recycling letters of the alphabet, while in other cases the Indo-Arabic figures are displaced by new or distinct symbols.

In what was referred to as the "general open-base form", "a heavier column-marker" could be to place the digits standing in the place of the same power of the base inside brackets to separate them from the digits representing the multiples or magnitudes of the other places. A style of brackets that has been used is the curly ones: {xx}. Another marker that could be used is an apostrophe, but this use is not the same as the "accent" replacing a digit one from the tens position or "in the high row".

wendy.krieger wrote:The Humphreys point (;) suffices to differentiate dozenal from decimal, which uses a lighter point (. or ,).  Likewise, twelfty is normally written with a heavy radix (:), with lighter comma markings.

If the semicolon can be confused with punctuation, perhaps a reversed ⁏ with Unicode 204F or some other orientation of semicolon could be used in printed format, with the semicolon as an acceptable fallback in swift communication.

The colon is likely to be interpreted as implicating a ratio of the two sets of numerical figures on either side of it. The lighter comma markers might be enough to show the meaning as a separator between the whole and fractional sections because of their grouping of the digits evenly rather than by threes. A ratio of twelfty numbers might thereafter be confined to another convention, such as the forward leaning slash / indicating a fraction or division.

"The span of an Equation"
What I have seen being used as an abbreviation in computation is to write only the multipliers of each power of the variable, which can be positive or negative.
https://math.stackexchange.com/questions/3137016/span-of-an-equation

Normally, signs of punctuation are followed by a space, while numeral dividing is not. So 5,832 has no space, so it's 18^3. But 5, 832 is a list of two numbers or a coordinate 5 by 832.

You are more likely to confuse close-set punctuation with mathematical operators, such as 5.832 = 4160 (aka 5 × 832).

The colon is already used to seperate sexagesimal time, eg 20:31, so it is already functioning as a column-marker in the time context. The use of colons in ratios (eg 2:3) is a little unsettling, but a ratio requires two numbers, and the colon is the heaviest separator.

The HP calculators already suppose using two points to allow fraction entry, eg 5.3.9 = 5 3/9, and you could do the same with ratios, supposing the : be red as 'as x then y", so :2:3.

I tend to avoid brackets, super-scripts and subscripts. They don't serve the eye well, and they have already a different meaning.

Superscripts and subscripts are useful for adding properties. So \(\mbox{name}_{\mbox{property}}\) already means that if the name descends into a subscript, then how might it be further subscripted.

In a product of variables, the powers might be written lining (eg a2 for a²), But this does not serve the need when the power is negative.

The whole point of these rules is not to be prescriptive on what symbols ought be used, but rather the notion behind the notation. The numner 5832 is already a list of four different operations, and three functions. And yet in large bases we are expecting it to function as a point-entry.

Fractions

The series of added and egyptian fractions, as a matter of computer entry, would require separate notations.

Egyptian fractions would require a symbol, which has the effect of #x = +(1/x), such that a fraction like 3 13/81 might be written 3 #9 #27 #81, or 3 #8 #36 #216 #324. The reason for this is that the egyptians did not see the fraction as a result of division, but as separate partitions.

Added fractions or continued numerator, already has a standard form, where the fractions are written one after another. We already see 18.33½, and thirds occupying the fraction too. The normal tex style is \( 3 \frac 18 \frac 2{15} \) for 3 17/120, writen as eights and added fifteenths. Here the 17/120 makes for (1 2/15)/8, but the whole is closely allied to eights and a second fraction, which is 15th.

Look-up and look-down fractions also have been looked at. A look-up fraction has the denominator preceeding the numerator, while the look-down fraction has the numerator first.

Look-up fractions more closely follow tables with columns headed with the denominator (eg £ s d), and the carry is decided not from a written denominator, but the column-head, so 17s 6d would be \(\frac{20}{17}\frac{12}{6}\), and written \20&17\12&6, where the backslash points up to the heading, and the ampersand points to the numerator. The corresponding look-down fraction is 17/20&6/12.

The rule for implied denominators is that the next denominator in the same direction applies. So something like \( 3 \frac 3{} \frac 4{ } \frac 6{12} \) makes 12 the denominator for the 3 and 4 as well as the six. For look-up fractions, the denominator is at the lead.

Since each numerator in an added fraction represents a column, the rules relevant to column partitions and unit-markers applies.

The span of an Equation

The stackexchange question you link to is my attempt to ask the mathematical community what this might be called. In order to avoid algebra, I treat equations as a carry condition, and the space that powers of the root make into a space.

The span of a set of vectors is \( S = \sum z_n v_n \) where \( z_n \in \mathbb Z\). The transfer comes from mapping powers of a root of the equation onto \(a^n => v_n\).

wendy.krieger wrote:Normally, signs of punctuation are followed by a space, while numeral dividing is not.

An exception where the semicolon at the end of a number might be followed by a space is in lists of enumerated or bulleted numbers. The semicolon is often used as a punctuation mark between elements in a list. Not all stylists may prescribe that items in a bulleted list should not end with a punctuation before the next bullet. Thus, a number followed at its end by a semicolon at the end of an entry in a list might not be recognised as dozenal.

It is a good point about the colon for digital clock times. Also, it is used as a separator between Biblical chapter or verse and line numbers. The colon notation for ratios is probably an outdated mode of representing numbers where fractions using the forward slash / would work nowadays. There can also be confusion about whether to look up or down with the ratio in converting to a fraction, for example in notation of frequencies of notes in musical theory. So, perhaps the colon as ratio notation should be abolished. However, the forward slash is also used as a separator in dates as xx/yy/zz for years, months, days.

The fractional point should never appear more than once with that meaning in the same number, so if a point does appear at least twice it could be interpreted as a multiplicative operator, for example in seven factorial as 7.6.5.4.3.2.1. This could be another reason not to use the point as a grouper of digits. An alternative to the point operator for multiplication unlikely to be mistaken for a fractional point is to place curved brackets () around the numbers being multiplied, so seven factorial would be (7)(6)(5)(4)(3)(2)(1). The point is also used as a divider between the numbers of sections and subsections in numerically headed paragraphs.

The avoidance of superscripts and subscripts for annotations of the base is sensible, because numbers which are already superscripted or subscripted would lead to very tiny text from further superscripts or subscripts within the superscripts or subscripts. Nonetheless, a subscript for annotation of the base can be placed on the same level as the rest of the line by following it after the underscore _.

Egyptian fractions have been called "unit fractions" elsewhere, because their numerators are one the unit. As partitions rather than divisions, they may be thought of as being formed by cutting off sequentially smaller parts that are always commensurate a whole number of times with the original segment. As there is not a unique unit fraction expression, there is a dilemma about partitioning greedily, by always cutting off at each stage as large a unit fraction portion as possible, or economically by a minimum number of partitions.

In contrast, the "added fractions" notation as mentioned has the next fraction contain a denominator or lower part of the fraction as the number of segments subdividing the previous fraction rather than the original or whole.

The message that can be gleaned is that there are many ways by which different bases being used together can be distinguished by adaptation of readily available and convenient symbols and formatting without having to be burdened by nuisance literal annotations of the base. As ever, it would still be necessary to declare the base being used at the start of a communication, unless a certain format becomes so well recognised amongst its adherents as to be deemed the norm.