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 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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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