# Numeric font

A number font is a writing system for the writing of numbers . Through the medium of the written (see also: glyph and Graph ) including this historical also techniques of scribing, notching, stamping and chiselling, is written numerals bordered on the one hand against the numeral systems (numerals) of natural languages and on the other hand against systems in which fingers - and body gestures , arithmetic stones , nodes , light signals or other, neither linguistic nor in the narrower sense written symbols are used to represent numbers .

## Basic structure of written expressions of natural numbers

A numerical expression formed in numerical form on the basis of a number system always indicates two things for a natural number:

• which powers of the base are contained in the number, beginning with the highest contained in it and
• how often these potencies are contained in it.

If the product of the value of the highest contained power and the frequency of its occurrence does not yet give the total value of the number to be represented, but still remains a remainder, then this remainder and the next lower power are processed according to the same principle, and so on until there is no more residue. The lowest power is not the base number itself (B 1 ), but the number one (B 0 ), according to the rule that the power of any base with the exponent zero always has the value one. Examples based on the number 1434:

• Roman (type cumulative-additive, base 10, without auxiliary base 5):
MCCCCXXXIIII
= 1000 + (100 + 100 + 100 + 100) + (10 + 10 + 10) + (1 + 1 + 1 +1)
• Milesian - Greek (type
= 1000 + 400 + 30 + 4
• Chinese (type multiplicative-additive, base 10):
一千 四百 三十 四
= (1 × 1000) + (4 × 100) + (3 × 10) + 4
• Babylonian (type cumulative-positional, cumulative auxiliary base 10, positional base 60): = (10 + 10 + 1 + 1 + 1) × 60 + (10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1) × 1

• Maya (number positional type, base 20): = (3 × 400) + (11 × 20) + (14 × 1)

• Indian Arabic (numeric positional type, base 10):
1434
= (1 × 1000) + (4 × 100) + (3 × 10) + (4 × 1)

## Typological differentiation of number fonts

Against the background of the particular appreciation of the Indo-Arabic numerals and the advantages of their place value system for the writing of large numbers and for the execution of written arithmetic operations, numerals are divided into "positional" and "additive" according to a widely used principle.

Positional numerals like the Indo-Arabic express the value of each power through the position and the frequency through the value of a single character. You therefore only need B-1 (in the decimal system: 10-1 = 9) numerals to specify the possible multiplications for any power up to the next higher power, and an additional space for the value zero, if within a compound expression no value has to be given for a certain power range, and are nevertheless suitable for displaying numbers of any size with such a limited inventory of characters, only limited by the available writing space.

Additive numerals, on the other hand, name each power of the base with its own character, which they repeat cumulatively according to the frequency of their occurrence (cumulative-additive), such as the Roman, or which they provide with a multiplying character, such as the Chinese (multiplicative-additive), or they name, as the Greek does, not only every power, but also every multiple of a power that is possible until the next higher power is reached, with its own symbol, so that the individual symbol always expresses the finished product of frequency and power value (numerical-additive).

The number range that can be represented with additive number fonts is in principle limited by the scope of the available character inventory, if this is not to grow at will through the constant introduction of new characters for higher powers, or if an additive principle is not combined with the positional. The latter is the case with the Babylonian-kuneiform number writing, which proceeds "cumulative-positionally" by writing the numbers 1–59 by cumulative repetition of two characters for 1 and 10, while the cumulative expressions formed in this way are in turn written down within an overarching place value system of the base 60, so that it is also possible to write numbers of any size, limited only by the available writing space.

## literature

• Stephen Chrisomalis: Numerical Notation. A Comparative History. Cambridge University Press, Cambridge [u. a.] 2010, ISBN 978-0-521-87818-0
• Geneviève Guitel : Histoire comparée des numérations écrites. Flammarion, Paris 1975
• Georges Ifrah: Histoire universelle des chiffres , Seghers, Paris 1981, reprint Editions Robert Laffont, Paris 1994, ISBN 2-221-07838-1 ; German translation: Universal history of numbers. Campus, Frankfurt / New York 1991, ISBN 3-88059-956-4
• Karl Menninger : number and number. A cultural history of the number. 2nd, revised and expanded edition, Vandenhoeck & Ruprecht, Göttingen 1958, reprint ibid 1998, ISBN 3-525-40701-7 digi20

## Individual evidence

1. Cf. Chrisomalis (2010), p. 9ff., From which the examples are also taken, and according to whose terminology the typological assignments are made (for Chrisomalis: "cumulative-additive", "multiplicative-additive", "ciphered-additive" , "cumulative-positional", "ciphered-positional")