Numerals. The invention of signs to represent numbers is doubtless much older than any form of writing. But the origin of counting, such as would involve the use of signs, is not so ancient as might be thought; the power of apprehending even comparatively small numbers comes but late in the development from savage to civilised life. Even yet the aborigines of Australia work with only the numbers 1 and 2; 3 being 2 and 1 or 1 and 2; 4 being 2 and 2; and, as a rule, no Australian black can count as high as 7. The earliest visible signs are doubtless the fingers held up; and the denary system of notation is due to the fact that we have ten fingers. The rude method of finger-counting has been developed into a highly-complicated system of reckoning, still in use in eastern Europe by pedlars; various positions and arrangements of the ten digits allowing of reckoning as high as 10,000. For permanent purposes a system of single strokes is the most obvious method; and series of strokes as high as four or five are found in various countries in old inscriptions. But strokes, when numerous, are inconvenient and confusing; hence additional symbols are found to make their appearance for 5, 10, 100, and 1000. In Babylonian inscriptions two Cuneiforms (q.v.) serve to express all the numbers from 1 to 99. The Egyptian scheme is explained and illustrated at HIEROGLYPHICS (Vol. V. p. 707); and from these hieroglyphs were derived the Phœnician, Palmyrene, and Syriac numerals.
After alphabetic writing was in use, the alphabetic signs obviously lent themselves to employment as numerals—either following the order of the letters, each having a successively greater value than its predecessor; or the initial letter of the word for the several numbers might be used. Thus, according to the latter method, the Greek inscriptions used I for 1, II (Πέντε) for 5, Δ (Δέκα) for 10, Η (the old sign for the rough breathing in "Ἔκατον" for 100, Χ (Χίλιοι) for 1000, and Μ (Μύρια) for 10,000. Then a Π with a Δ inscribed in it stood for 50 (), and with Η inscribed () for 500. In this connection the capitals or uncials were used of course. Otherwise, following simply the order of the letters, the twenty-four letters of the Ionic alphabet were used for the numbers 1 to 24; the books of the Iliad, for example, are often thus numbered. But a more ingenious method was soon adopted by the Greeks, as also by the Hebrews. The alphabet (cursive) was divided into three groups, of which the first did duty for the units, the second for the tens, the next for hundreds. The Hebrew square character had twenty-two distinct letters, and double forms for five of them, so that three groups, each of nine characters, were available. The Greek alphabet, as ultimately arranged, had twenty-four letters; the three additional signs required to make up three nines were obtained by keeping two of the old Phœnician letters or (see DIGAMMA) for 6, and 5 or 4 (koph) for 90, and adding the superfluous sibilant (sampi) for 900. Then to were 1 to 9; from to koph were 10 to 90; to sampi were 100 to 900. The thousands were made by subscribing an beneath the units; thus was 1000; is 1891. Sometimes a sort of algebraic method was employed for larger numbers; .
The cumbrous Roman method of using the capitals is familiar enough to ourselves yet. The C has been understood to be the initial of centum, and M of mille. But some (as Canon Taylor) contend that the Latins, when they dropped the Greek phi, chi, and theta as phonetic signs, retained them as numerals, with arbitrary values. In this case the C would be originally , assimilated to C, because C was the initial of centum. The old , used for 1000, came to be written CI, afterwards confounded with or M, the initial of mille. The L would be a derivative from an old Chalcidian form of chi, inscribed for lapidary purposes , and then simplified. Those who do not accept the theory of the dropped Greek letters suppose that M is from a circle with a vertical stroke, the C a circle with a horizontal stroke or a cross, . The X, V, and L might all come from this letter. In any case, X is twice V (whether or not the latter originated in the hand held with the thumb to one side and the other fingers together); and D (for 500) or IO is half CI. See the articles in this work on the letters C, D, L, and M.
It is doubtful how far the Abacus (q.v.) has to do with the development of the system of numerals, in which the value of the cipher depends on its position. There were abacus boards so arranged that the first column meant units, the second tens, the third hundreds, the fourth thousands; or, conversely, a method of writing numbers derived from this was actually used in Europe in the middle ages; we show the columnar arrangement simplifying the reading in the several cases, 654, 650, 604, 54.
In the decimal scheme of figures as now used by us, the nine numerals with the zero, which enables the value of the position to be secured without abacus or columnar arrangement, are known as the Arabic numerals, but are unquestionably of Indian origin. From India they were apparently brought to Bagdad after the middle of the 8th century, and their value and use was set forth early next century by the Arab mathematician Abu Ja'far Mohammed Ben Mnsa, or Al-Kharizmi ('native of Khwarizm'—Khiva); whence the system came to be known in Europe, where it became familiar in the 12th century as Algorism (erroneously Algorithm). The earliest European forms of these characters are found in MSS. of the 12th century; by the 14th they were practically of the same shape as now. The 12th century numerals are evidently forms of the Gobar or western Arabic numerals used in a १ २ ३ ४ ५ ६ ७ ८ ९ ०
b 1 2 3 4 5 6 7 8 9 .
c 1 2 3 4 5 6 7 8 9 0 a, Indian, 10th century; b, Gobar, 10th century;
c, European, 12th century.
Persia in the 10th century. These can be traced to the contemporary Indian Devanagari numerals, which again are as certainly based on an old series of characters used in cave-inscriptions in the 1st and 2d centuries. These Canon Taylor contends are (mainly, at least) degraded forms of the Indo-Bactrian alphabet. See ALPHABET, Vol. I. p. 188. The modern arithmetic was not practised in England till about the middle of the 16th century, and for a long time after its introduction was taught only in the universities.
The decimal system, possessing only nine symbols—viz. 1, 2, 3, 4, 5, 6, 7, 8, 9 (called the nine digits)—adopts the principle of giving to each symbol or 'figure' two values, one the absolute value, and the other a value depending upon its position, a figure moved one place to the left being held to be increased in value ten times. When such a number as 6473 is analysed, it is seen to mean ; and 6004 becomes . In this latter instance the peculiar importance of the figure 0 is seen (see DECIMAL SYSTEM).
It should be mentioned that European nations do not all read the numerals in the same way, as regards larger numbers. Let us take the figures 56,084,763,204,504; these, read after the fashion of the French and other continental arithmeticians, are fifty-six trillions, eighty-four billions, seven-hundred-and-sixty-three millions, two-hundred-and-four thousands, five-hundred-and-four units; and so also in America. In Britain, instead of billions, we have, according to the current usage, thousands of millions; after this, tens of thousands of millions and hundreds of thousands of millions, and then billions, which are the same as the French trillions. The above number, according to the British mode, would be read fifty-six billions, eighty-four-thousand-seven-hundred-and-sixty-three millions, two-hundred-and-four thousands, five-hundred-and-four units. The British trillion has nineteen figures, the continental has thirteen.
As to the Indian origin of our numerals, see Canon Taylor, The Alphabet (1883; vol. ii. p. 263-268); Woepke, Memoire sur la Propagation des Chiffres Indiens (1863); Burnell, South Indian Palaeography (1874); Treutlein, Geschichte unsrer Zahlzeichen (1875).