REPRESENTATION OF NUMBERS BY SYMBOLS. Aside from primitive number-pictures, such as the Egyptian hieroglyphics and the Babylonian cuneiform symbols, the ancients commonly used the letters of their alphabets to represent num bers—e.g., the a, /3, y, 4, of the Greeks. Some what more refined is the system of the Romans, who used only a limited set of letters, combin ing these according to simple additive and sub tractive principles. But even in the Roman notation no extended calculations were possible without the aid of some registering instrument; hence the early and extensive use of the abacus. (See CALCULATING MACHINES.) The notation in use at present, which consists in combining tell digits according to a simple position-system, originated with the Hindus, was transmitted to the Arabs, and came to the knowledge of Euro peans chiefly through the labors of Leonardo of Pisa, about A.D. 1200. This powerful system freed arithmetic from the reign of the abacus.
As to fractional numbers, the Ahmes papyius, which is at least thirty-six centuries old, shows that the Egyptians had a knowledge of fractions at a very remote date. But while the concept
and symbolism of the common fraction are thus very old, the decimal fraction, the decimal point, and other improvements in notation. are com paratively recent, dating from the Sixteenth and Seventeenth centuries.
Besides the decimal scale, fractions have also been written on various other scales, such as the binary (scale of two), ternary (scale of three), . . duodenary (scale of twelve), and nota bly the sexagesimal (scale of sixty) by the Egyptians and Babylonians. At present, the scale of ten is generally recognized as the most convenient. The scale of twelve. how ever, has the advantage of producing simpler tractional forms. E.g., on the scale of ten the fractions %, are written, respectively, 0 333.... 0.25. 0.125: on the scale of twelve they are written, 0.4. 0.3, 0.16.