numeration

numeration, in mathematics, process of designating numbers according to any particular system; the number designations are in turn called numerals. In any place value system of numeration, a base number must be specified, and groupings are then made by powers of the base number. The position of a numeral in a grouping indicates which power of the base it is to be multiplied by. The most widely used system of numeration is the decimal system, which uses base 10. Thus, in the decimal system, the numeral 342 means (3×10²)+(4×10¹)+(2×10⁰), or 300+40+2. The binary system uses base 2 and is important because of its application to modern computers. Whereas the decimal system uses the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, the binary system uses only the two digits 0 and 1. In the binary system, the numeral 111, for example, means (1×2²)+(1×2¹)+(1×2⁰), i.e., 4+2+1, or 7, in the decimal system. The decimal numeral 7 and the binary numeral 111 are thus designations for the same number. The duodecimal system uses 12 as a base and has some advantages arising from the fact that 12 is divisible by four different numbers-2, 3, 4, 6-other than 1 and 12 itself. The base 12 requires the use of 12 different digits. Thus, in addition to the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, the symbols X (called "dek") and E (called "el") to represent the numbers 10 and 11 have been suggested by the Duodecimal Society of America. The duodecimal numeral 24E, for example, means (2×12²)+(4×12¹)+(11×12⁰), i.e., (2×144)+(4×12)+(11×1), or 347, in the decimal system. The hexadecimal system, or base 16, uses the digits 0 through 9 and the letters A through F (or a through f) to represent 16 different digits. Hexadecimal numeration is often used in computing because it more readily translates the binary system used by computers than decimal numeration does. A computer byte, which is composed of 8 bits (binary digits), must be represented by the numbers 0 through 255 in the decimal system, but in the hexadecimal system it is represented by 00 through FF. The decimal, binary, duodecimal, and hexadecimal systems of numeration constitute only four examples. The ancient Babylonians used a system of base 60, which still survives in our smaller divisions both of time and of angle, i.e., minutes and seconds. In general, any integer n greater than one can be used as the base of a numeration system, and the system will employ n different digits.