fraction
fraction
[′frak·shən]Fraction
in arithmetic, a quantity consisting of an integral number of parts of a unit. A fraction is represented by the symbol m/n, where n, the denominator of the fraction, indicates the number of parts into which the unit is to be divided and m, the numerator of the fraction, indicates the number of such parts taken. A fraction may be viewed as the quotient obtained by dividing one integer (m) by another (n). If m is divisible by n without a remainder, then the quotient m\n denotes an integer (for example, 6/3 = 2, 33/11 = 3). The numerator and denominator of a fraction may be simultaneously multiplied or divided by the same number without changing the value of the fraction. Any fraction can be represented in reduced form, that is, as a fraction whose numerator and denominator do not have common factors; for example, 16/72 is not in reduced form [16/72 = 2x8/9x 8 = 2/9], but 27/64 is. To add fractions with the same denominator, we add their numerators and take the same denominator: (a/b) + (c/b) + (d/b) = (a + c + d)/b. To add several fractions with different denominators, it is necessary to bring them to a common denominator. Subtraction of fractions is done in the same way. To multiply several fractions, we divide the product of their numerators by the product of their denominators: (a/b) x (c/d) = ac/bd. Defining division as an inverse operation of multiplication implies the following rule for division: (a/b) ÷ (c/d) = ad/bc. If the numerator of a fraction is less than the denominator, the fraction is called a proper fraction; if the opposite is true, it is called an improper fraction. An improper fraction may be shown to be the sum of an integer and a proper fraction (a mixed number). For this it is necessary to divide the numerator (with remainder) by the denominator; for example,
This proposition of elementary arithmetic can be extended to all real numbers: a real number x can be represented uniquely as x = n + d, where n is an integer and 0 ≤ d < 1. The integer n is called the integral part of x and is denoted by [x]. The number d = x - [x] is called the fractional part of x.
Decimal fractions are fractions whose denominator is a power of 10. Such fractions are written without denominators; for example, 5,481,475/10,000 = 548.1475 and 23/1,000 = 0.023.
Operations with fractions are encountered in the ancient Egyptian Ahmes papyrus (c. 2000 B.C.) where the only admissible fractions are fractions of the type l/n (aliquot fractions). Hence the distinctive “Egyptian” problem of representing any fraction as the sum of unequal fractions of the type l/n (in addition to aliquot fractions, the Egyptians had a special symbol for the fraction 2/3); for example, 7/29 = (1/5) + (1/29) + (1/145).
In ancient Babylonian manuscripts we encounter so-called sexagesimal fractions, that is, fractions having a denominator that is a power of 60. The number 60 played a significant role in classical arithmetic; the division of a unit into 60 and 3,600 = 602 parts has been preserved to the present day in the division of an hour or a degree into 60 minutes (1/60) and of a minute into 60 seconds. The ancient Hindus, apparently, were the first to conceive the modern symbol for a fraction.
REFERENCES
Entsiklopediia elementarnoi matematiki, book 1: Arifmetika. Moscow-Leningrad, 1951.Depman, I. la. Istoriia arifmetiki, 2nd ed. Moscow, 1965.
Fraction
a portion of a granular or lumpy solid (such as crushed rock, sand, or powder) or of a liquid mixture (such as petroleum) isolated according to a specific criterion. In sieve analysis, fractions are isolated by particle or grain size; in gravity concentration, by density; and in petroleum distillation, by boiling point.