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Notation for Fractions
Characteristics of Fractions
Arithmetic Operations Involving Fractions
When fractions having the same denominator, as 3-10 and 4-10, are added, only the numerators are added, and their sum is then written over the common denominator: 3-10+4-10=7-10. Fractions having unlike denominators, e.g., 1-4 and 1-6, must first be converted into fractions having a common denominator, a denominator into which each denominator may be divided, before addition may be performed. In the case of 1-4 and 1-6, for example, the lowest number into which both 4 and 6 are divisible is 12. When both fractions are converted into fractions having this number as a denominator, then 1-4 becomes 3-12, and 1-6 becomes 2-12. The change is accomplished in the same way in both cases—the denominator is divided into the 12 and the numerator is multiplied by the result of this division. The addition then is performed as in the case of fractions having the same denominator: 1-4+1-6=3-12+2-12=5-12. In subtraction, the numerator and the denominator are subjected to the same preliminary procedure, but then the numerators of the converted fractions are subtracted: 1-4−1-6=3-12−2-12=1-12.
In multiplication the numerators of the fractions are multiplied together as are the denominators without needing change: 2-3×3-5=6-15. It should be noted that the result, here 6-15, may be reduced to 2-5 by dividing both numerator and denominator by 3. The division of one fraction by another, e.g., 3-5÷1-2, is performed by inverting the divisor and multiplying: 3-5÷1-2=3-5×2-1=6-5. The same rules apply to the addition, subtraction, multiplication, and division of fractions in which the numerators and denominators are algebraic expressions.
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.
REFERENCESEntsiklopediia elementarnoi matematiki, book 1: Arifmetika. Moscow-Leningrad, 1951.
Depman, I. la. Istoriia arifmetiki, 2nd ed. Moscow, 1965.
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.