Fraction (mathematics)

A fraction (from Latin fractus, 'broken') represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: 1 2 {displaystyle { frac {1}{2}}} and 17 3 {displaystyle { frac {17}{3}}} ) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.'The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).' A fraction (from Latin fractus, 'broken') represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: 1 2 {displaystyle { frac {1}{2}}} and 17 3 {displaystyle { frac {17}{3}}} ) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. We begin with positive common fractions, where the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero because zero parts can never make up a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates 3 4 {displaystyle { frac {3}{4}}} or ​3⁄4 of a cake. A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10−2 all equal the fraction 1/100. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four). The non-zero denominator in the case using a fraction to represent division is an example of the rule that division by zero is undefined. We can also write negative fractions, which represent the opposite of a positive fraction. For example if 1/2 represents a half dollar profit, then −1/2 represents a half dollar loss. Because of the rules of division of signed numbers, which require that, for example, negative divided by positive is negative, −1/2, -1/2 and 1/-2, all represent the same fraction, negative one-half. Because a negative divided by a negative produces a positive, -1/-2 represents positive one-half. In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational). In a fraction, the number of equal parts being described is the numerator (from Latin numerātor, 'counter' or 'numberer'), and the type or variety of the parts is the denominator (from Latin dēnōminātor, 'thing that names or designates'). As an example, the fraction ​8⁄5 amounts to eight parts, each of which is of the type named 'fifth'. In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor. Informally, the numerator and denominator may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The fraction bar may be horizontal (as in 1/3), oblique (as in 1/5), or diagonal (as in ​1⁄9). These marks are respectively known as the horizontal bar, the slash (US) or stroke (UK), and the fraction slash. In typography, horizontal fractions are also known as 'en' or 'nut fractions' and diagonal fractions as 'em fractions', based on the width of a line they take up. The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. (For example, ​2⁄5 and ​3⁄5 are both read as a number of 'fifths'.) Exceptions include the denominator 2, which is always read 'half' or 'halves', the denominator 4, which may be alternatively expressed as 'quarter'/'quarters' or as 'fourth'/'fourths', and the denominator 100, which may be alternatively expressed as 'hundredth'/'hundredths' or 'percent'. When the denominator is 1, it may be expressed in terms of 'wholes' but is more commonly ignored, with the numerator read out as a whole number. (For example, 3/1 may be described as 'three wholes' or as simply 'three'.) When the numerator is one, it may be omitted. (For example, 'a tenth' or 'each quarter'.)