A fraction 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 consists of a numerator displayed above a line, and a non-zero denominator, displayed below that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. In positive common fractions, 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 the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 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 are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one. Other uses for fractions are to represent ratios and division. Thus the fraction can also be used to represent the ratio 3:4, and the division 3 ÷ 4. The non-zero denominator rule, which applies when representing a division as a fraction, 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 represents a half dollar profit, then − represents a half dollar loss. Because of the rules of division of signed numbers, − and all represent the same fraction — negative one-half. And because a negative divided by a negative produces a positive, represents positive one-half. In mathematics the set of all numbers that can be expressed in the form 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. A number is a rational number precisely when it can be written in that form. However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions, and expressions that contain irrational numbers, such as \frac. This notation uses two or more lines of ordinary text, and results in a variation in spacing between lines when included within other text. While large and legible, these can be disruptive, particularly for simple fractions or within complex fractions. History The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used Egyptian fractions BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems. The Greeks used unit fractions and continued fractions. Followers of the Greek philosopher Pythagoras discovered that the square root of two cannot be expressed as a fraction of integers. In Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions. A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta, Brahmagupta, and Bhaskara. Their works form fractions by placing the numerators over the denominators, but without a bar between them. which is the equivalent of and would be written in modern notation as 6 1 and 2 − . The horizontal fraction bar is first attested in the work of Al-Hassār, The same fractional notation—with the fraction given before the integer In discussing the origins of decimal fractions, Dirk Jan Struik states: "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, 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." While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. In formal education Pedagogical tools In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper, pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboards, counters and computer software. Documents for teachers Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form where a is a whole number and b is a positive whole number. " The document itself also refers to negative fractions. See also Cross multiplication 0.999... Multiple FRACTRAN Notes References External links