Fraction
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: and ) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, 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 3/4 of a cake.
For other uses, see Fraction (disambiguation).
Other uses for fractions are to represent ratios and division.[1] Thus the fraction 3/4 can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).
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 states in part that negative divided by positive is negative), −1/2, −1/2 and 1/−2 all represent the same fraction – negative one-half. And 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 or , which stands for quotient. A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as (see Square root of 2) and π/4 (see Proof that π is irrational).
Forms of fractions
Simple, common, or vulgar fractions
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a/b or , where a and b are both integers.[9] As with other fractions, the denominator (b) cannot be zero. Examples include 1/2, −8/5, −8/5, and 8/−5. The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.[10]
Common fractions can be positive or negative, and they can be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not common fractions; though, unless irrational, they can be evaluated to a common fraction.
Built-up fractions: . 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.
In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half). See the article on Number Forms for information on doing this in Unicode.
Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:[29]
In formal education
Pedagogical tools
In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), 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 is a whole number and is a positive whole number. (The word fraction in these standards always refers to a non-negative number.)"[43] The document itself also refers to negative fractions.
