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A fraction (from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
', "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: \tfrac and \tfrac) consists of a numerator displayed above a line (or before a slash like ), and a non-zero 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. In positive common fractions, the numerator and denominator are
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. 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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
. That same number can also be represented as a
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
, 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 An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). Other uses for fractions are to represent
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
s and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
. Thus the fraction 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). The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 (which states in part that negative divided by positive is negative), −, 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 In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
. 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 fraction In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmeti ...
s (quotients of algebraic expressions), and expressions that contain
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, such as \frac (see
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
) and (see
proof that π is irrational In the 1760s, Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French language, French; 26 or 28 August 1728 – 25 September 1777) was a Switzerland, Swiss polymath who made important contributions to the subjects of ma ...
).


Vocabulary

In a fraction, the number of equal parts being described is the numerator (from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
', "counter" or "numberer"), and the type or variety of the parts is the denominator (from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
', "thing that names or designates"). As an example, the fraction amounts to eight parts, each of which is of the type named "fifth". In terms of
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, the numerator corresponds to the
dividend A dividend is a distribution of profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit ...
, and the denominator corresponds to the
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in ), oblique (as in 2/5), or diagonal (as in ). These marks are respectively known as the horizontal bar; the virgule,
slash Slash may refer to: * Slash (punctuation), the "/" character Arts and entertainment Fictional characters * Slash (Marvel Comics) * Slash (Teenage Mutant Ninja Turtles), Slash (''Teenage Mutant Ninja Turtles'') Music * Slash (musician), stage n ...
( US), or
stroke A stroke is a medical condition A disease is a particular abnormal condition that negatively affects the structure or function (biology), function of all or part of an organism, and that is not due to any immediate external injury. Di ...
( UK); and the fraction bar, solidus, or
fraction slash The slash is an oblique slanting line #Conjunction, punctuation mark . Once used to mark full stop, periods and commas, the slash is now most often used to represent #XOR, exclusive or #And, inclusive or, #Division, division and #Fractions, fra ...
. In
typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by humans, including ...

typography
, fractions stacked vertically are also known as " en" or "
nut Nut often refers to: * Nut (fruit), a fruit composed of a hard shell and a seed * Nut (food), collective noun for dry and edible fruits or seeds * Nut (hardware), a fastener used with a bolt Nut or Nuts may also refer to: Places * Nomenclature of ...
fractions", and diagonal ones as " em" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow ''en'' square, or a wider ''em'' square.. In traditional
typefounding Movable Type is a weblog publishing system developed by the company Six Apart Six Apart Ltd., sometimes abbreviated 6A, is a software company known for creating the Movable Type Movable Type is a blog software, weblog publishing system develop ...
, a piece of type bearing a complete fraction (e.g. ) was known as a "case fraction," while those representing only part of fraction were called "piece fractions." The denominators of English fractions are generally expressed as
ordinal numbers In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

ordinal numbers
, in the plural if the numerator is not 1. (For example, and 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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, may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter"). The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction and "two fifths" is the same fraction understood as 2 instances of .) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
. (For example, may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a
slash mark The slash is an oblique slanting line punctuation mark . Once used to mark periods and comma The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe The apost ...
. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are ''not'' powers of ten are often rendered in this fashion (e.g., as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., as "six-millionths", "six millionths", or "six one-millionths").


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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
written as ''a''/''b'' or \tfrac, where ''a'' and ''b'' are both
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

integers
. As with other fractions, the denominator (''b'') cannot be zero. Examples include \tfrac, -\tfrac, \tfrac, and \tfrac. The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy. ''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. * A
unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was original ...
is a common fraction with a numerator of 1 (e.g., \tfrac). Unit fractions can also be expressed using negative exponents, as in 2−1, which represents 1/2, and 2−2, which represents 1/(22) or 1/4. * A
dyadic fraction 300px, Dyadic rationals in the interval from 0 to 1. In mathematics, a dyadic rational is a number that can be expressed as a fraction whose denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic l ...

dyadic fraction
is a common fraction in which the denominator is a
power of two A power of two is a number of the form where is an integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Comm ...
, e.g. \tfrac=\tfrac. In Unicode, precomposed fraction characters are in the
Number Forms Number Forms is a Unicode blockA Unicode block is one of several contiguous ranges of numeric character codes ( code points) of the Unicode Unicode is an information technology Technical standard, standard for the consistent character encodin ...
block.


Proper and improper fractions

Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook ''
The Ground of Arts ''The Ground of Arts'' Robert Recorde's ''Arithmetic: or, The Ground of Arts'' was one of the first printing, printed English language, English textbooks on arithmetic and the most popular of its time. ''The Ground of Arts'' appeared in London in 1 ...
''. In general, a common fraction is said to be a proper fraction, if the
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3.


Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of \tfrac, for instance, is \tfrac. The product of a fraction and its reciprocal is 1, hence the reciprocal is the
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. When the numerator and denominator of a fraction are equal (for example, \tfrac), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as \tfrac, where 1 is sometimes referred to as the ''invisible denominator''. Therefore, every fraction or integer, except for zero, has a reciprocal. For example. the reciprocal of 17 is \tfrac.


Ratios

A
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group ''n''". For example, if a car lot had 12 vehicles, of which * 2 are white, * 6 are red, and * 4 are yellow, then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1. A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that of the cars or of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
that it would be yellow.


Decimal fractions and percentages

A
decimal fraction The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of a
decimal separator A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...

decimal separator
, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see
decimal separator A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...
). Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, ''viz.'' 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the
fractional part The fractional part or decimal part of a non‐negative real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republi ...
of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, 3\tfrac. Decimal fractions can also be expressed using
scientific notation Scientific notation is a way of expressing numbers A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or coul ...
with negative exponents, such as , which represents 0.0000006023. The represents a denominator of . Dividing by moves the decimal point 7 places to the left. Decimal fractions with infinitely many digits to the right of the decimal separator represent an
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. For example, = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... . Another kind of fraction is the
percentage In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

percentage
(Latin ''per centum'' meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100. The related concept of ''
permille Per mille (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
'' or ''parts per thousand'' (ppt) has an implied denominator of 1000, while the more general
parts-per notation In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and predictio ...
, as in 75 ''parts per million'' (ppm), means that the proportion is 75/1,000,000. Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By
mental calculation Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies (such as pencil and paper) or devices such as a calculator. People may use mental calculation when computing tools are not availab ...
, it is easier to
multiply Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on Computer, computers, by an asterisk ) is one of the four Elementary arithmetic, eleme ...

multiply
16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more
accurate In a set of measurements, accuracy is closeness of the measurements to a specific value, while precision is the closeness of the measurements to each other. ''Accuracy'' has two definitions: # More commonly, it is a description of ''systematic err ...
to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6.


Mixed numbers

A mixed numeral (also called a ''mixed fraction'' or ''mixed number'') is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign). It is used primarily in measurement: 2\tfracinches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers. The sum is implied without the use of a visible operator such as the appropriate "+". For example, in referring to two entire cakes and three quarters of another cake, the numerals denoting the integer part and the fractional part of the cakes are written next to each other as 2\tfracinstead of the unambiguous notation 2+\tfrac. Negative mixed numerals, as in -2\tfrac, are treated like \scriptstyle -\left(2+\frac\right). Any such sum of a ''whole'' plus a ''part'' can be converted to an
improper fraction A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
by applying the rules of adding unlike quantities. This tradition is, formally, in conflict with the notation in algebra where adjacent symbols, without an explicit
infix operator Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of Operator (mathematics), operators between operands—"infixed operators"—such as the plus sign in 2 + 2. U ...
, denote a product. In the expression 2x, the "understood" operation is multiplication. If is replaced by, for example, the fraction \tfrac, the "understood" multiplication needs to be replaced by explicit multiplication, to avoid the appearance of a mixed number. When multiplication is intended, 2 \tfrac may be written as : 2 \cdot \frac,\quad or \quad 2 \times \frac,\quad or \quad 2 \left(\frac\right),\;\ldots An improper fraction can be converted to a mixed number as follows: # Using
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
(division with remainder), divide the numerator by the denominator. In the example, \tfrac, divide 11 by 4. 11 ÷ 4 = 2 remainder 3. # The
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
(without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part. # The new denominator is the same as the denominator of the improper fraction. In the example, it is 4. Thus \tfrac =2\tfrac.


Historical notions


Egyptian fraction

An
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
is the sum of distinct positive unit fractions, for example \tfrac+\tfrac. This definition derives from the fact that the
ancient Egypt Ancient Egypt was a civilization  A civilization (or civilisation) is a that is characterized by , , a form of government, and systems of communication (such as ). Civilizations are intimately associated with additional char ...

ancient Egypt
ians expressed all fractions except \tfrac, \tfrac and \tfrac in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, \tfrac can be written as \tfrac + \tfrac + \tfrac. Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write \tfrac are \tfrac+\tfrac+\tfrac and \tfrac+\tfrac+\tfrac+\tfrac.


Complex and compound fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, \frac and \frac are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example: :\frac=\tfrac\times\tfrac=\tfrac :\frac = 12\tfrac \cdot \tfrac = \tfrac \cdot \tfrac = \tfrac \cdot \tfrac = \tfrac :\frac5=\tfrac\times\tfrac=\tfrac :\frac=8\times\tfrac=24. If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations, e.g. as :5/(10/(20/40)) = \frac = \frac\quad or as \quad (5/10)/(20/40) = \frac = 1 A compound fraction is a fraction of a fraction, or any number of fractions connected with the word ''of'', corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...
). For example, \tfrac of \tfrac is a compound fraction, corresponding to \tfrac \times \tfrac = \tfrac. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction \tfrac \times \tfrac is equivalent to the complex fraction \tfrac.) Nevertheless, "complex fraction" and "compound fraction" may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts".


Arithmetic with fractions

Like whole numbers, fractions obey the
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
,
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and distributive laws, and the rule against
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
.


Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number n, the fraction \tfrac equals 1. Therefore, multiplying by \tfrac is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction \tfrac. When the numerator and denominator are both multiplied by 2, the result is \tfrac, which has the same value (0.5) as \tfrac. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (\tfrac) make up half the cake (\tfrac).


Simplifying (reducing) fractions

Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction \tfrac are divisible by c, then they can be written as a=cd and b=ce, and the fraction becomes \tfrac, which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction \tfrac. If one takes for the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

greatest common divisor
of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
s. One says that the fraction has been reduced to its ''
lowest terms An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English ...
''. If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be '' irreducible'', ''reduced'', or ''in simplest terms''. For example, \tfrac is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, \tfrac ''is'' in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1. Using these rules, we can show that \tfrac = \tfrac = \tfrac = \tfrac, for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction \tfrac can be reduced to lowest terms by dividing the numerator and denominator by 21: :\tfrac = \tfrac= \tfrac The
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
gives a method for finding the greatest common divisor of any two integers.


Comparing fractions

Comparing fractions with the same positive denominator yields the same result as comparing the numerators: :\tfrac>\tfrac because , and the equal denominators 4 are positive. If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions: :\tfrac<\tfrac \text \tfrac= \tfrac \text -3 < -2. If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. One way to compare fractions with different numerators and denominators is to find a common denominator. To compare \tfrac and \tfrac, these are converted to \tfrac and \tfrac (where the dot signifies multiplication and is an alternative symbol to ×). Then ''bd'' is a common denominator and the numerators ''ad'' and ''bc'' can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare ''ad'' and ''bc'', without evaluating ''bd'', e.g., comparing \tfrac ? \tfrac gives \tfrac>\tfrac. For the more laborious question \tfrac ? \tfrac, multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding \tfrac ? \tfrac. It is not necessary to calculate 18 \times 17 – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is \tfrac>\tfrac. Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.


Addition

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows: :\tfrac24+\tfrac34=\tfrac54=1\tfrac14.


Adding unlike quantities

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator 1. For adding quarters to thirds, both types of fraction are converted to twelfths, thus: : \frac14\ + \frac13=\frac\ + \frac=\frac3\ + \frac4=\frac7. Consider adding the following two quantities: :\frac35+\frac23 First, convert \tfrac35 into fifteenths by multiplying both the numerator and denominator by three: \tfrac35\times\tfrac33=\tfrac9. Since \tfrac33 equals 1, multiplication by \tfrac33 does not change the value of the fraction. Second, convert \tfrac23 into fifteenths by multiplying both the numerator and denominator by five: \tfrac23\times\tfrac55=\tfrac. Now it can be seen that: :\frac35+\frac23 is equivalent to: :\frac9+\frac=\frac=1\frac4 This method can be expressed algebraically: :\frac + \frac = \frac This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding \tfrac and \tfrac the single denominators have a common factor 2, and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator. :\begin \frac34+\frac56 &= \frac+\frac=\frac + \frac&=\frac\\ &=\frac+\frac =\frac + \frac&=\frac \end The smallest possible denominator is given by the
least common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'a ...

least common multiple
of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. This is called the least common denominator.


Subtraction

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, :\tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16


Multiplication


Multiplying a fraction by another fraction

To multiply fractions, multiply the numerators and multiply the denominators. Thus: :\frac \times \frac = \frac To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths. A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example: :\frac \times \frac = \frac \times \frac = \frac \times \frac = \frac A two is a common
factor FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry". FACTOR was founded in 1982 by r ...

factor
in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.


Multiplying a fraction by a whole number

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. :6 \times \tfrac = \tfrac \times \tfrac = \tfrac This method works because the fraction 6/1 means six equal parts, each one of which is a whole.


Multiplying mixed numbers

When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction. For example: :3 \times 2\frac = 3 \times \left (\frac + \frac \right ) = 3 \times \frac = \frac = 8\frac In other words, 2\tfrac is the same as \tfrac + \tfrac, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is 8\tfrac, since 8 cakes, each made of quarters, is 32 quarters in total.


Division

To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, \tfrac \div 5 equals \tfrac and also equals \tfrac = \tfrac, which reduces to \tfrac. To divide a number by a fraction, multiply that number by the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
of that fraction. Thus, \tfrac \div \tfrac = \tfrac \times \tfrac = \tfrac = \tfrac.


Converting between decimals and fractions

To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change to a decimal, divide by (" into "), to obtain . To change to a decimal, divide by (" into "), and stop when the desired accuracy is obtained, e.g., at decimals with . The fraction can be written exactly with two decimal digits, while the fraction cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus 12.3456 = \tfrac.


Converting repeating decimals to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite
repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...
is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. The preferred way to indicate a repeating decimal is to place a bar (known as a
vinculum Vinculum may refer to: * Vinculum (ligament), a band of connective tissue, similar to a ligament, that connects a flexor tendon to a phalanx bone * Vinculum (symbol), a horizontal line used in mathematical notation for a specific purpose * Vinculum ...
) over the digits that repeat, for example = 0.789789789... For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. For example: : = 5/9 : = 62/99 : = 264/999 : = 6291/9999 In case
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s precede the pattern, the nines are suffixed by the same number of
trailing zeroIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s: : = 5/90 : = 392/999000 : = 12/9900 In case a non-repeating set of decimals precede the pattern (such as ), we can write it as the sum of the non-repeating and repeating parts, respectively: :0.1523 + Then, convert both parts to fractions, and add them using the methods described above: :1523 / 10000 + 987 / 9990000 = 1522464 / 9990000 Alternatively, algebra can be used, such as below: # Let ''x'' = the repeating decimal: #: ''x'' = # Multiply both sides by the power of 10 just great enough (in this case 104) to move the decimal point just before the repeating part of the decimal number: #: 10,000''x'' = # Multiply both sides by the power of 10 (in this case 103) that is the same as the number of places that repeat: #: 10,000,000''x'' = # Subtract the two equations from each other (if ''a'' = ''b'' and ''c'' = ''d'', then ''a'' − ''c'' = ''b'' − ''d''): #: 10,000,000''x'' − 10,000''x'' = − # Continue the subtraction operation to clear the repeating decimal: #: 9,990,000''x'' = 1,523,987 − 1,523 #: 9,990,000''x'' = 1,522,464 # Divide both sides by 9,990,000 to represent ''x'' as a fraction #: ''x'' =


Fractions in abstract mathematics

In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair (a,b) of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s a and b \ne 0, for which the operations
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
are defined as follows: :(a,b) + (c,d) = (ad+bc,bd) \, :(a,b) - (c,d) = (ad-bc,bd) \, :(a,b) \cdot (c,d) = (ac,bd) :(a,b) \div (c,d) = (ad,bc) \quad(\text c \ne 0) These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as: :\begin -(a,b) &= (-a, b) & & \text \\ &&&\text (0,b) \text\\ (a,b)^ &= (b,a) & & \text a \ne 0, \\ &&&\text (b,b) \text. \end Furthermore, the relation, specified as :(a, b) \sim (c, d)\quad \iff \quad ad=bc, is an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of fractions. Each fraction from one equivalence class may be considered as a
representative Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someone ...
for the whole class, and each whole class may be considered as one abstract fraction. This equivalence is preserved by the above defined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions :(a,b) \sim (a',b')\quad and \quad (c,d) \sim (c',d') \quad imply ::((a,b) + (c,d)) \sim ((a',b') + (c',d')) and similarly for the other operations. In the case of fractions of integers, the fractions with and
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
and are often taken as uniquely determined representatives for their ''equivalent'' fractions, which are considered to be the ''same'' rational number. This way the fractions of integers make up the field of the rational numbers. More generally, ''a'' and ''b'' may be elements of any
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
''R'', in which case a fraction is an element of the
field of fractions In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
of ''R''. For example,
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s in one indeterminate, with coefficients from some integral domain ''D'', are themselves an integral domain, call it ''P''. So for ''a'' and ''b'' elements of ''P'', the generated ''field of fractions'' is the field of
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s (also known as the field of
rational function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

rational function
s).


Algebraic fractions

An algebraic fraction is the indicated
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
of two
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s. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same
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properties as arithmetic fractions. If the numerator and the denominator are
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s, as in \frac, the algebraic fraction is called a ''rational fraction'' (or ''rational expression''). An ''irrational fraction'' is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in \frac. The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as \frac, is called a complex fraction. The field of rational numbers is the
field of fractions In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
of the integers, while the integers themselves are not a field but rather an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. Similarly, the
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s with coefficients in a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients,
radical expression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s representing numbers, such as \textstyle \sqrt/2, are also rational fractions, as are a
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s such as \pi/2, since all of \sqrt,\pi, and 2 are
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s, and thus considered as coefficients. These same numbers, however, are not rational fractions with ''integer'' coefficients. The term
partial fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction \frac can be decomposed as the sum of two fractions: \frac + \frac. This is useful for the computation of
antiderivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
s of
rational function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

rational function
s (see
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for more).


Radical expressions

A fraction may also contain radicals in the numerator or the denominator. If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a
monomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator: : \frac = \frac \cdot \frac = \frac The process of rationalization of binomial denominators involves multiplying the top and the bottom of a fraction by the
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
of the denominator so that the denominator becomes a rational number. For example: :\frac = \frac \cdot \frac = \frac = \frac = - \frac :\frac = \frac \cdot \frac = \frac = \frac = - \frac Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator.


Typographical variations

In computer displays and
typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by humans, including ...

typography
, simple fractions are sometimes printed as a single character, e.g. ½ (
one half One half is the irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integer An integer (from the Latin wikt:integer#Lati ...
). See the article on
Number Forms Number Forms is a Unicode blockA Unicode block is one of several contiguous ranges of numeric character codes ( code points) of the Unicode Unicode is an information technology Technical standard, standard for the consistent character encodin ...
for information on doing this in
Unicode Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

Unicode
. Scientific publishing distinguishes four ways to set fractions, together with guidelines on use: * special fractions: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts. * case fractions: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them ''upright''. An example would be \tfrac, but rendered with the same height as other characters. Some sources include all rendering of fractions as ''case fractions'' if they take only one typographical space, regardless of the direction of the bar. * shilling or solidus fractions: 1/2, so called because this notation was used for pre-decimal British currency (
£sd £sd (occasionally written Lsd, spoken as "pounds, shillings and pence" or pronounced ) is the popular name for the pre-decimal currencies A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversi ...
), as in 2/6 for a half crown, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions ( complex fractions) or within exponents to increase legibility. Fractions written this way, also known as ''piece fractions'', are written all on one typographical line, but take 3 or more typographical spaces. * built-up fractions: \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 of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s: ancient symbols representing one part of two, one part of three, one part of four, and so on. The
Egyptians Egyptians ( arz, المصريين, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group of people originating from the country of Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a spanning t ...
used
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, ...
s  BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with
unit fractionA unit fraction is a rational number written as a fraction where the numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was original ...
s. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for
dyadic fraction 300px, Dyadic rationals in the interval from 0 to 1. In mathematics, a dyadic rational is a number that can be expressed as a fraction whose denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic l ...
s in the
Akhmim Wooden Tablet The Akhmim wooden tablets, also known as the Cairo wooden tablets (Cairo Cat. 25367 and 25368), are two wooden writing tablets from ancient Egypt, solving arithmetical problems. They each measure around and are covered with plaster. The tablets are ...
and several
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum The British Museum, in the Bloomsbury Bloomsbury is a district in the West End of London The West End of London (commonly referred to as the West End ...

Rhind Mathematical Papyrus
problems. The
Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group An ethnic group or ethnicity is a grouping of people A people is any plurality of person A person (plural people or persons) is a being that has cer ...
used unit fractions and (later)
continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. Followers of the
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

Pythagoras
( BC) discovered that the
square root of two The square root of 2, or the one-half power of 2, written in mathematics as \sqrt or 2^, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the principal square root of 2, to di ...
cannot be expressed as a fraction of integers. (This is commonly though probably erroneously ascribed to
Hippasus Hippasus of Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...
of
Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia Magna Graecia (, ; Latin meaning "Greater Greece", grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Roman ...
, who is said to have been executed for revealing this fact.) In
Jain Jainism (), traditionally known as ''Jain Dharma'', is an ancient Indian religion Indian religions, sometimes also termed Dharmic religions or Indic religions, are the religions that originated in the Indian subcontinent. These religion ...

Jain
mathematicians in
India India, officially the Republic of India (Hindi Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, ...
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 Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

Brahmagupta
(), and Bhaskara (). Their works form fractions by placing the numerators ( sa, amsa) over the denominators (), but without a bar between them. In
Sanskrit literature Sanskrit literature broadly comprises texts composed in the earliest attested descendant of the Proto-Indo-Aryan language Proto-Indo-Aryan (sometimes Proto-Indic) is the Linguistic reconstruction, reconstructed proto-language of the Indo-Aryan ...

Sanskrit literature
, fractions were always expressed as an addition to or subtraction from an integer. The integer was written on one line and the fraction in its two parts on the next line. If the fraction was marked by a small circle or cross , it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, Bhaskara I writes: : ६        १        २ : १        १        १ : ४        ५        ९ which is the equivalent of : 6        1        2 : 1        1        −1 : 4        5        9 and would be written in modern notation as 6, 1, and 2 −  (i.e., 1). The horizontal fraction bar is first attested in the work of Al-Hassār (), a Muslim mathematician from
Fez Fez most often refers to: * Fez (hat) The fez (, ), also called tarboosh ( ar, طربوش, translit=ṭarbūš, derived from fa, سرپوش, translit=sarpuš, lit=cap), is a felt headdress in the shape of a short cylindrical peakless hat, usuall ...

Fez
,
Morocco ) , image_map = Morocco (orthographic projection, WS claimed).svg , map_caption = Location of Morocco in northwest Africa.Dark green: Undisputed territory of Morocco.Lighter green: Western Sahara, a United Nations lis ...

Morocco
, who specialized in
Islamic inheritance jurisprudence Islamic Inheritance jurisprudence is a field of Islamic jurisprudence ( ar, فقه) that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called ''Mīrāth'', and its branch of Islamic law is technicall ...
. In his discussion he writes, "... for example, if you are told to write three-fifths and a third of a fifth, write thus, \frac." The same fractional notation—with the fraction given before the integer—appears soon after in the work of
Leonardo Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

Leonardo Fibonacci
in the 13th century. In discussing the origins of
decimal fractions The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the H ...
,
Dirk Jan Struik Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Netherlands, Dutch-born United States, American (since 1934) mathematician, historian of mathematics and Marxist philosophy, Marxian theoretician (Marxism), theoretician who spent ...
states:
"The introduction of decimal fractions as a common computational practice can be dated back to the
Flemish Flemish (''Vlaams'') is a Low Franconian Low Franconian, Low Frankish, NetherlandicSarah Grey Thomason, Terrence Kaufman: ''Language Contact, Creolization, and Genetic Linguistics'', University of California Press, 1991, p. 321. (Calling i ...
pamphlet ''De Thiende'', published at
Leyden Leiden ( , ; in English language, English and Archaism, archaic Dutch language, Dutch also Leyden) is a List of cities in the Netherlands by province, city and List of municipalities of the Netherlands, municipality in the Provinces of the N ...

Leyden
in 1585, together with a French translation, ''La Disme'', by the Flemish mathematician
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
(1548–1620), then settled in the Northern
Netherlands ) , national_anthem = ( en, "William of Nassau") , image_map = EU-Netherlands.svg , map_caption = , image_map2 = BES islands location map.svg , map_caption2 = , image_map3 ...

Netherlands
. It is true that decimal fractions were used by the
Chinese Chinese can refer to: * Something related to China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's most populous country, with a populat ...
many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
fractions with great ease in his ''Key to arithmetic'' (
Samarkand fa, سمرقند , native_name_lang = , settlement_type = City , image_skyline = , image_alt = , image_caption = Clockwise from the top: The Reg ...

Samarkand
, early fifteenth century)."
While the
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, Persian people, the majority ethnic group in Iran, not to be conflated with the Iranian peoples ** Persian language, an Iranian ...

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
Baghdad Baghdad (; ar, بَغْدَاد ) is the capital of Iraq Iraq ( ar, الْعِرَاق, translit=al-ʿIrāq; ku, عێراق, translit=Êraq), officially the Republic of Iraq ( ar, جُمْهُورِيَّة ٱلْعِرَاق '; ku, ...

Baghdad
i mathematician
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab Mathematics in medieval Islam, mathematician, who was active in Damascus and Baghdad. He wrote the earliest survivi ...
as early as the 10th century.


In formal education


Pedagogical tools

In
primary school A primary school (in Ireland, the United Kingdom, Australia, New Zealand, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school A school is ...

primary school
s, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting),
pattern block Pattern Blocks are a set of mathematical manipulatives developed in the 1960s. The six shapes are both a play resource and a tool for learning in mathematics, which serve to develop spatial reasoning skills that are fundamental to the learning of ma ...
s, 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 word ''fraction'' in these standards always refers to a non-negative number.)" The document itself also refers to negative fractions.


See also

* Cross multiplication * 0.999... * Multiple (mathematics), Multiple * FRACTRAN


Notes


References


External links

* * * * {{DEFAULTSORT:Fraction (Mathematics) Fractions (mathematics), Division (mathematics) Egyptian inventions Elementary arithmetic Numbers