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A fraction (from la, 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: $\tfrac$ 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
. That same number can also be represented as a
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic num ...
, a percent, or with a negative
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
. For example, 0.01, 1%, and 10−2 are all equal to the fraction 1/100. An
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 shows 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 the ...
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 ...
. Thus the fraction can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division (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, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there ...
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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
. 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 number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s, such as $\frac$ (see
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the prin ...
) and (see proof that π is irrational).

# Vocabulary

In a fraction, the number of equal parts being described is the numerator (from la, numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from la, dēnōminātor, "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 ...
, the numerator corresponds to the
dividend A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-i ...
, and the denominator corresponds to the
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
. 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 ( US), or
stroke A stroke is a medical condition in which poor blood flow to the brain causes cell death. There are two main types of stroke: ischemic, due to lack of blood flow, and hemorrhagic, due to bleeding. Both cause parts of the brain to stop func ...
( UK); and the fraction bar, solidus, or fraction slash. In
typography Typography is the art and technique of arranging type to make written language legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing ( leading), a ...
, fractions stacked vertically are also known as " en" or " nut 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 (US English; moveable type in British English) is the system and technology of printing and typography that uses movable components to reproduce the elements of a document (usually individual alphanumeric characters or punctuatio ...
, 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, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
, 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, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also use ...
". 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 In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
. (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. (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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
written as ''a''/''b'' or $\tfrac$, where ''a'' and ''b'' are both
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. 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 fraction 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 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, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
, e.g. $\tfrac=\tfrac$. In Unicode, precomposed fraction characters are in the
Number Forms Number Forms is a Unicode block containing Unicode compatibility characters that have specific meaning as numbers, but are constructed from other characters. They consist primarily of vulgar fractions and Roman numerals. In addition to the c ...
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''. In general, a common fraction is said to be a proper fraction, if the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
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 shows 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 the ...
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 concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
that it would be yellow.

## Decimal fractions and percentages

A
decimal fraction The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
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 part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
, the appearance of which (e.g., a period, an
interpunct An interpunct , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin script. (Word-separating spaces did n ...
(·), a comma) depends on the locale (for examples, see
decimal separator A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...
). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 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 x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part c ...
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 that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, ...
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, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. For example, = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... Another kind of fraction is the
percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also u ...
(from la, 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'' or ''parts per thousand'' (ppt) has an implied denominator of 1000, while the more general
parts-per notation In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, th ...
, 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 avai ...
, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate 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: $2\tfrac$inches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers. The sum can be 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 can be written next to each other as $2\tfrac$instead of the unambiguous notation $2+\tfrac.$ Negative mixed numerals, as in $-2\tfrac$, are treated like $\scriptstyle -\left\left(2+\frac\right\right).$ Any such sum of a ''whole'' plus a ''part'' can be converted to an improper fraction 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, 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 $\quad 2 \left\left(\frac\right\right),\;\ldots$ 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 dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller tha ...
(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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
(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 fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from ea ...
is the sum of distinct positive unit fractions, for example $\tfrac+\tfrac$. This definition derives from the fact that the ancient Egyptians 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 :$5/\left(10/\left(20/40\right)\right) = \frac = \frac\quad$ or as $\quad \left(5/10\right)/\left(20/40\right) = \frac = 1$ 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 mathematical operations of arithmetic, with the other ones being ad ...
). 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, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
,
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, and distributive laws, and the rule against
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there ...
.

## 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 ...
of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
s. One says that the fraction has been reduced to its '' lowest terms''. 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 e ...
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.

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: :$\tfrac24+\tfrac34=\tfrac54=1\tfrac14$.

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 and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by b ...
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 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: :$3 \times 2\frac = 3 \times \left \left(\frac + \frac \right \right) = 3 \times \frac = \frac = 8\frac$ 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 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 digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if ...
is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example: : = 5/9 : = 62/99 : = 264/999 : = 6291/9999 If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: : = 5/90 : = 392/999000 : = 12/9900 If a non-repeating set of decimals precede the pattern (such as ), one may write the number 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 $\left(a,b\right)$ of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s $a$ and $b \ne 0,$ for which the operations
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
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 ...
,
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 mathematical operations of arithmetic, with the other ones being ad ...
, 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 ...
are defined as follows: :$\left(a,b\right) + \left(c,d\right) = \left(ad+bc,bd\right) \,$ :$\left(a,b\right) - \left(c,d\right) = \left(ad-bc,bd\right) \,$ :$\left(a,b\right) \cdot \left(c,d\right) = \left(ac,bd\right)$ :$\left(a,b\right) \div \left(c,d\right) = \left(ad,bc\right) \quad\left(\text c \ne 0\right)$ 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: :$\begin -\left(a,b\right) &= \left(-a, b\right) & & \text \\ &&&\text \left(0,b\right) \text\\ \left(a,b\right)^ &= \left(b,a\right) & & \text a \ne 0, \\ &&&\text \left(b,b\right) \text. \end$ Furthermore, the relation, specified as :$\left(a, b\right) \sim \left(c, d\right)\quad \iff \quad ad=bc,$ is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
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, someon ...
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 :$\left(a,b\right) \sim \left(a\text{'},b\text{'}\right)\quad$ and $\quad \left(c,d\right) \sim \left(c\text{'},d\text{'}\right) \quad$ imply ::$\left(\left(a,b\right) + \left(c,d\right)\right) \sim \left(\left(a\text{'},b\text{'}\right) + \left(c\text{'},d\text{'}\right)\right)$ and similarly for the other operations. In the case of fractions of integers, the fractions with and
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
''R'', in which case a fraction is an element of the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
of ''R''. For example,
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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 fractions (also known as the field of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s).

# Algebraic fractions

An algebraic fraction is the indicated
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of two
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For ...
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 field properties as arithmetic fractions. If the numerator and the denominator are
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
of the integers, while the integers themselves are not a field but rather an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such as $\textstyle \sqrt/2,$ are also rational fractions, as are a
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classe ...
s such as $\pi/2,$ since all of $\sqrt,\pi,$ and $2$ are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, and thus considered as coefficients. These same numbers, however, are not rational fractions with ''integer'' coefficients. The term partial fraction 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, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolic ...
s of
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s (see partial fraction decomposition for more).

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, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
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 Binomial may refer to: In mathematics * Binomial (polynomial), a polynomial with two terms *Binomial coefficient, numbers appearing in the expansions of powers of binomials * Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition ...
denominators involves multiplying the top and the bottom of a fraction by the conjugate 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 legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing ( leading), a ...
, simple fractions are sometimes printed as a single character, e.g. ½ (
one half One half ( : halves) is the irreducible fraction resulting from dividing one by two or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or "halving"; conversely, ...
). See the article on
Number Forms Number Forms is a Unicode block containing Unicode compatibility characters that have specific meaning as numbers, but are constructed from other characters. They consist primarily of vulgar fractions and Roman numerals. In addition to the c ...
for information on doing this in
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
. 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 once common throughout Europe, especially in the British Isles and hence in several countries of the ...
), 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 reciprocals of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s: ancient symbols representing one part of two, one part of three, one part of four, and so on. The
Egyptians Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile Valley in Egypt. Egyptian identi ...
used
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from ea ...
s  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 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 ar ...
and several
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchas ...
problems. The
Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, ot ...
used unit fractions and (later)
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its intege ...
s. Followers of the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek t ...
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
( BC) discovered that the square root of two cannot be expressed as a fraction of integers. (This is commonly though probably erroneously ascribed to
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes ...
of
Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus and the Casuentus (modern Basento). It was distant about 20 km fr ...
, who is said to have been executed for revealing this fact.) In
Jain Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle bein ...
mathematicians in
India India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on th ...
wrote the "
Sthananga Sutra Sthananga Sutra (Sanskrit: Sthānāṅgasūtra Prakrit: Ṭhāṇaṃgasutta) (c. 3rd-4th century CE) forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala as per the ...
", 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 Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (w ...
(),
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tre ...
(), 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 all literature in the Sanskrit language. This includes texts composed in the earliest attested descendant of the Proto-Indo-Aryan language known as Vedic Sanskrit, texts in Classical Sanskrit as well as ...
, 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), a type of felt hat commonly worn in the Ottoman Empire * Fez, Morocco (or Fes), the second largest city of Morocco Fez or FEZ may also refer to: Media * ''Fez'' (Frank Stella), a 1964 painting by the moder ...
,
Morocco Morocco (),, ) officially the Kingdom of Morocco, is the westernmost country in the Maghreb region of North Africa. It overlooks the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to ...
, 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 technically ...
. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, The same fractional notation—with the fraction given before the integer—appears soon after in the work of 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 denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
, Dirk Jan Struik states:
The introduction of decimal fractions as a common computational practice can be dated back to the
Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Southern Dutch (). Flemish is native to Flanders, a historical region in northern Belgiu ...
pamphlet ''De Thiende'', published at
Leyden Leiden (; in English and archaic Dutch also Leyden) is a city and municipality in the province of South Holland, Netherlands. The municipality of Leiden has a population of 119,713, but the city forms one densely connected agglomeratio ...
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, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
(1548–1620), then settled in the Northern
Netherlands ) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Neth ...
. It is true that decimal fractions were used by the
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **'' Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of v ...
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 with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified for ...
fractions with great ease in his ''Key to arithmetic'' (
Samarkand fa, سمرقند , native_name_lang = , settlement_type = City , image_skyline = , image_caption = Clockwise from the top: Registan square, Shah-i-Zinda necropolis, Bibi-Khanym Mosque, view inside Shah-i-Zi ...
, early fifteenth century).
While the
Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...
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 and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesi ...
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 mathematician, who was active in Damascus and Baghdad. He wrote the earliest surviving book on the positional use ...
as early as the 10th century.

# In formal education

## Pedagogical tools

In
primary school A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary ...
s, fractions have been demonstrated through
Cuisenaire rods Cuisenaire rods are mathematics learning aids for students that provide an interactive, hands-on way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divis ...
, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper,
geoboard A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygon In geometry, a polygon () is a plane figure that is described by a ...
s, counters and computer software.

## Documents for teachers

Several states in the United States have adopted learning trajectories from the
Common Core State Standards Initiative The Common Core State Standards Initiative, also known as simply Common Core, is an educational initiative from 2010 that details what K–12 students throughout the United States should know in English language arts and mathematics at the con ...
'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.