Fraction

For other uses, seeFraction (disambiguation).

Afraction(fromLatin: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. Acommon,vulgar,orsimplefraction (examples:⁠1/2⁠and⁠17/3⁠) consists of an integernumerator,displayed above a line (or before a slash like1⁄2), and anon-zerointegerdenominator,displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction⁠3/4⁠,the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates⁠3/4⁠of a cake.

A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by thefraction1⁄4.

Fractions can be used to representratiosanddivision.Thus the fraction⁠3/4⁠can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division3 ÷ 4(three divided by four).

We can also write negative fractions, which represent the opposite of a positive fraction. For example, if⁠1/2⁠represents a half-dollar profit, then −⁠1/2⁠represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −⁠1/2⁠,⁠−1/2⁠and⁠1/−2⁠all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive,⁠−1/−2⁠represents positive one-half.

In mathematics arational numberis a number that can be represented by a fraction of the form⁠a/b⁠,whereaandbare integers andbis not zero; the set of all rational numbers is commonly represented by the symbol⁠$\mathbb {Q}$⁠orQ,which stands forquotient.The termfractionand the notation⁠a/b⁠can also be used for mathematical expressions that do not represent a rational number (for example$\textstyle {\frac {\sqrt {2}}{2}}$), and even do not represent any number (for example therational fraction$\textstyle {\frac {1}{x}}$).

Vocabulary

See also:Numeral (linguistics) § Fractional numbers,English numerals § Fractions and decimals,andUnicode subscripts and superscripts § Fraction slash

In a fraction, the number of equal parts being described is thenumerator(fromLatin:numerātor,"counter" or "numberer" ), and the type or variety of the parts is thedenominator(fromLatin:dēnōminātor,"thing that names or designates" ).As an example, the fraction⁠8/5⁠amounts to eight parts, each of which is of the type namedfifth.In terms ofdivision,the numerator corresponds to thedividend,and the denominator corresponds to thedivisor.

Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by afraction bar.The fraction bar may be horizontal (as in⁠1/3⁠), oblique (as in 2/5), or diagonal (as in4⁄9).These marks are respectively known as the horizontal bar; the virgule,slash(US), orstroke(UK); and the fraction bar, solidus,orfraction slash.Intypography,fractions stacked vertically are also known asenornutfractions,and diagonal ones asemormutton fractions,based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrowensquare, or a wideremsquare.In traditionaltypefounding,a piece of type bearing a complete fraction (e.g.⁠1/2⁠) was known as acase fraction,while those representing only parts of fractions were calledpiece fractions.

The denominators of English fractions are generally expressed asordinal numbers,in the plural if the numerator is not 1. (For example,⁠2/5⁠and⁠3/5⁠are both read as a number offifths.) Exceptions include the denominator 2, which is always readhalforhalves,the denominator 4, which may be alternatively expressed asquarter/quartersor asfourth/fourths,and the denominator 100, which may be alternatively expressed ashundredth/hundredthsorpercent.

When the denominator is 1, it may be expressed in terms ofwholesbut is more commonly ignored, with the numerator read out as a whole number. For example,⁠3/1⁠may be described asthree wholes,or simply asthree.When the numerator is 1, it may be omitted (as ina tenthoreach 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-fifthsis the fraction⁠2/5⁠andtwo fifthsis the same fraction understood as 2 instances of⁠1/5⁠.) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numeratoroverthe denominator, with the denominator expressed as acardinal number.(For example,⁠3/1⁠may also be expressed asthree over one.) The termoveris used even in the case of solidus fractions, where the numbers are placed left and right of aslash mark.(For example, 1/2 may be readone-half,one half,orone over two.) Fractions with large denominators that arenotpowers of ten are often rendered in this fashion (e.g.,⁠1/117⁠asone over one hundred seventeen), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g.,⁠6/1000000⁠assix-millionths,six millionths,orsix one-millionths).

Forms of fractions

Simple, common, or vulgar fractions

Asimple fraction(also known as acommon fractionorvulgar fraction)is arational numberwritten asa/bor⁠${\tfrac {a}{b}}$⁠,whereaandbare bothintegers.As with other fractions, the denominator (b) cannot be zero. Examples include⁠1/2⁠,−⁠8/5⁠,⁠−8/5⁠,and⁠8/−5⁠.The term was originally used to distinguish this type of fraction from thesexagesimal fractionused in astronomy.

Common fractions can be positive or negative, and they can beproperorimproper(see below). Compound fractions, complex fractions, mixed numerals, and decimal expressions (see below) are notcommon fractions;though, unless irrational, they can be evaluated to a common fraction.

• Aunit fractionis a common fraction with a numerator of 1 (e.g.,⁠1/7⁠). 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.
• Adyadic fractionis a common fraction in which the denominator is apower of two,e.g.⁠1/8⁠=⁠1/2³⁠.

In Unicode, precomposed fraction characters are in theNumber Formsblock.

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 animproper fractionis a late development, with the terminology deriving from the fact thatfractionmeanspiece,so a proper fraction must be less than 1.This was explained in the 17th century textbookThe Ground of Arts.

In general, a common fraction is said to be aproper fractionif theabsolute valueof 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 animproper fraction,or sometimestop-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

Thereciprocalof a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of⁠3/7⁠,for instance, is⁠7/3⁠.The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is themultiplicative inverseof 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,⁠7/7⁠), 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 as⁠17/1⁠,where 1 is sometimes referred to as theinvisible denominator.Therefore, every fraction and every integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is⁠1/17⁠.

Ratios

Aratiois 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 groupn".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 that⁠4/12⁠of the cars or⁠1/3⁠of 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 orprobabilitythat it would be yellow.

Decimal fractions and percentages

Adecimal fractionis a fraction whose denominator is an integer power of ten, commonly expressed using decimal notation, in which the denominator is not given explicitly but is implied by the number ofdigitsto the right of adecimal separator.The separator can be a period ⟨.⟩,interpunct⟨·⟩, or comma ⟨,⟩, depending on locale. (For examples, seeDecimal separator.) 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), thefractional partof the number is expressed by the digits to the right of the separator (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+75/100⁠.

Decimal fractions can also be expressed usingscientific notationwith negative exponents, such as6.023×10⁻⁷,a convenient alternative to the unwieldy 0.0000006023. The10⁻⁷represents a denominator of10⁷.Dividing by10⁷moves the decimal point seven places to the left.

A decimal fraction with infinitely many digits to the right of the decimal separator represents aninfinite series.For example,⁠1/3⁠= 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 +....

Another kind of fraction is thepercentage(fromLatin:per centum,meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means51⁄100.Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% means311⁄100and −27% means−27⁄100.

The related concept ofpermille,orparts per thousand(ppt), means a denominator of 1000, and thisparts-pernotationis commonly used with larger denominators, such asmillionandbillion,e.g.75 parts per million(ppm) means that the proportion is⁠75/1000000⁠.

The choice between fraction and decimal notation is often a matter of taste and context. Fractions are used most often when the denominator is relatively small. Bymental calculation,it is easier tomultiply16 by3⁄16than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is moreprecise(exact, in fact) to multiply 15 by1⁄3,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 digits after the decimal separator, 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", commonly readthree and six,meansthree shillings and sixpenceand has no relationship to the fractionthree sixths.

Mixed numbers

Amixed number(also called amixed fractionormixed numeral) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (orconcatenation) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them.

As a basic example, two entire cakes and three quarters of another cake might be written as$2{\tfrac {3}{4}}$cakes or$2\ \,3/4$cakes, with the numeral$2$representing the whole cakes and the fraction${\tfrac {3}{4}}$representing the additional partial cake juxtaposed; this is more concise than the more explicit notation$2+{\tfrac {3}{4}}$cakes. The mixed number⁠2+3/4⁠is spokentwo and three quartersortwo and three fourths,with the integer and fraction portions connected by the wordand.Subtraction or negation is applied to the entire mixed numeral, so$-2{\tfrac {3}{4}}$means$-{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.$

Any mixed number can be converted to animproper fractionby applying the rules ofadding unlike quantities.For example,$2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.$Conversely, an improper fraction can be converted to a mixed number usingdivision with remainder,with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over,${\tfrac {11}{4}}=2+{\tfrac {3}{4}}.$

In primary school, teachers often insist that every fractional result should be expressed as a mixed number.Outside school, mixed numbers are commonly used for describing measurements, for instance⁠2+1/2⁠hours or 5 3/16inches,and remain widespread in daily life and in trades, especially in regions that do not use the decimalizedmetric system.However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as arational number,the quotient⁠p/q⁠of integers, leaving behind the concepts ofimproper fractionandmixed number.College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition inalgebraic expressionsmeans multiplication.

Historical notions

Egyptian fraction

AnEgyptian fractionis the sum of distinct positive unit fractions, for example${\tfrac {1}{2}}+{\tfrac {1}{3}}$.This definition derives from the fact that theancient Egyptiansexpressed all fractions except${\tfrac {1}{2}}$,${\tfrac {2}{3}}$and${\tfrac {3}{4}}$in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example,${\tfrac {5}{7}}$can be written as${\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.$Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write${\tfrac {13}{17}}$are${\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}$and${\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}$.

Complex and compound fractions

Not to be confused withComplex numbers.

In acomplex fraction,either the numerator, or the denominator, or both, is a fraction or a mixed number,corresponding to division of fractions. For example,${\tfrac {1/2}{1/3}}$and${\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26$are complex fractions. To interpret nested fractions writtenstackedwith a horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at§ Division.For example:

${\begin{aligned}{\frac {\;\!{\tfrac {1}{2}}\;\!}{\tfrac {1}{3}}}&={\frac {1}{2}}\div {\frac {1}{3}}={\frac {1}{2}}\times {\frac {3}{1}}={\frac {3}{2}},\qquad {\frac {\;\!{\tfrac {3}{2}}\;\!}{5}}={\frac {3}{2}}\div 5={\frac {3}{2}}\times {\frac {1}{5}}={\frac {3}{10}},\\[10mu]{\frac {12{\tfrac {3}{4}}}{26}}&={\frac {12\times 4+3}{4}}\div 26={\frac {12\times 4+3}{4}}\times {\frac {1}{26}}={\frac {51}{104}}.\end{aligned}}$

A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression$5/10/20$could be plausibly interpreted as either${\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}$or as$5{\big /}{\tfrac {10}{20}}=10.$The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance$(5/10){\big /}20$or$5{\big /}(10/20).$

Acompound fractionis a fraction of a fraction, or any number of fractions connected with the wordof,corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see§ Multiplication). For example,${\tfrac {3}{4}}$of${\tfrac {5}{7}}$is a compound fraction, corresponding to${\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}$.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${\tfrac {3}{4}}\times {\tfrac {5}{7}}$is equivalent to the complex fraction⁠${\tfrac {3/4}{7/5}}$⁠.)

Nevertheless,complex fractionandcompound fractionmay both be considered outdatedand now used in no well-defined manner, partly even taken as synonymous with each otheror withmixed numerals.They have lost their meaning as technical terms and the attributescomplexandcompoundtend to be used in their everyday meaning ofconsisting of parts.

Arithmetic with fractions

Like whole numbers, fractions obey thecommutative,associative,anddistributivelaws, and the rule againstdivision by zero.

Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts.

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${\tfrac {n}{n}}$equals 1. Therefore, multiplying by${\tfrac {n}{n}}$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${\tfrac {1}{2}}$⁠.When the numerator and denominator are both multiplied by 2, the result is⁠2/4⁠,which has the same value (0.5) as⁠1/2⁠.To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (⁠2/4⁠) make up half the cake (⁠1/2⁠).

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${\tfrac {a}{b}}$are divisible by⁠$c$⁠,then they can be written as$a=cd$,$b=ce$,and the fraction becomes⁠cd/ce⁠,which can be reduced by dividing both the numerator and denominator bycto give the reduced fraction⁠d/e⁠.

If one takes forcthegreatest common divisorof the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowestabsolute values.One says that the fraction has been reduced to itslowest 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 beirreducible,reduced,orin simplest terms.For example,${\tfrac {3}{9}}$is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast,${\tfrac {3}{8}}$isin lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.

Using these rules, we can show that⁠5/10⁠=⁠1/2⁠=⁠10/20⁠=⁠50/100⁠,for example.

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction⁠63/462⁠can be reduced to lowest terms by dividing the numerator and denominator by 21:

${\frac {63}{462}}={\frac {63\,\div \,21}{462\,\div \,21}}={\frac {3}{22}}.$

TheEuclidean algorithmgives 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:

${\tfrac {3}{4}}>{\tfrac {2}{4}}$because3 > 2,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:

${\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.$

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${\tfrac {a}{b}}$and${\tfrac {c}{d}}$,these are converted to${\tfrac {a\cdot d}{b\cdot d}}$and${\tfrac {b\cdot c}{b\cdot d}}$(where the dot signifies multiplication and is an alternative symbol to ×). Thenbdis a common denominator and the numeratorsadandbccan be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compareadandbc,without evaluatingbd,e.g., comparing${\tfrac {2}{3}}$?${\tfrac {1}{2}}$gives${\tfrac {4}{6}}>{\tfrac {3}{6}}$.

For the more laborious question${\tfrac {5}{18}}$?${\tfrac {4}{17}},$multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding${\tfrac {5\times 17}{18\times 17}}$?${\tfrac {18\times 4}{18\times 17}}$.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${\tfrac {5}{18}}>{\tfrac {4}{17}}$⁠.

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:

${\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}$.

If${\tfrac {1}{2}}$of a cake is to be added to${\tfrac {1}{4}}$of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

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 theinvisible denominator1.

For adding quarters to thirds, both types of fraction are converted to twelfths, thus:

${\frac {1}{4}}+{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}+{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}+{\frac {4}{12}}={\frac {7}{12}}.$

Consider adding the following two quantities:

${\frac {3}{5}}+{\frac {2}{3}}.$

First, convert${\tfrac {3}{5}}$into fifteenths by multiplying both the numerator and denominator by three:⁠${\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}$⁠.Since⁠3/3⁠equals 1, multiplication by⁠3/3⁠does not change the value of the fraction.

Second, convert⁠2/3⁠into fifteenths by multiplying both the numerator and denominator by five:⁠${\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}$⁠.

Now it can be seen that

${\frac {3}{5}}+{\frac {2}{3}}$

is equivalent to

${\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}.$

This method can be expressed algebraically:

${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}.$

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${\tfrac {3}{4}}$and${\tfrac {5}{6}}$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.

${\begin{alignedat}{3}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot 6}{4\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {20}{24}}&&={\frac {19}{12}}\\[10mu]&={\frac {3\cdot 3}{4\cdot 3}}+{\frac {2\cdot 5}{2\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&&={\frac {19}{12}}.\end{alignedat}}$

The smallest possible denominator is given by theleast common multipleof 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,

${\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}.$

To subtract a mixed number, an extra one can be borrowed from the minuend, for instance

$4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.$

Multiplication

Multiplying a fraction by another fraction

To multiply fractions, multiply the numerators and multiply the denominators. Thus:

${\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}.$

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 calledcancellation.Effectively the answer is reduced to lowest terms during multiplication. For example:

${\frac {2}{3}}\times {\frac {3}{4}}={\frac {{\color {RoyalBlue}{\cancel {\color {Black}2}}}^{~1}}{{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}}\times {\frac {{\color {RedOrange}{\cancel {\color {Black}3}}}^{~1}}{{\color {RoyalBlue}{\cancel {\color {Black}4}}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}.$

A two is a commonfactorin 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. For example,

$6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}.$

This method works because the fraction 6/1 means six equal parts, each one of which is a whole.

Multiplying mixed numbers

The product of mixed numbers can be computed by converting each to an improper fraction.For example:

$3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}.$

Alternately, mixed numbers can be treated as sums, andmultiplied as binomials.In this example,

$3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.$

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,${\tfrac {10}{3}}\div 5$equals${\tfrac {2}{3}}$and also equals${\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}$,which reduces to${\tfrac {2}{3}}$.To divide a number by a fraction, multiply that number by thereciprocalof that fraction. Thus,${\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\times {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}$.

Converting between fractions and decimal notation

To change a common fraction to decimal notation, do a long division of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator" ), and round the result to the desired precision. For example, to change⁠1/4⁠to a decimal expression, divide1by4( "4into1"), to obtain exactly0.25.To change⁠1/3⁠to a decimal expression, divide1...by3( "3into1..."), and stop when the desired precision is obtained, e.g., at four places after thedecimal separator(ten-thousandths) as0.3333.The fraction⁠1/4⁠is expressed exactly with only two digits after the decimal separator, while the fraction⁠1/3⁠cannot be written exactly as a decimal with a finite number of digits. A decimal expression can be converted to a fraction by removing the decimal separator, using the result as the numerator, and using1followed by the same number of zeroes as there are digits to the right of the decimal separator as the denominator. Thus,$1.23={\tfrac {123}{100}}.$

Converting repeating digits in decimal notation to fractions

Further information:Decimal representation § Conversion to fraction

See also:Repeating decimal

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infiniterepeating decimalis required to reach the same precision. Thus, it is often useful to convert repeating digits into fractions.

A conventional way to indicate a repeating decimal is to place a bar (known as avinculum) over the digits that repeat, for example 0.789= 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:

0.5= 5/9

0.62= 62/99

0.264= 264/999

0.6291= 6291/9999

Ifleading zerosprecede the pattern, the nines are suffixed by the same number oftrailing zeros:

0.05= 5/90

0.000392= 392/999000

0.0012= 12/9900

If a non-repeating set of digits precede the pattern (such as 0.1523987), one may write the number as the sum of the non-repeating and repeating parts, respectively:

0.1523 + 0.0000987

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:

1. Letx= the repeating decimal:

   x= 0.1523987

2. 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:

   10,000x= 1,523.987

3. Multiply both sides by the power of 10 (in this case 10³) that is the same as the number of places that repeat:

   10,000,000x= 1,523,987.987

4. Subtract the two equations from each other (ifa=bandc=d,thena−c=b−d):

   10,000,000x− 10,000x= 1,523,987.987− 1,523.987

5. Continue the subtraction operation to clear the repeating decimal:

   9,990,000x= 1,523,987 − 1,523

   9,990,000x= 1,522,464

6. Divide both sides by 9,990,000 to representxas a fraction

   x=⁠1522464/9990000⁠

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 areconsistent and reliable.Mathematicians define a fraction as an ordered pair$(a,b)$ofintegers$a$and$b\neq 0,$for which the operationsaddition,subtraction,multiplication,anddivisionare defined as follows:

$(a,b)+(c,d)=(ad+bc,bd)\,$

$(a,b)-(c,d)=(ad-bc,bd)\,$

$(a,b)\cdot (c,d)=(ac,bd)$

$(a,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)$

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, theinversefractions with respect to addition and multiplication might be defined as:

${\begin{aligned}-(a,b)&=(-a,b)&&{\text{additive inverse fractions,}}\\&&&{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-1}&=(b,a)&&{\text{multiplicative inverse fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}$

Furthermore, therelation,specified as

$(a,b)\sim (c,d)\quad \iff \quad ad=bc,$

is anequivalence relationof fractions. Each fraction from one equivalence class may be considered as arepresentativefor 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 fractions⁠a/b⁠withaandbcoprimeandb> 0are often taken as uniquely determined representatives for theirequivalentfractions, which are considered to be thesamerational number. This way the fractions of integers make up the field of the rational numbers.

More generally,aandbmay be elements of anyintegral domainR,in which case a fraction is an element of thefield of fractionsofR.For example,polynomialsin one indeterminate, with coefficients from some integral domainD,are themselves an integral domain, call itP.So foraandbelements ofP,the generatedfield of fractionsis the field ofrational fractions(also known as the field ofrational functions).

Algebraic fractions

Main article:Algebraic fraction

An algebraic fraction is the indicatedquotientof twoalgebraic expressions.As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are${\frac {3x}{x^{2}+2x-3}}$and⁠${\frac {\sqrt {x+2}}{x^{2}-3}}$⁠.Algebraic fractions are subject to the samefieldproperties as arithmetic fractions.

If the numerator and the denominator arepolynomials,as in⁠${\frac {3x}{x^{2}+2x-3}}$⁠,the algebraic fraction is called arational fraction(orrational expression). Anirrational fractionis one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in⁠${\frac {\sqrt {x+2}}{x^{2}-3}}$⁠.

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${\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}$⁠,is called acomplex fraction.

The field of rational numbers is thefield of fractionsof the integers, while the integers themselves are not a field but rather anintegral domain.Similarly, therational fractionswith coefficients in afieldform the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients,radical expressionsrepresenting numbers, such as⁠$\textstyle {\sqrt {2}}/2$⁠,are also rational fractions, as are atranscendental numberssuch as${\textstyle \pi /2,}$since all of${\sqrt {2}},\pi,$and$2$arereal numbers,and thus considered as coefficients. These same numbers, however, are not rational fractions withintegercoefficients.

The termpartial fractionis used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction${\frac {2x}{x^{2}-1}}$can be decomposed as the sum of two fractions:⁠${\frac {1}{x+1}}+{\frac {1}{x-1}}$⁠.This is useful for the computation ofantiderivativesofrational functions(seepartial fraction decompositionfor more).

Radical expressions

Main articles:Nth rootandRationalization (mathematics)

A fraction may also containradicalsin the numerator or the denominator. If the denominator contains radicals, it can be helpful torationalizeit (compareSimplified 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 amonomialsquare root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator:

${\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}.$

The process of rationalization ofbinomialdenominators involves multiplying the top and the bottom of a fraction by theconjugateof the denominator so that the denominator becomes a rational number. For example:

${\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}},$

${\frac {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}.$

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

See also:Slash § Encoding

In computer displays andtypography,simple fractions are sometimes printed as a single character, e.g. ½ (one half). See the article onNumber Formsfor information on doing this inUnicode.

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 themupright.An example would be⁠1/2⁠,but rendered with the same height as other characters. Some sources include all rendering of fractions ascase fractionsif 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), as in "2/6" for ahalf crown,meaning two shillings and six pence. While the notationtwo shillings and six pencedid not represent a fraction, the 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 aspiece fractions,are written all on one typographical line but take three or more typographical spaces.
• Built-up fractions:${\frac {1}{2}}$.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 werereciprocalsofintegers:ancient symbols representing one part of two, one part of three, one part of four, and so on.The Egyptians usedEgyptian fractionsc. 1000BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples withunit fractions.Their methods gave the same answer as modern methods.The Egyptians also had a different notation fordyadic fractions,used for certain systems of weights and measures.

TheGreeksused unit fractions and (later)simple continued fractions.Followersof theGreekphilosopherPythagoras(c. 530BC) discovered that thesquare root of twocannot be expressed as a fraction of integers.(This is commonly though probably erroneously ascribed toHippasusofMetapontum,who is said to have been executed for revealing this fact.) In150 BCJainmathematicians inIndiawrote theSthananga Sutra,which contains work on the theory of numbers, arithmetical operations, and operations with fractions.

A modern expression of fractions known asbhinnarasiseems to have originated in India in the work ofAryabhatta(c. AD 500),Brahmagupta(c. 628), andBhaskara(c. 1150).Their works form fractions by placing the numerators (Sanskrit:amsa) over the denominators (cheda), but without a bar between them.InSanskrit 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⟨०⟩or cross⟨+⟩,it is subtracted from the integer; if no such sign appears, it is understood to be added. For example,Bhaskara Iwrites:

६  १  २

१  १  १_०

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 6⁠1/4⁠,1⁠1/5⁠,and 2 −⁠1/9⁠(i.e., 1⁠8/9⁠).

The horizontal fraction bar is first attested in the work ofAl-Hassār(fl. 1200),aMuslim mathematicianfromFez,Morocco,who specialized inIslamic inheritance jurisprudence.In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus,${\frac {3\quad 1}{5\quad 3}}$".The same fractional notation—with the fraction given before the integer—appears soon after in the work ofLeonardo Fibonacciin the 13th century.

In discussing the origins ofdecimal fractions,Dirk Jan Struikstates:

The introduction of decimal fractions as a common computational practice can be dated back to theFlemishpamphletDe Thiende,published atLeydenin 1585, together with a French translation,La Disme,by the Flemish mathematicianSimon Stevin(1548–1620), then settled in the NorthernNetherlands.It is true that decimal fractions were used by theChinesemany centuries before Stevin and that the Persian astronomerAl-Kāshīused both decimal andsexagesimalfractions with great ease in hisKey to arithmetic(Samarkand,early fifteenth century).

While thePersianmathematicianJamshīd al-Kāshīclaimed 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 theBaghdadimathematicianAbu'l-Hasan al-Uqlidisias early as the 10th century.

In formal education

Primary schools

Inprimary schools,fractions have been demonstrated throughCuisenaire rods,Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting),pattern blocks,pie-shaped pieces, plastic rectangles, grid paper,dot paper,geoboards,counters and computer software.

Documents for teachers

Several states in the United States have adopted learning trajectories from theCommon 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$a$⁄$b$where$a$is a whole number and$b$is a positive whole number. (The wordfractionin these standards always refers to a non-negative number.) "The document itself also refers to negative fractions.

See also

• Cross multiplication
• 0.999...
• Multiple
• FRACTRAN

Number systems Complex${\displaystyle:\;\mathbb {C} }$ Real${\displaystyle:\;\mathbb {R} }$ Rational${\displaystyle:\;\mathbb {Q} }$ Integer${\displaystyle:\;\mathbb {Z} }$ Natural${\displaystyle:\;\mathbb {N} }$ Zero:0 One:1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Irrational period Transcendental Imaginary

References

Weisstein, Eric(2003). "CRC Concise Encyclopedia of Mathematics, Second Edition".CRC Concise Encyclopedia of Mathematics.Chapman & Hall/CRC. p. 1925.ISBN1-58488-347-2.

External links

• "Fraction, arithmetical".The Online Encyclopaedia of Mathematics.
• "Fraction".Encyclopædia Britannica.5 January 2024.