Fraction (mathematics)

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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.




Other uses for fractions are to represent ratios and to represent division.[1] Thus the fraction ¾ is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).



In a fraction, the number of equal parts being described is the numerator (equivalent to the dividend in division) and the number of equal parts that make up a whole is the denominator (equivalent to the divisor). Informally, they may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The fraction bar may be horizontal (as in −1/3), oblique (as in 1/5 or −1∕7), or diagonal (as in 19).[2] These marks are respectively known as the horizontal bar, the slash (US) or stroke (UK), the division slash, and the fraction slash.[n 1] In typography, horizontal fractions are also known as "en" or "nut fractions" and diagonal fractions as "em fractions", based on the width of a line they take up.[2]

The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. (For example, 25 and 35 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". 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, 3/1 may be described as "three wholes" or as simply "three".) When the numerator is one, it may be omitted. (For example, "a tenth" or "each quarter".) A 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 2/5 and "two fifths" is the same fraction understood as 2 instances of 15.) 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. (For example, 3/1 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, ½ 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., 1/117 as "one over a hundred and seventeen") while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., 6/1000000 as "six-millionths", "six millionths", or "six one-millionths").

Forms of fractions[edit]

Simple, common, or vulgar fractions[edit]

A simple fraction (also known as a common fraction or vulgar fraction) is a rational number written as a/b or , where a and b are both integers.[6] As with other fractions, the denominator (b) cannot be zero. Examples include , , , , and 3/17. Simple fractions can be positive or negative, proper, or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not simple fractions, though, unless irrational, they can be evaluated to a simple fraction.

Proper and improper fractions[edit]

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.[7][8] In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.[9][10] It is said to be an improper fraction, or sometimes top-heavy fraction,[11] 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; examples of improper fractions are 9/4, –4/3, and 3/3.

Mixed numbers[edit]

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a non-zero integer and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: .

This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". In algebra, for example is not a mixed number. Instead, multiplication is understood where .

To avoid confusion, the multiplication is often explicitly expressed. So may be written as



  1. Write the mixed number as a sum .


  1. Divide the numerator by the denominator. In the example, , divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.

Mixed numbers can also be negative, as in , which equals .



  • 2 are white,
  • 6 are red, and
  • 4 are yellow,



Reciprocals and the "invisible denominator"[edit]

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of , for instance, is . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction.

Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is .

Complex fractions[edit]

Not to be confused with fractions involving complex numbers

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number,[12][13] corresponding to division of fractions. For example, and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:

If, in a complex fraction, there is no clear way to tell which fraction lines takes precedence, then the expression is obviously improperly formed, because of ambiguity. So 5/10/20/40 is a poorly constructed mathematical expression, with multiple possible meanings, e.g.:


Compound fractions[edit]

A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of,[12][13] corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication). For example, of is a compound fraction, corresponding to . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other.

Decimal fractions and percentages[edit]



Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, ⅓ = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .




Special cases[edit]


Arithmetic with fractions[edit]


Equivalent fractions[edit]

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 , the fraction . Therefore, multiplying by is equivalent to 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 . When the numerator and denominator are both multiplied by 2, the result is , which has the same value (0.5) as . To picture this visually, imagine cutting a cake into four pieces; two of the pieces together () make up half the cake ().

Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime (that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible, in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, 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 = = = .

A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, as the greatest common divisor of 63 and 462 is 21, the fraction can be reduced to lowest terms by dividing the numerator and denominator by 21:


Comparing fractions[edit]

Comparing fractions with the same denominator only requires comparing the numerators.

because 3>2.


One way to compare fractions with different numerators and denominators is to find a common denominator. To compare and , these are converted to and . Then bd is a common denominator and the numerators ad and bc can be compared.

 ? gives









If of a cake is to be added to of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

Adding unlike quantities[edit]



Consider adding the following two quantities:

First, convert into fifteenths by multiplying both the numerator and denominator by three: . Since equals 1, multiplication by does not change the value of the fraction.

Second, convert into fifteenths by multiplying both the numerator and denominator by five: .

Now it can be seen that:

is equivalent to:

This method can be expressed algebraically:

And for expressions consisting of the addition of three fractions:

This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add and the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12.




Multiplying a fraction by another fraction[edit]



A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example:


Multiplying a fraction by a whole number[edit]

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply.


Multiplying mixed numbers[edit]

When multiplying mixed numbers, it is considered preferable[citation needed] to convert the mixed number into an improper fraction. For example:

In other words, is the same as , making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is , since 8 cakes, each made of quarters, is 32 quarters in total.


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, equals and also equals , which reduces to . To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, .

Converting between decimals and fractions[edit]

To change a common fraction to a decimal, divide the denominator into the numerator. Round the answer to the desired accuracy. For example, to change ¼ to a decimal, divide 4 into 1.00, to obtain 0.25. To change ⅓ to a decimal, divide 3 into 1.0000..., and stop when the desired accuracy is obtained. Note that ¼ can be written exactly with two decimal digits, while ⅓ cannot be written exactly with any finite number of decimal digits.


Converting repeating decimals to fractions[edit]



0.5 = 5/9
0.62 = 62/99
0.264 = 264/999
0.6291 = 6291/9999


0.05 = 5/90
0.000392 = 392/999000
0.0012 = 12/9900


0.1523 + 0.0000987


1523/10000 + 987/9990000 = 1522464/9990000

Alternatively, algebra can be used, such as below:

  1. Let x = 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 (if a = b and c = d, then ac = bd):
    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 to represent x as a fraction
    x = 1522464/9990000

Fractions in abstract mathematics[edit]


In addition, an equivalence relation is specified as follows: if and only if .



Algebraic fractions[edit]

Main article: Algebraic fraction

An algebraic fraction is the indicated quotient of two algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions.

If the numerator and the denominator are polynomials, as in , the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent or root, as in .



The term partial fraction is used when decomposing rational expressions into sums. The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree. For example, the rational expression can be rewritten as the sum of two fractions: + . This is useful in many areas such as integral calculus and differential equations.

Radical expressions[edit]




Typographical variations[edit]


Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:[15]



The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on.[18] The Egyptians used Egyptian fractions c. 1000 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.[19] The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.

The Greeks used unit fractions and (later) continued fractions. Followers of the Greek philosopher Pythagoras (c. 530 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 of Metapontum, who is said to have been executed for revealing this fact.) In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions.

The modern expression of fractions seems to have originated in India in the work of Aryabhatta (c.AD 500),[citation needed] Brahmagupta (c. 628), and Bhaskara (c. 1150).[20] Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (cheda), but without a bar between them.[20] In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer.[citation needed] 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 I writes[21]

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 61/4, 11/5, and 2−1/9 (i.e., 18/9).

The horizontal fraction bar is first attested in the work of Al-Hassār (fl. 1200),[20] a Muslim mathematician from Fez, Morocco, who specialized in Islamic 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, ."[22] The same fractional notation—with the fraction given before the integer[20]—appears soon after in the work of Leonardo Fibonacci in the 13th century.[23]

In discussing the origins of decimal fractions, Dirk Jan Struik states:[24]



In formal education[edit]

Pedagogical tools[edit]


Documents for teachers[edit]

Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form where is a whole number and is a positive whole number. (The word fraction in the standards always refers to a non-negative number.)"[28] The document itself also refers to negative fractions.

See also[edit]


  1. ^ Some typographers such as Bringhurst mistakenly distinguish the slash ⟨/⟩ as the virgule and the fraction slash ⟨⟩ as the solidus,[3] although in fact both are synonyms for the standard slash.[4][5]


  1. ^ H. Wu, "The Mis-Education of Mathematics Teachers", Notices of the American Mathematical Society, Volume 58, Issue 03 (March 2011), page 374
  2. ^ a b Ambrose, Gavin; et al. (2006). The Fundamentals of Typography, (2nd ed.). Lausanne: AVA Publishing. p. 74. ISBN 978-2-940411-76-4. .
  4. ^ "virgule, n.". Oxford English Dictionary (1st ed.). Oxford: Oxford University Press. 1917. 
  5. ^ "solidus, n.1". Oxford English Dictionary (1st ed.). Oxford: Oxford University Press. 1913. .
  6. ^ Weisstein, Eric W. "Common Fraction". MathWorld. 
  7. ^ "World Wide Words: Vulgar fractions". World Wide Words. Retrieved 2014-10-30. 
  8. ^ Weisstein, Eric W. "Improper Fraction". MathWorld. 
  9. ^ Laurel (31 March 2004). "Math Forum – Ask Dr. Math:Can Negative Fractions Also Be Proper or Improper?". Retrieved 2014-10-30. 
  10. ^ "New England Compact Math Resources". 
  11. ^ Greer, A. (1986). New comprehensive mathematics for 'O' level (2nd ed., reprinted. ed.). Cheltenham: Thornes. p. 5. ISBN 9780859501590. Retrieved 2014-07-29. 
  14. ^ "Fraction". Encyclopedia of Mathematics. 2012-04-06. Retrieved 2012-08-15. 
  15. ^ Galen, Leslie Blackwell (March 2004). "Putting Fractions in Their Place" (PDF). American Mathematical Monthly. 111 (3). 
  16. ^ "built fraction". glossary. Retrieved 2013-06-18. 
  17. ^ "piece fraction". glossary. Retrieved 2013-06-18. 
  18. ^ Eves, Howard (1990). An introduction to the history of mathematics (6th ed.). Philadelphia: Saunders College Pub. ISBN 0-03-029558-0. 
  20. ^ a b c d Miller, Jeff (22 December 2014). "Earliest Uses of Various Mathematical Symbols". Retrieved 15 February 2016. .
  21. ^ Filliozat (2004), p. 152
  22. ^ Cajori, Florian (1928). A History of Mathematical Notations (PDF). 1. La Salle, Illinois: Open Court Publishing Company. p. 269. 
  23. ^ Cajori (1928), pg.89
  27. ^ "MacTutor's al-Uqlidisi biography". Retrieved 2011-11-22.
  28. ^ "Common Core State Standards for Mathematics" (PDF). Common Core State Standards Initiative. 2010. p. 85. Retrieved 2013-10-10. 

External links[edit]