Fraction (mathematics)

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A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: \tfrac{1}{2} and 17/3) consists of an integer numerator, displayed above a line (or before a slash), and a non-zero integer 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.

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates \tfrac{3}{4} or 3/4 of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10−2 respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.[1] Thus the fraction 3/4 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).



When reading fractions it is customary in English to pronounce the denominator using the corresponding ordinal number, in plural if the numerator is not one, as in "fifths" for fractions with a 5 in the denominator. Thus, 3/5 is rendered as three fifths and 5/32 as five thirty-seconds. This generally applies to whole number denominators greater than 2, though large denominators that are not powers of ten are often rendered using the cardinal number. Thus, 5/123 might be rendered as "five one-hundred-twenty-thirds", but is often "five over one hundred twenty-three". In contrast, because one million is a power of ten, 6/1,000,000 is usually expressed as "six millionths" or "six one-millionths", rather than as "six over one million".

The denominators 1, 2, and 4 are special cases. The fraction 3/1 may be spoken of as three wholes. The denominator 2 is expressed as half (plural halves); "−32" is minus three-halves or negative three-halves. The fraction 3/4 may be either "three fourths" or "three quarters". Furthermore, since most fractions in prose function as adjectives, the fractional modifier is hyphenated. This is evident in standard prose in which one might write about "every two-tenths of a mile", "the quarter-mile run", or the Three-Fifths Compromise. When the fraction's numerator is 1, then the word one may be omitted, such as "every tenth of a second" or "during the final quarter of the year".

In the examples 2/5 and 7/3, the slanting line is called a solidus or forward slash. In the examples \tfrac{2}{5} and \tfrac{7}{3}, the horizontal line is called a "fraction bar". When the solidus is encountered in a fraction, a speaker will sometimes parse it by pronouncing it over as in the examples above.

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 \tfrac{a}{b}, where a and b are both integers.[2] As with other fractions, the denominator (b) cannot be zero. Examples include \tfrac{1}{2}, -\tfrac{8}{5}, \tfrac{-8}{5}, \tfrac{8}{-5}, 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.[3][4] 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.[5][6] It is said to be an improper fraction, or sometimes top-heavy fraction,[7] 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: 2+\frac{3}{4}=2\tfrac{3}{4}.

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,  a \tfrac{b}{c} for example is not a mixed number. Instead, multiplication is understood where  a \tfrac{b}{c}  = a \times \tfrac{b}{c}.

To avoid confusion, the multiplication is often explicitly expressed. So  a \tfrac{b}{c} may be written as

 a \times \tfrac{b}{c},

 a \cdot \tfrac{b}{c}, or

 a (\tfrac{b}{c}).

An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:

  1. Write the mixed number 2\tfrac{3}{4} as a sum 2+\tfrac{3}{4}.
  2. Convert the whole number to an improper fraction with the same denominator as the fractional part, 2=\tfrac{8}{4}.
  3. Add the fractions. The resulting sum is the improper fraction. In the example, 2\tfrac{3}{4}=\tfrac{8}{4}+\tfrac{3}{4}=\tfrac{11}{4}.

Similarly, an improper fraction can be converted to a mixed number as follows:

  1. Divide the numerator by the denominator. In the example, \tfrac{11}{4}, divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.
  3. The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus \tfrac{11}{4} =2\tfrac{3}{4}.

Mixed numbers can also be negative, as in -2\tfrac{3}{4}, which equals  -(2+\tfrac{3}{4}) = -2-\tfrac{3}{4}.



  • 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 \tfrac{3}{7}, for instance, is \tfrac{7}{3}. 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 \tfrac{17}{1}, 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 \tfrac{1}{17}.

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,[8][9] corresponding to division of fractions. For example, \frac{\tfrac{1}{2}}{\tfrac{1}{3}} and \frac{12\tfrac{3}{4}}{26} are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:

\frac{12\tfrac{3}{4}}{26} = 12\tfrac{3}{4} \cdot \tfrac{1}{26} = \tfrac{12 \cdot 4 + 3}{4} \cdot \tfrac{1}{26} = \tfrac{51}{4} \cdot \tfrac{1}{26} = \tfrac{51}{104}

If, in a complex fraction, there is no clear way to tell which fraction lines takes precedence, then the expression is improperly formed, and ambiguous. Thus 5/10/20/40 is a poorly constructed mathematical expression, with multiple possible values.

Compound fractions[edit]

A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of,[8][9] 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, \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.

Decimal fractions and percentages[edit]


Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×107, which represents 0.0000006023. The 107 represents a denominator of 107. Dividing by 107 moves the decimal point 7 places to the left.



The related concept of permille or parts per thousand has an implied denominator of 1000, while the more general parts-per notation, as in 75 parts per million, means that the proportion is 75/1,000,000.


Special cases[edit]

  • A unit fraction is a vulgar fraction with a numerator of 1, e.g. \tfrac{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.
  • An Egyptian fraction is the sum of distinct positive unit fractions, for example \tfrac{1}{2}+\tfrac{1}{3}. This definition derives from the fact that the ancient Egyptians expressed 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}.
  • A dyadic fraction is a vulgar fraction in which the denominator is a power of two, e.g. \tfrac{1}{8}.

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

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, \tfrac{3}{9} is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, \tfrac{3}{8} 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{5}{10} = \tfrac{1}{2} = \tfrac{10}{20} = \tfrac{50}{100}.

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 \tfrac{63}{462} can be reduced to lowest terms by dividing the numerator and denominator by 21:

\tfrac{63}{462} = \tfrac{63 \div 21}{462 \div 21}= \tfrac{3}{22}


Comparing fractions[edit]

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

\tfrac{3}{4}>\tfrac{2}{4} because 3>2.


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{ad}{bd} and \tfrac{bc}{bd}. Then bd is a common denominator and the numerators ad and bc can be compared.

\tfrac{2}{3} ? \tfrac{1}{2} gives \tfrac{4}{6}>\tfrac{3}{6}

It is not necessary to determine the value of the common denominator to compare fractions. This short cut is known as "cross multiplying" – you can just compare ad and bc, without computing the denominator.

\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:

\tfrac{5 \times 17}{18 \times 17} ? \tfrac{4 \times 18}{17 \times 18}

The denominators are now the same, but it is not necessary to calculate their value – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), \tfrac{5}{18}>\tfrac{4}{17}.

Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction.



If \tfrac12 of a cake is to be added to \tfrac14 of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

Adding unlike quantities[edit]


For adding quarters to thirds, both types of fraction are converted to twelfths, thus: \tfrac14\ + \tfrac13=\tfrac{1*3}{4*3}\ + \tfrac{1*4}{3*4}=\tfrac3{12}\ + \tfrac4{12}=\tfrac7{12}.

Consider adding the following two quantities:


First, convert \tfrac35 into fifteenths by multiplying both the numerator and denominator by three: \tfrac35\times\tfrac33=\tfrac9{15}. 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{10}{15}.

Now it can be seen that:


is equivalent to:


This method can be expressed algebraically:

\tfrac{a}{b} + \tfrac {c}{d} = \tfrac{ad+cb}{bd}

And for expressions consisting of the addition of three fractions:

\tfrac{a}{b} + \tfrac {c}{d} + \tfrac{e}{f} = \tfrac{a(df)+c(bf)+e(bd)}{bdf}

This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add \tfrac{3}{4} and \tfrac{5}{12} 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]


\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12}



\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{\cancel{2} ^{~1}}{\cancel{3} ^{~1}} \times \tfrac{\cancel{3} ^{~1}}{\cancel{4} ^{~2}} = \tfrac{1}{1} \times \tfrac{1}{2} = \tfrac{1}{2}


Multiplying a fraction by a whole number[edit]


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


Multiplying mixed numbers[edit]


3 \times 2\tfrac{3}{4} = 3 \times \left (\tfrac{8}{4} + \tfrac{3}{4} \right ) = 3 \times \tfrac{11}{4} = \tfrac{33}{4} = 8\tfrac{1}{4}

In other words, 2\tfrac{3}{4} is the same as \tfrac{8}{4} + \tfrac{3}{4}, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is 8\tfrac{1}{4}, 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, \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 the reciprocal of 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 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 1/4 to a decimal, divide 4 into 1.00, to obtain 0.25. To change 1/3 to a decimal, divide 3 into 1.0000..., and stop when the desired accuracy is obtained. Note that 1/4 can be written exactly with two decimal digits, while 1/3 cannot be written exactly with any finite number of decimal digits.

To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits in the original decimal, omitting the decimal point. Thus 12.3456 = 123456/10000.

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
    10,000x = 1,523.987
    10,000,000x = 1,523,987.987
  4. Subtract the two equations from each other (if a = b and c = d, then a - 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 to represent x as a fraction
    x = 1522464/9990000

Fractions in abstract mathematics[edit]

In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair (a, b) of integers a and b ≠ 0, for which the operations addition, subtraction, multiplication, and division are defined as follows:[10]

(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)     (when c ≠ 0)

In addition, an equivalence relation is specified as follows: (a, b) ~ (c, d) if and only if ad=bc.


More generally, a and b may be elements of any integral domain R, in which case a fraction is an element of the field of fractions of R. For example, when a and b are polynomials in one indeterminate, the field of fractions is the field of rational fractions (also known as the field of rational functions). When a and b are integers, the field of fractions is the field of rational numbers.

Algebraic fractions[edit]

Main article: Algebraic fraction

An algebraic fraction is the indicated quotient of two algebraic expressions. 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 same laws as arithmetic fractions.

If the numerator and the denominator are polynomials, as in \frac{3x}{x^2+2x-3}, 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 \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 a complex fraction.

Rational numbers are the quotient field of integers. Rational expressions are the quotient field of the polynomials (over some integral domain). Since a coefficient is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction. Another example (over the reals) is \textstyle{\tfrac{\pi}{2}}, the radian measure of a right angle.

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 \textstyle{2x \over x^2-1} can be rewritten as the sum of two fractions: \textstyle{1 \over x+1} + \textstyle{1 \over x-1}. This is useful in many areas such as integral calculus and differential equations.

Radical expressions[edit]


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


\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}


Typographical variations[edit]


Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:[11]

  • 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{1}{2}, 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.[12]
  • 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.


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.[14] The Egyptians used Egyptian fractions ca. 1000 BC. About 4,000 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.[15] 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 and followers of the Greek philosopher Pythagoras, ca. 530 BC, discovered that the square root of two cannot be expressed as a fraction of integers. In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, operations with fractions.

The method of putting one number below the other and computing fractions first appeared in Aryabhatta's work around AD 499.[citation needed] In Sanskrit literature, fractions, or rational numbers were always expressed by an integer followed by a fraction. When the integer is written on a line, the fraction is placed below it and is itself written on two lines, the numerator called amsa part on the first line, the denominator called cheda “divisor” on the second below. If the fraction is written without any particular additional sign, one understands that it is added to the integer above it. If it is marked by a small circle or a cross (the shape of the “plus” sign in the West) placed on its right, one understands that it is subtracted from the integer. For example (to be read vertically), Bhaskara I writes[16]

६        १        २
१        १        १
४        ५        ९

That is,

6        1        2
1        1        1
4        5        9

to denote 6+1/4, 1+1/5, and 2–1/9.

Al-Hassār, a Muslim mathematician from Fez, Morocco specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. 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}."[17] This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.[18]

In discussing the origins of decimal fractions, Dirk Jan Struik states:[19]



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 \tfrac{a}{b} where a is a whole number and b is a positive whole number. (The word fraction in the standards always refers to a non-negative number.)"[23] The document itself also refers to negative fractions.

See also[edit]


  1. ^ H. Wu, The Mis-Education of Mathematics Teachers, Notices of the American Mathematical Society, Volume 58, Issue 03 (March 2011), page 374
  2. ^ Weisstein, Eric W., "Common Fraction", MathWorld.
  4. ^ Weisstein, Eric W., "Improper Fraction", MathWorld.
  6. ^ "New England Compact Math Resources". 
  10. ^ "Fraction – Encyclopedia of Mathematics". 2012-04-06. Retrieved 2012-08-15. 
  11. ^ Galen, Leslie Blackwell (March 2004), "Putting Fractions in Their Place" (PDF), American Mathematical Monthly 111 (3) 
  12. ^ "built fraction". glossary. Retrieved 2013-06-18. 
  13. ^ "piece fraction". glossary. Retrieved 2013-06-18. 
  14. ^ Eves, Howard ; with cultural connections by Jamie H. (1990). An introduction to the history of mathematics (6th ed. ed.). Philadelphia: Saunders College Pub. ISBN 0-03-029558-0. 
  16. ^ (Filliozat 2004, p. 152)
  17. ^ Cajori, Florian (1928), A History of Mathematical Notations (Vol.1) (PDF), La Salle, Illinois: The Open Court Publishing Company, p. 269 
  18. ^ (Cajori 1928, pg.89)
  22. ^ While there is some disagreement among history of mathematics scholars as to the primacy of al-Uqlidisi's contribution, there is no question as to his major contribution to the concept of decimal fractions. [1] "MacTutor's al-Uqlidisi biography". Retrieved 2011-11-22.

External links[edit]