# Fraction

(Redirected from Denominator)

For other uses, see Fraction (disambiguation). Fraction_sentence_0

A common fraction is a numeral which represents a rational number. Fraction_sentence_1

That same number can also be represented as a decimal, a percent, or with a negative exponent. Fraction_sentence_2

For example, 0.01, 1%, and 10 are all equal to the fraction 1/100. Fraction_sentence_3

An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). Fraction_sentence_4

Other uses for fractions are to represent ratios and division. Fraction_sentence_5

Thus the fraction 3/4 can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four). Fraction_sentence_6

The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that division by zero is undefined. Fraction_sentence_7

We can also write negative fractions, which represent the opposite of a positive fraction. Fraction_sentence_8

For example, if 1/2 represents a half dollar profit, then −1/2 represents a half dollar loss. Fraction_sentence_9

Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −1/2, –1/2 and 1/–2 all represent the same fraction—negative one-half. Fraction_sentence_10

And because a negative divided by a negative produces a positive, –1/–2 represents positive one-half. Fraction_sentence_11

In mathematics the set of all numbers that can be expressed in the form a/b, 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. Fraction_sentence_12

A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). Fraction_sentence_13

However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Fraction_sentence_14

Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational). Fraction_sentence_15

## Vocabulary Fraction_section_0

See also: Numeral (linguistics) § Fractional numbers, and English numerals § Fractions and decimals Fraction_sentence_16

In a fraction, the number of equal parts being described is the numerator (from Latin numerātor, "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin dēnōminātor, "thing that names or designates"). Fraction_sentence_17

As an example, the fraction ​⁄5 amounts to eight parts, each of which is of the type named "fifth". Fraction_sentence_18

In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor. Fraction_sentence_19

Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. Fraction_sentence_20

The fraction bar may be horizontal (as in 1/3), oblique (as in 2/5), or diagonal (as in ​⁄9). Fraction_sentence_21

These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash. Fraction_sentence_22

In typography, 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. Fraction_sentence_23

In traditional typefounding, a piece of type bearing a complete fraction (e.g. 1/2) was known as a "case fraction," while those representing only part of fraction were called "piece fractions." Fraction_sentence_24

The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. Fraction_sentence_25

(For example, ​⁄5 and ​⁄5 are both read as a number of "fifths".) Fraction_sentence_26

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". Fraction_sentence_27

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. Fraction_sentence_28

For example, 3/1 may be described as "three wholes", or simply as "three". Fraction_sentence_29

When the numerator is one, it may be omitted (as in "a tenth" or "each quarter"). Fraction_sentence_30

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. Fraction_sentence_31

(For example, "two-fifths" is the fraction 2/5 and "two fifths" is the same fraction understood as 2 instances of ​⁄5.) Fraction_sentence_32

Fractions should always be hyphenated when used as adjectives. Fraction_sentence_33

Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number. Fraction_sentence_34

(For example, 3/1 may also be expressed as "three over one".) Fraction_sentence_35

The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. Fraction_sentence_36

(For example, 1/2 may be read "one-half", "one half", or "one over two".) Fraction_sentence_37

Fractions with large denominators that are not powers of ten are often rendered in this fashion (e.g., 1/117 as "one over one hundred 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"). Fraction_sentence_38

## Forms of fractions Fraction_section_1

### Simple, common, or vulgar fractions Fraction_section_2

Common fractions can be positive or negative, and they can be proper or improper (see below). Fraction_sentence_39

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. Fraction_sentence_40

### Proper and improper fractions Fraction_section_3

Common fractions can be classified as either proper or improper. Fraction_sentence_41

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. Fraction_sentence_42

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. Fraction_sentence_43

This was explained in the 17th century textbook The Ground of Arts. Fraction_sentence_44

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. Fraction_sentence_45

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. Fraction_sentence_46

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. Fraction_sentence_47

### Ratios Fraction_section_5

A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Fraction_sentence_48

Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Fraction_sentence_49

Ratios are expressed as "group 1 to group 2 ... to group n". Fraction_sentence_50

For example, if a car lot had 12 vehicles, of which Fraction_sentence_51

Fraction_unordered_list_0

• 2 are white,Fraction_item_0_0
• 6 are red, andFraction_item_0_1
• 4 are yellow,Fraction_item_0_2

then the ratio of red to white to yellow cars is 6 to 2 to 4. Fraction_sentence_52

The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1. Fraction_sentence_53

A ratio is often converted to a fraction when it is expressed as a ratio to the whole. Fraction_sentence_54

In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. Fraction_sentence_55

We can convert these ratios to a fraction, and say that ​⁄12 of the cars or ​⁄3 of the cars in the lot are yellow. Fraction_sentence_56

Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow. Fraction_sentence_57

### Decimal fractions and percentages Fraction_section_6

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10, which represents 0.0000006023. Fraction_sentence_58

The 10 represents a denominator of 10. Fraction_sentence_59

Dividing by 10 moves the decimal point 7 places to the left. Fraction_sentence_60

Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. Fraction_sentence_61

For example, 1/3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... . Fraction_sentence_62

Another kind of fraction is the percentage (Latin per centum meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Fraction_sentence_63

Thus, 51% means 51/100. Fraction_sentence_64

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. Fraction_sentence_65

The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 parts per million (ppm), means that the proportion is 75/1,000,000. Fraction_sentence_66

Whether common fractions or decimal fractions are used is often a matter of taste and context. Fraction_sentence_67

Common fractions are used most often when the denominator is relatively small. Fraction_sentence_68

By mental calculation, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). Fraction_sentence_69

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. Fraction_sentence_70

Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example \$3.75. Fraction_sentence_71

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. Fraction_sentence_72

### Mixed numbers Fraction_section_7

An improper fraction can be converted to a mixed number as follows: Fraction_sentence_73

### Historical notions Fraction_section_8

#### Complex and compound fractions Fraction_section_10

Not to be confused with Complex numbers. Fraction_sentence_74

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. Fraction_sentence_75

So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations, e.g. as Fraction_sentence_76

Nevertheless, "complex fraction" and "compound fraction" may both by considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. Fraction_sentence_77

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". Fraction_sentence_78

## Arithmetic with fractions Fraction_section_11

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero. Fraction_sentence_79

### Equivalent fractions Fraction_section_12

#### Simplifying (reducing) fractions Fraction_section_13

The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers. Fraction_sentence_80

### Comparing fractions Fraction_section_14

Comparing fractions with the same positive denominator yields the same result as comparing the numerators: Fraction_sentence_81

If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions: Fraction_sentence_82

If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. Fraction_sentence_83

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. Fraction_sentence_84

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. Fraction_sentence_85

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. Fraction_sentence_86

This allows, together with the above rules, to compare all possible fractions. Fraction_sentence_87

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Fraction_sentence_88

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. Fraction_sentence_89

Since four quarters is equivalent to one (dollar), this can be represented as follows: Fraction_sentence_90

For adding quarters to thirds, both types of fraction are converted to twelfths, thus: Fraction_sentence_91

Consider adding the following two quantities: Fraction_sentence_92

Now it can be seen that: Fraction_sentence_93

is equivalent to: Fraction_sentence_94

This method can be expressed algebraically: Fraction_sentence_95

The smallest possible denominator is given by the least common multiple of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. Fraction_sentence_96

This is called the least common denominator. Fraction_sentence_97

### Subtraction Fraction_section_17

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. Fraction_sentence_98

The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. Fraction_sentence_99

For instance, Fraction_sentence_100

### Multiplication Fraction_section_18

#### Multiplying a fraction by another fraction Fraction_section_19

To multiply fractions, multiply the numerators and multiply the denominators. Fraction_sentence_101

Thus: Fraction_sentence_102

To explain the process, consider one third of one quarter. Fraction_sentence_103

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. Fraction_sentence_104

Therefore, a third of a quarter is a twelfth. Fraction_sentence_105

Now consider the numerators. Fraction_sentence_106

The first fraction, two thirds, is twice as large as one third. Fraction_sentence_107

Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. Fraction_sentence_108

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. Fraction_sentence_109

Thus two thirds times three quarters is six twelfths. Fraction_sentence_110

A short cut for multiplying fractions is called "cancellation". Fraction_sentence_111

Effectively the answer is reduced to lowest terms during multiplication. Fraction_sentence_112

For example: Fraction_sentence_113

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. Fraction_sentence_114

Three is a common factor of the left denominator and right numerator and is divided out of both. Fraction_sentence_115

#### Multiplying a fraction by a whole number Fraction_section_20

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. Fraction_sentence_116

This method works because the fraction 6/1 means six equal parts, each one of which is a whole. Fraction_sentence_117

#### Multiplying mixed numbers Fraction_section_21

When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction. Fraction_sentence_118

For example: Fraction_sentence_119

### Converting between decimals and fractions Fraction_section_23

#### Converting repeating decimals to fractions Fraction_section_24

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Fraction_sentence_121

Sometimes an infinite repeating decimal is required to reach the same precision. Fraction_sentence_122

Thus, it is often useful to convert repeating decimals into fractions. Fraction_sentence_123

The preferred way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0.789 = 0.789789789... For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. Fraction_sentence_124

For example: Fraction_sentence_125

Fraction_description_list_1

• 0.5 = 5/9Fraction_item_1_3
• 0.62 = 62/99Fraction_item_1_4
• 0.264 = 264/999Fraction_item_1_5
• 0.6291 = 6291/9999Fraction_item_1_6

In case leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: Fraction_sentence_126

Fraction_description_list_2

• 0.05 = 5/90Fraction_item_2_7
• 0.000392 = 392/999000Fraction_item_2_8
• 0.0012 = 12/9900Fraction_item_2_9

In case a non-repeating set of decimals precede the pattern (such as 0.1523987), we can write it as the sum of the non-repeating and repeating parts, respectively: Fraction_sentence_127

Fraction_description_list_3

• 0.1523 + 0.0000987Fraction_item_3_10

Then, convert both parts to fractions, and add them using the methods described above: Fraction_sentence_128

Fraction_description_list_4

• 1523 / 10000 + 987 / 9990000 = 1522464 / 9990000Fraction_item_4_11

Alternatively, algebra can be used, such as below: Fraction_sentence_129

Fraction_ordered_list_5

1. Let x = the repeating decimal:Fraction_item_5_12
• x = 0.1523987Fraction_item_5_13
2. Multiply both sides by the power of 10 just great enough (in this case 10) to move the decimal point just before the repeating part of the decimal number:Fraction_item_5_14
• 10,000x = 1,523.987Fraction_item_5_15
3. Multiply both sides by the power of 10 (in this case 10) that is the same as the number of places that repeat:Fraction_item_5_16
• 10,000,000x = 1,523,987.987Fraction_item_5_17
4. Subtract the two equations from each other (if a = b and c = d, then a − c = b − d):Fraction_item_5_18
• 10,000,000x − 10,000x = 1,523,987.987 − 1,523.987Fraction_item_5_19
5. Continue the subtraction operation to clear the repeating decimal:Fraction_item_5_20
• 9,990,000x = 1,523,987 − 1,523Fraction_item_5_21
• 9,990,000x = 1,522,464Fraction_item_5_22
6. Divide both sides by 9,990,000 to represent x as a fractionFraction_item_5_23
• x = 1522464 / 9990000Fraction_item_5_24

## Fractions in abstract mathematics Fraction_section_25

These definitions agree in every case with the definitions given above; only the notation is different. Fraction_sentence_130

Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as: Fraction_sentence_131

Furthermore, the relation, specified as Fraction_sentence_132

is an equivalence relation of fractions. Fraction_sentence_133

Each fraction from one equivalence class may be considered as a representative for the whole class, and each whole class may be considered as one abstract fraction. Fraction_sentence_134

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. Fraction_sentence_135

Formally, for addition of fractions Fraction_sentence_136

and similarly for the other operations. Fraction_sentence_137

In the case of fractions of integers, the fractions a/b with a and b coprime and b > 0 are often taken as uniquely determined representatives for their equivalent fractions, which are considered to be the same rational number. Fraction_sentence_138

This way the fractions of integers make up the field of the rational numbers. Fraction_sentence_139

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, polynomials 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 functions). Fraction_sentence_140

## Algebraic fractions Fraction_section_26

Main article: Algebraic fraction Fraction_sentence_141

Main articles: Nth root and Rationalization (mathematics) Fraction_sentence_142

A fraction may also contain radicals in the numerator and/or the denominator. Fraction_sentence_143

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. Fraction_sentence_144

It is also more convenient if division is to be done manually. Fraction_sentence_145

When the denominator is a monomial square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator: Fraction_sentence_146

The process of rationalization of binomial 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. Fraction_sentence_147

For example: Fraction_sentence_148

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. Fraction_sentence_149

## Typographical variations Fraction_section_28

In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half). Fraction_sentence_151

See the article on Number Forms for information on doing this in Unicode. Fraction_sentence_152

Scientific publishing distinguishes four ways to set fractions, together with guidelines on use: Fraction_sentence_153

## History Fraction_section_29

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. Fraction_sentence_154

The Egyptians used Egyptian fractions c. 1000 BC. Fraction_sentence_155

About 4000 years ago, Egyptians divided with fractions using slightly different methods. Fraction_sentence_156

They used least common multiples with unit fractions. Fraction_sentence_157

Their methods gave the same answer as modern methods. Fraction_sentence_158

The Egyptians also had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems. Fraction_sentence_159

The Greeks used unit fractions and (later) continued fractions. Fraction_sentence_160

Followers of the Greek philosopher Pythagoras (c. 530 BC) discovered that the square root of two cannot be expressed as a fraction of integers. Fraction_sentence_161

(This is commonly though probably erroneously ascribed to Hippasus of Metapontum, who is said to have been executed for revealing this fact.) Fraction_sentence_162

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. Fraction_sentence_163

A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta (c. AD 500), Brahmagupta (c. 628), and Bhaskara (c. 1150). Fraction_sentence_164

Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (cheda), but without a bar between them. Fraction_sentence_165

In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer. Fraction_sentence_166

The integer was written on one line and the fraction in its two parts on the next line. Fraction_sentence_167

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. Fraction_sentence_168

For example, Bhaskara I writes: Fraction_sentence_169

Fraction_description_list_6

• ६        १        २Fraction_item_6_25
• १        १        १०Fraction_item_6_26
• ४        ५        ९Fraction_item_6_27

which is the equivalent of Fraction_sentence_170

Fraction_description_list_7

• 6        1        2Fraction_item_7_28
• 1        1        −1Fraction_item_7_29
• 4        5        9Fraction_item_7_30

and would be written in modern notation as 61/4, 11/5, and 2 − 1/9 (i.e., 18/9). Fraction_sentence_171

In discussing the origins of decimal fractions, Dirk Jan Struik states: Fraction_sentence_172

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. Fraction_sentence_173

## In formal education Fraction_section_30

### Pedagogical tools Fraction_section_31

In primary schools, fractions have been demonstrated through Cuisenaire 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. Fraction_sentence_174