Fraction
A fraction (from Latin ', "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: formula_1 and formula_2) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator"', displayed below (or after) that line. Numerators and denominators are also used in fractions that are not "common", including compound fractions, complex fractions, and mixed numerals. In  positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates of a cake. A common fraction is a numeral which represents a rational number. That same number  can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10 are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1). Other uses for fractions are to represent ratios and division. Thus the fraction 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). 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. We can also write negative fractions, which represent the opposite of a positive fraction. For example, if represents a half dollar profit, then − represents a half dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −, and all represent the same fraction — negative one-half. And because a negative divided by a negative produces a positive, represents positive one-half. In mathematics the set of all numbers that can be expressed in the form , where  a and "b" are integers and "b" is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word "fraction" can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as formula_3 (see square root of 2) and (see proof that π is irrational). Vocabulary. In a  fraction, the number of equal parts being described is the numerator (from Latin ', "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin ', "thing that names or designates"). As an example, the fraction amounts to eight parts, each of which is of the type named "fifth". In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor. Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in  ), oblique (as in 2/5), or diagonal (as in ). These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus, or fraction slash. 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. In traditional typefounding, a piece of type bearing a complete fraction (e.g. ) was known as a "case fraction," while those representing only  part of fraction were called "piece fractions." The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not 1. (For example, and are both read as a number of "fifths".) Exceptions include the denominator 2, which is always read "half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "percent". When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read out as  a whole number. For example, may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (as in "a tenth" or "each quarter"). The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction and "two fifths" is the same fraction understood as 2 instances of .) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading  it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number. (For example, may also be expressed as "three over one".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are "not" powers of ten are often rendered in this fashion (e.g., as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal  ordinal fashion (e.g., as "six-millionths", "six millionths", or "six one-millionths"). Forms of fractions. Simple, common, or vulgar fractions. A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as "a"/"b" or formula_4, where "a" and "b" are both integers. As with other fractions, the denominator ("b") cannot be zero. Examples include formula_1, formula_6, formula_7, and formula_8. The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy. "Common fractions" can be positive or negative, and they can  be proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not "common fractions;" though, unless irrational, they can be evaluated to a common fraction. In Unicode, precomposed fraction characters are in the Number Forms block. Proper and improper fractions. Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction  means a piece", so a proper fraction must be less than 1. This was explained in the 17th century textbook "The Ground of Arts". In general, a common fraction is said to be a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of  improper fractions are 9/4, −4/3, and 3/3. Reciprocals and the "invisible denominator". The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of formula_11, for instance, is formula_12. The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a proper fraction. When the numerator and denominator of a fraction are equal (for example, formula_13), its value  is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as formula_14, where 1 is sometimes referred to as the "invisible denominator". Therefore, every fraction or integer, except for zero, has a reciprocal. For example. the reciprocal of 17 is formula_15. Ratios. A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in  a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to group 2 ... to group "n"". For example, if a car lot had 12 vehicles, of which then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1. A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on  the lot is 4:12 or 1:3. We can convert these ratios to a fraction, and say that of the cars or of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow. Decimal fractions and percentages. A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of  digits to the right of a decimal separator, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, "viz." 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as  an improper fraction, 375/100, or as a mixed number, formula_16. Decimal fractions can also be expressed using scientific notation with negative exponents, such as , which represents 0.0000006023. The represents a denominator of . Dividing by moves the decimal point 7 places to the left. Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... . 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. Thus , 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100. The related concept of "permille" or "parts per thousand" (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 "parts per million" (ppm), means that the proportion is 75/1,000,000. Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation, it is easier to multiply 16 by 3/16  than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to  the fraction 3/6. Mixed numbers. A mixed numeral (also called a "mixed fraction" or "mixed number") is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign). It is used primarily in measurement: formula_17inches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers. The sum can be implied without the use of a visible operator such as the appropriate "+". For example, in referring to two entire cakes and three quarters of another cake, the numerals denoting the integer part and the fractional part of the cakes  can be written next to each other as formula_18instead of the unambiguous notation formula_19 Negative mixed numerals, as in formula_20, are treated like formula_21 Any such sum of a "whole" plus a "part" can be converted to an improper fraction by applying the rules of adding unlike quantities. This tradition is, formally, in conflict with the notation in algebra where adjacent symbols, without an explicit infix operator, denote a product. In the expression formula_22, the "understood" operation is multiplication. If is replaced by, for example, the fraction formula_23, the "understood" multiplication needs to be replaced by explicit multiplication, to avoid  the appearance of a mixed number. When multiplication is intended, formula_24 may be written as An improper fraction can be converted to a mixed number as follows: Historical notions. Egyptian fraction. An Egyptian fraction is the sum of distinct positive unit fractions, for example formula_30. This definition derives from the fact that the ancient Egyptians expressed all fractions except formula_1, formula_32 and formula_33 in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, formula_34 can be written as formula_35 Any positive rational number can be written as a sum of unit fractions in infinitely  many ways. Two ways to write formula_36 are formula_37 and formula_38. Complex and compound fractions. In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, formula_39 and formula_40 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 unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression , because of multiple possible interpretations, e.g. as A compound fraction is a fraction of a fraction, or any number of fractions connected with the word "of", corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication). For example, formula_33 of formula_34 is a compound fraction, corresponding to formula_49. 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 formula_50 is equivalent to the complex fraction formula_51.) Nevertheless, "complex fraction" and  compound fraction may both be considered outdated and now used in no well-defined manner, partly even taken synonymously for each other or for mixed numerals. They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day meaning of "consisting of parts". Arithmetic with fractions. Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero. 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 formula_52, the fraction formula_53 equals formula_54. Therefore, multiplying by formula_53 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 formula_1. When the numerator and denominator are both multiplied by 2, the result is formula_57, which has the same value (0.5) as formula_1. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (formula_57) make up half the cake (formula_1). 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 formula_4 are divisible by formula_62 then they can be written as formula_63 and formula_64 and the fraction becomes formula_65, which can be reduced by dividing both the numerator and denominator by formula_66  to give the reduced fraction formula_67 If one takes for the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its "lowest terms". If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be "irreducible", "reduced", or "in simplest terms". For example, formula_68 is not in lowest terms because both 3 and 9 can be exactly divided by 3 . In contrast, formula_69 "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 formula_70, for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction formula_71 can be reduced to lowest terms by dividing the numerator and denominator by 21: The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers. Comparing fractions. Comparing fractions with the same positive denominator yields the same result as comparing the numerators: If the equal denominators are  negative, then the opposite result of comparing the numerators holds for the fractions: 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 formula_4 and formula_77, these are converted to formula_78 and formula_79 (where the dot signifies multiplication and is an alternative symbol to ×). Then "bd" is a common denominator and the numerators "ad" and "bc" can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare "ad" and "bc", without evaluating "bd", e.g., comparing formula_32 ? formula_1 gives formula_82. For the more laborious question formula_83 ? formula_84 multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding formula_85 ? formula_86. It  is not necessary to calculate formula_87 – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is formula_88. 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: 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  the invisible denominator formula_90 For adding quarters to thirds, both types of fraction are converted to twelfths, thus: Consider adding the following two quantities: First, convert formula_93 into fifteenths by multiplying both the numerator and denominator by three: formula_94. Since formula_95 equals 1, multiplication by formula_95 does not change the value of the fraction. Second, convert formula_97 into fifteenths by multiplying both the numerator and denominator by five: formula_98. Now it can be seen that: is equivalent to: This method can be expressed algebraically: 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 formula_33 and formula_103 the single denominators have a common factor formula_104 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. 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 . 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, Multiplication. Multiplying a fraction by another fraction. To multiply fractions, multiply the numerators and multiply the denominators. Thus: To explain the process, consider one third of one quarter. Using the example of a cake, if three  small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two  thirds times three quarters is six twelfths. A short cut for multiplying fractions is called "cancellation". Effectively the answer is reduced to lowest terms during multiplication. For example: A two is a common factor in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both. Multiplying a fraction by a whole number. Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. This method  works because the fraction 6/1 means six equal parts, each one of which is a whole. Multiplying mixed numbers. When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction. For example: In other words, formula_18 is the same as formula_112, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is formula_113, since 8 cakes, each made of quarters, is 32 quarters in total. Division. To divide a fraction by a whole number, you may either divide the numerator by the number, if it  goes evenly into the numerator, or multiply the denominator by the number. For example, formula_114 equals formula_32 and also equals formula_116, which reduces to formula_32. To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, formula_118. Converting between decimals and fractions. To change a common fraction to a decimal, do a long division of the decimal representations of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the answer to the desired accuracy. For example, to change to a decimal, divide by (" into  ), to obtain . To change to a decimal, divide by ( into "), and stop when the desired accuracy is obtained, e.g., at decimals with . The fraction can be written exactly with two decimal digits, while the fraction cannot be written exactly as a decimal with a finite number of digits. To change a decimal to a fraction, write in the denominator a followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus formula_119 Converting repeating decimals to  fractions. Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions. A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines  as a denominator. For example: If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: If a non-repeating set of decimals precede the pattern (such as ), one may write the number as the sum of the non-repeating and repeating parts, respectively: Then, convert both parts to fractions, and add them using the methods described above: Alternatively, algebra can be used, such as below: Fractions in abstract mathematics. In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are  consistent and reliable. Mathematicians define a fraction as an ordered pair formula_120 of integers formula_121 and formula_122 for which the operations addition, subtraction, multiplication, and division are defined as follows: These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with respect to addition and multiplication might be defined as: Furthermore, the relation, specified as is an equivalence relation of fractions. 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. 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 and similarly for the other operations. In the case of fractions of integers, the fractions with and coprime and are often taken as uniquely determined representatives for their "equivalent" fractions, which are considered to be the "same" rational number. This way the fractions of integers make up the field of the rational numbers. More generally, "a" and "b" may be elements of  any integral domain "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). Algebraic fractions. An algebraic fraction is the indicated quotient of two algebraic expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions  are formula_132 and formula_133. Algebraic fractions are subject to the same field properties as arithmetic fractions. If the numerator and the denominator are polynomials, as in formula_132, the algebraic fraction is called a "rational fraction" (or "rational expression"). An "irrational fraction" is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in formula_133. 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 formula_136, is called a complex fraction. The field of rational numbers is the field of fractions of the integers, while the integers themselves are not a field but rather an integral domain. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such as formula_137 are also rational fractions, as are a transcendental numbers such as formula_138 since all of formula_139  and formula_140 are real numbers, and thus considered as coefficients. These same numbers, however, are not rational fractions with "integer" coefficients. The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction formula_141 can be decomposed as the sum of two fractions: formula_142 This is useful for the computation of antiderivatives of rational functions (see partial fraction decomposition for more). Radical expressions. A fraction may also contain radicals in the numerator or the denominator. If the denominator contains radicals, it can be helpful to rationalize it (compare Simplified form of a  radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a monomial square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator: 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. For example: