Fraction

A fraction (from , "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 simple fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the 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. Fractions can be used to represent ratios and division. Thus the fraction can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division (three divided by four). Negative fractions 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.…

A fraction (from , "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 simple fraction (examples: and ) consists of an integer numerator, displayed above a line (or before a slash like ), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the 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. Fractions can be used to represent ratios and division. Thus the fraction can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division (three divided by four). Negative fractions 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 a rational number is a number that can be represented by a fraction of the form , where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol or Q, which stands for quotient. The term fraction and the notation can also be used for mathematical expressions that do not represent a rational number (for example \textstyle \frac \sqrt 2 2 ), or even do not represent any number (for example the rational fraction \textstyle \frac 1 x ). A rational number, expressed as \frac{p}{q} where p and q are coprime integers and is in base {b} , has a terminating representation in base {b} if and only if q divides a power of b, or \frac{p}{q} = \frac{C}{b^n} ,for some {C} and some integer {n} > 0. By cross multiplying, the equality is equivalent to {qC}={pb^n} . Because q doesn't divide p , it must divide b^n , and the expansion will not continue. === Simple, common, or vulgar fractions === A simple fraction (also known as a common fraction or vulgar fraction) is a rational number written as a \div b or , where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Examples include , − , , and . 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 decimal expressions (see below) are not common fractions though, unless irrational, they can be evaluated to a common fraction. A unit fraction is a common fraction with a numerator of 1 (e.g., ). Unit fractions can also be expressed using negative exponents, as in 2^{-1} , which represents \frac{1}{2^{1}} , which also equals \frac{1}{2} , and 2^{-2} which represents \frac{1}{2^{2}} or \frac{1}{4} . A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. = . In Unicode, precomposed fraction characters are in the Number Forms block. A decimal fraction with infinitely many digits to the right of the decimal separator represents an infinite series. For example, = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... Another kind of fraction is the percentage (from , meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means . Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% means and −27% means . The related concept of permille, or parts per thousand (ppt), means a denominator of 1000, and this parts-per notation is commonly used with larger denominators, such as million and billion, e.g. 75 parts per million (ppm) means that the proportion is . The choice between fraction and decimal notation is often a matter of taste and context. Fractions are used most often when the denominator is relatively small. By mental calculation, it is easier to multiply 16 by than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more precise (exact, in fact) to multiply 15 by , for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions with denominator 100, i.e., with two digits after the decimal separator, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the form (but not the meaning) of a fraction, as, for example, "3/6", commonly read three and six, means three shillings and sixpence and has no relationship to the fraction three sixths. Any mixed number can be converted to an improper fraction by applying the rules of adding unlike quantities: change the measure of one of the unlike quantities so it has the same measure as the other. For example, 2 + \tfrac34 = \tfrac84 + \tfrac34 = \tfrac{11}4. Alternatively, one can multiply the whole number part by the denominator, then add the numerator, to get the numerator of the improper fraction, which keeps the same denominator as the improper fraction. Conversely, an improper fraction can be converted to a mixed number using division with remainder, with the mixed number consisting the whole number part as the quotient and the fraction part (if any) being the remainder divided by denominator of the original improper fraction. For example, since 4 goes into 11 twice, with 3 left over, \tfrac{11}4 = 2 + \tfrac{3}{4}. In primary school, teachers often insist that every fractional result should be expressed as a mixed number. Outside school, mixed numbers are commonly used for describing measurements, for instance hours or 5 3/16 inches, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized metric system. However, scientific measurements typically use the metric system, which is based on decimal fractions, and starting from the secondary school level, mathematics pedagogy treats every fraction uniformly as a rational number, the quotient of integers, leaving behind the concepts of improper fraction and mixed number. College students with years of mathematical training are sometimes confused when re-encountering mixed numbers because they are used to the convention that juxtaposition in algebraic expressions means multiplication. &= \frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}, \qquad \frac{\;\!\tfrac{3}{2}\;\!}5 = \frac{3}{2} \div 5 = \frac{3}{2}\times\frac{1}{5} = \frac{3}{10}, \\[10mu] \frac{12\tfrac{3}{4}}{26} &= \frac{12 \times 4 + 3}{4} \div 26 = \frac{12 \times 4 + 3}{4} \times \frac{1}{26} = \frac{51}{104}. \end{align} A complex fraction should never be written without an obvious marker showing which fraction is nested inside the other, as such expressions are ambiguous. For example, the expression 5/10/20 could be plausibly interpreted as either \tfrac{5}{10} \big/ 20 = \tfrac{1}{40} or as 5 \big/ \tfrac{10}{20} = 10. The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance (5/10)\big/20 or 5\big/(10/20). 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 ). For example, \tfrac{3}{4} of \tfrac{5}{7} is a compound fraction, corresponding to \tfrac{3}{4} \times \tfrac{5}{7} = \tfrac{15}{28} . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction \tfrac{3}{4} \times \tfrac{5}{7} is equivalent to the complex fraction .) Nevertheless, complex fraction and compound fraction may both be considered outdated and now used in no well-defined manner, partly even taken as synonymous with each other or with mixed numerals. They have lost their meaning as technical terms and the attributes complex and compound tend to be used in their everyday meaning of consisting of parts. ==== Simplifying (reducing) fractions ==== Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction \tfrac{a}{b} are divisible by , then they can be written as a=cd , b=ce , and the fraction becomes , which can be reduced by dividing both the numerator and denominator by c to give the reduced fraction . 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, \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 , for example. As another example, since 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: \frac{63}{462} = \frac{63 \,\div\, 21}{462 \,\div\, 21}= \frac{3}{22}. The Euclidean algorithm gives a method for finding the greatest common divisor of any two integers. \tfrac{3}{-4} If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger. One way to compare fractions with different numerators and denominators is to find a common denominator. To compare \tfrac{a}{b} and \tfrac{c}{d} , these are converted to \tfrac{a\cdot d}{b\cdot d} and \tfrac{b\cdot c}{b\cdot d} (where the dot signifies multiplication and is an alternative symbol to ×). 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 \tfrac{2}{3} ? \tfrac{1}{2} gives \tfrac{4}{6}>\tfrac{3}{6} . For the more laborious question \tfrac{5}{18} ? \tfrac{4}{17}, multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding \tfrac{5 \times 17}{18 \times 17} ? \tfrac{18 \times 4}{18 \times 17} . It is not necessary to calculate 18 \times 17 – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is . 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. \tfrac24+\tfrac34=\tfrac54=1\tfrac14 . \frac14 + \frac13=\frac{1\times3}{4\times3} + \frac{1\times4}{3\times4}=\frac3{12} + \frac4{12}=\frac7{12}. Consider adding the following two quantities: \frac35+\frac23. First, convert \tfrac35 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 \frac35+\frac23 is equivalent to \frac9{15}+\frac{10}{15}=\frac{19}{15}=1\frac4{15}. This method can be expressed algebraically: \frac{a}{b} + \frac {c}{d} = \frac{ad+cb}{bd}. This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding \tfrac{3}{4} and \tfrac{5}{6} the single denominators have a common factor 2, and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator. \begin{alignat}{3} \frac34 + \frac56 &= \frac{3 \cdot 6}{4 \cdot 6} + \frac{4 \cdot 5}{4\cdot 6} = \frac{18}{24} + \frac{20}{24} &&= \frac{19}{12} \\[10mu] &=\frac{3 \cdot 3}{4 \cdot 3} + \frac{2 \cdot 5}{2 \cdot 6} = \frac{9}{12} + \frac{10}{12} &&=\frac{19}{12}. \end{alignat} 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. \tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16. To subtract a mixed number, an extra one can be borrowed from the minuend, for instance 4 - 2\tfrac34 = (4-2-1) + \bigl(1 - \tfrac34\bigr) = 1\tfrac14. \frac{2}{3} \times \frac{3}{4} = \frac{6}{12}. To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths. A short cut for multiplying fractions is called cancellation. Effectively the answer is reduced to lowest terms during multiplication. For example: \frac{2}{3} \times \frac{3}{4} = \frac{{\color{RoyalBlue}\cancel{\color{Black}2}} ^{~1}} {{\color{RedOrange}\cancel{\color{Black}3}} ^{~1}} \times \frac{{\color{RedOrange}\cancel{\color{Black}3}} ^{~1}} {{\color{RoyalBlue}\cancel{\color{Black}4}} ^{~2}} 6 \times \tfrac{3}{4} = \tfrac{6}{1} \times \tfrac{3}{4} = \tfrac{18}{4}. This method works because the fraction 6/1 means six equal parts, each one of which is a whole. 3 \times 2\frac{3}{4} = \frac31 \times \frac{2\times 4 + 3}{4} = \frac{33}{4} = 8\frac{1}{4}. Alternately, mixed numbers can be treated as sums, and multiplied as binomials. In this example, 3 \times 2\frac34 = 3 \times 2 + 3 \times \frac34 = 6 + \frac94 = 8\frac14. 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} . To change a common fraction to decimal notation, do a long division of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the result to the desired precision. For example, to change to a decimal expression, divide by (" into "), to obtain exactly . To change to a decimal expression, divide by (" into "), and stop when the desired precision is obtained, e.g., at four places after the decimal separator (ten-thousandths) as . The fraction is expressed exactly with only two digits after the decimal separator, while the fraction cannot be written exactly as a decimal with a finite number of digits. A decimal expression can be converted to a fraction by removing the decimal separator, using the result as the numerator, and using followed by the same number of zeroes as there are digits to the right of the decimal separator as the denominator. Thus, 1.23 = \tfrac{123}{100}. 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: = 5/9 = 62/99 = 264/999 = 6291/9999 If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: = 5/90 = 392/999000 = 12/9900 If a non-repeating set of digits precede the pattern (such as ), one may write the number as the sum of the non-repeating and repeating parts, respectively: 0.1523 + Then, convert both parts to fractions, and add them using the methods described above: 1523 / 10000 + 987 / 9990000 = 1522464 / 9990000 Alternatively, algebra can be used, such as below: Let x = the repeating decimal: x = Multiply both sides by the power of 10 just great enough (in this case 10 4 ) to move the decimal point just before the repeating part of the decimal number: 10,000x = Multiply both sides by the power of 10 (in this case 10 3 ) that is the same as the number of places that repeat: 10,000,000x = Subtract the two equations from each other (if and , then ): 10,000,000x − 10,000x = − Continue the subtraction operation to clear the repeating decimal: 9,990,000x = 1,523,987 − 1,523 9,990,000x = 1,522,464 Divide both sides by 9,990,000 to represent x as a fraction x = (a,b) + (c,d) = (ad+bc,bd) \, (a,b) - (c,d) = (ad-bc,bd) \, (a,b) \cdot (c,d) = (ac,bd) (a,b) \div (c,d) = (ad,bc) \quad(\text{with, additionally, } c \ne 0) These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead of defining subtraction and division as operations, the inverse fractions with respect to addition and multiplication might be defined as: \begin{align} -(a,b) &= (-a, b) & & \text{additive inverse fractions,} \\ &&&\text{with } (0,b) \text{ as additive unities, and}\\ (a,b)^{-1} &= (b,a) & & \text{multiplicative inverse fractions, for } a \ne 0, \\ &&&\text{with } (b,b) \text{ as multiplicative unities}. \end{align} Furthermore, the relation, specified as (a, b) \sim (c, d)\quad \iff \quad ad=bc, 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 (a,b) \sim (a',b')\quad and \quad (c,d) \sim (c',d') \quad imply ((a,b) + (c,d)) \sim ((a',b') + (c',d')) and similarly for the other operations. In the case of fractions of integers, the 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). 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 \frac{3x}{x^2+2x-3} and {{tmath|1= \frac{\sqrt{x+2} }{x^2-3} }}. Algebraic fractions are subject to the same field properties 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 is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in {{tmath|1= \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 {{tmath|1= \frac{1 + \tfrac{1}{x} }{1 - \tfrac{1}{x} } }}, 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 , are also rational fractions, as are a transcendental numbers such as \pi/2, since all of \sqrt{2},\pi, and 2 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 \frac{2x}{x^2-1} can be decomposed as the sum of two fractions: . This is useful for the computation of antiderivatives of rational functions (see partial fraction decomposition for more). \frac{3}{\sqrt{7}} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{3\sqrt{7}}{7}. 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: \frac{3}{3-2\sqrt{5}} = \frac{3}{3-2\sqrt{5}} \cdot \frac{3+2\sqrt{5}}{3+2\sqrt{5}} = \frac{3(3+2\sqrt{5})}{{3}^2 - (2\sqrt{5})^2} = \frac{ 3 (3 + 2\sqrt{5} ) }{ 9 - 20 } = - \frac{ 9+6 \sqrt{5} }{11}, \frac{3}{3+2\sqrt{5}} = \frac{3}{3+2\sqrt{5}} \cdot \frac{3-2\sqrt{5}}{3-2\sqrt{5}} = \frac{3(3-2\sqrt{5})}{{3}^2 - (2\sqrt{5})^2} = \frac{ 3 (3 - 2\sqrt{5} ) }{ 9 - 20 } = - \frac{ 9-6 \sqrt{5} }{11}. Even if this process results in the numerator being irrational, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator.

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