Fraction - Knowledge

2805: 12318: 2473: 7936: 5734: 2800:{\displaystyle {\begin{aligned}{\frac {\;\!{\tfrac {1}{2}}\;\!}{\tfrac {1}{3}}}&={\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}},\\{\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{aligned}}} 9827: 4729: 12287: 40: 7727: 5482: 9677: 9413: 9723:: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts. 6041:

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.

6206: 7931:{\displaystyle {\begin{aligned}-(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\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}} 5729:{\displaystyle {\begin{aligned}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot 6}{4\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {20}{24}}&={\frac {19}{12}}\\&={\frac {3\cdot 3}{4\cdot 3}}+{\frac {2\cdot 5}{2\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&={\frac {19}{12}}\end{aligned}}} 9419: 9155: 4957: 3488:

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

9771:, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions ( 8034:

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

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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

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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

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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

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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

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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,

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Record's Arithmetick: Or, the Ground of Arts: Teaching the Perfect Work and Practise of Arithmetick ... Made by Mr. Robert Record ... Afterward Augmented by Mr. John Dee. And Since Enlarged with a Third Part of Rules of Practise ... By John Mellis. And Now Diligently Perused, Corrected ... and

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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:

9136: 6423: 1397:, 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. 1928:

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

9812:. 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. 2462:

are complex fractions. To interpret nested fractions written "stacked" with a horizontal fraction bars, treat shorter bars as nested inside longer bars. Complex fractions can be simplified using multiplication by the reciprocal, as described below at

6538: 11574:, with little classroom experience with mixed numbers, so that for them, when returning to mixed number forms, they apply their recent knowledge of multiplication as the implied operation in concatenation to the 'new' situation of mixed numbers. 77:, "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 5966: 5842: 9672:{\displaystyle {\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}}} 9408:{\displaystyle {\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}}} 6304: 169:, 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 3911: 1290:

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

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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

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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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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.

1309:). Thus, for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, namely, 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the 7715: 7010:= 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: 1816: 5333: 4722: 2336: 728:

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

10069:, fractions were always expressed as an addition to or subtraction from an integer. The integer was written on one line and the fraction in its two parts on the next line. If the fraction was marked by a small circle 10376:'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 8279: 5210: 3167: 5096: 6035: 1434:

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.

1904:. 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 5411: 2174: 2262: 10926:

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.

6201:{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {{\cancel {2}}^{~1}}{{\cancel {3}}^{~1}}}\times {\frac {{\cancel {3}}^{~1}}{{\cancel {4}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}} 8727: 6700: 2478: 2460: 761:.) 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 9048:), 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 2898: 6215:

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.

6662: 1458:) of the two parts, without the use of an intermediate plus (+) or minus (−) sign. When the fraction is written horizontally, a space is added between the integer and fraction to separate them. 1877:

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

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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

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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

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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.

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Enlarged; with an Appendix of Figurative Numbers ... with Tables of Board and Timber Measure ... the First Calculated by R. C. But Corrected, and the Latter ... Calculated by Ro. Hartwell ...

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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,

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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

6323: 1095:, 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. 6548:

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,

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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

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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

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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

8862: 6438: 4465: 3034: 2203: 1668: 1610: 4433: 2987: 1491: 145: 10696: 10658: 10620: 10582: 10544: 6691: 6610: 5473: 5444: 5035: 4788: 4759: 4357: 4328: 4185: 4156: 3738: 3709: 3516: 3412: 3381: 3352: 3108: 3079: 2112: 2083: 2054: 2025: 1575: 887: 116: 9810: 396: 185:, 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 4581: 5743:

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.

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Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts.

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

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College students have had many years of high school and perhaps college experience in which multiplication has been the implied operation in concatenations such as 4

5758: 7304: 6229: 8829: 3598: 3569: 4952:{\displaystyle {\frac {1}{4}}\ +{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}\ +{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}\ +{\frac {4}{12}}={\frac {7}{12}}.} 3837: 3169:. 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 10466: 10446: 10422: 10398: 8882: 7327: 3997: 3538: 3323: 1546: 1070:

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

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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,

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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

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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

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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:

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In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are

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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), −

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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

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

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containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

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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.

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as "one over one hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g.,

12190: 5042: 3046:, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see 2357:, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, 10184: 1822:, with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, 725:

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").

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

10977: 10357: 9941: BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with 5987: 2945:

The meaning can be made explicit by writing the fractions using distinct separators or by adding explicit parentheses, in this instance

508: 11373: 6810:{\displaystyle {\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\times {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}} 5344: 2117: 634:" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow 315:

all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive,

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form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients,

2208: 1008:). Unit fractions can also be expressed using negative exponents, as in 2, which represents 1/2, and 2, which represents 1/(2) or 1/4. 6934:

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

9729:: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them 8671: 1272:

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

12067: 2409: 11762: 8415:. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are 4289:

can be compared. It is not necessary to determine the value of the common denominator to compare fractions – one can just compare

11306: 2849: 10373: 6615: 4583:– only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is 1164:), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. 4115:{\displaystyle {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.} 12147: 12108: 12040: 11900: 11727: 11493: 11434: 11367: 11263: 11213: 11118: 11065: 11007: 7951: 4126:

numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

9131:{\displaystyle {\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}} 3172: 1673: 1087:

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

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multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding

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is one that is not rational, as, for example, one that contains the variable under a fractional exponent or root, as in

4515: 4470: 4362: 3934: 12256: 10206:. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, 9891: 8951: 6941: 3655:

of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest

785:. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are 431: 43:

A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the

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Lee, Mary A; Messner, Shelley J. (2000). "Analysis of concatenations and order of operations in written mathematics".

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

666: 11928: 11528: 11405: 10951: 10203: 9910: 9863: 6418:{\displaystyle 3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}} 1953: 11993: 10472:

in these standards always refers to a non-negative number.)" The document itself also refers to negative fractions.

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

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One way to compare fractions with different numerators and denominators is to find a common denominator. To compare

2903: 12470: 12024: 9931:: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used 9052:

square root, it can be rationalized by multiplying both the top and the bottom of the fraction by the denominator:

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Bringhurst, Robert (2002). "5.2.5: Use the Virgule with Words and Dates, the Solidus with Split-level Fractions".

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Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write

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Filliozat, Pierre-Sylvain (2004). "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature". In

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is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions.

1187:. Therefore, every fraction or integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is 662:) was known as a "case fraction", while those representing only part of fraction were called "piece fractions". 500: 9848: 9015: 8899:

is used when decomposing rational fractions into sums of simpler fractions. For example, the rational fraction

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Multiply both sides by the power of 10 (in this case 10) that is the same as the number of places that repeat:

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Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as

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

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numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction

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16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more

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

1900:, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized 235:

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

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The same fractional notation—with the fraction given before the integer—appears soon after in the work of

9877: 6533:{\displaystyle 3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.} 1577:

representing the additional partial cake juxtaposed; this is more concise than the more explicit notation

504: 11946: 11711: 10077:, it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, 10011:", which contains work on the theory of numbers, arithmetical operations, and operations with fractions. 3218: 2360: 11656: 11634: 9945:. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for 12162: 11866: 11841: 10328: 8834: 4438: 2992: 2179: 1638: 1580: 6317:

The product of mixed numbers can be computed by converting each to an improper fraction. For example:

4409: 2948: 1464: 1454:) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or 121: 12429: 11110: 10675: 10637: 10599: 10561: 10523: 9859: 9844: 9768: 6667: 6586: 5449: 5420: 5011: 4764: 4735: 4333: 4304: 4161: 4132: 3714: 3685: 3492: 3388: 3357: 3328: 3084: 3055: 2088: 2085:

in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example,

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is pronounced "two and three quarters", with the integer and fraction portions connected by the word

1551: 863: 92: 31: 17: 9788: 8888:, and thus considered as coefficients. These same numbers, however, are not rational fractions with 11784: 1105: 428:

can also be used for mathematical expressions that do not represent a rational number (for example

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can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division

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The introduction of decimal fractions as a common computational practice can be dated back to the

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of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of

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

4560: 3652: 3448:. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ( 1819: 1362:

Decimal fractions with infinitely many digits to the right of the decimal separator represent an

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in the numerator or the denominator. If the denominator contains radicals, it can be helpful to

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

5961:{\displaystyle 4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.} 12299: 10973: 10491: 9924: 8331:

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

7942: 7371: 7081:), one may write the number as the sum of the non-repeating and repeating parts, respectively: 6694: 5837:{\displaystyle {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}} 1142: 209: 11280: 7332: 6299:{\displaystyle 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}} 1496: 12230: 11561: 11395: 11253: 11181: 11057: 10829: 9142: 8797: 5740: 3287: 2813: 11481: 11422: 11357: 3906:{\displaystyle {\tfrac {63}{462}}={\tfrac {63\,\div \,21}{462\,\div \,21}}={\tfrac {3}{22}}} 1461:

As a basic example, two entire cakes and three quarters of another cake might be written as

12412: 11955: 11685: 10481: 8412: 8027: 7277: 3670: 3001: 2971: 2912: 2867: 2444: 1930: 10998:

The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English

3574: 3545: 8: 10292: 10066: 9146: 8803: 8751: 8526: 3917: 1336: 12057: 11959: 9775:) or within exponents to increase legibility. Fractions written this way, also known as 5847:

To subtract a mixed number, an extra one can be borrowed from the minuend, for instance

3815:

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction

12375: 12360: 12218: 11809: 11758: 11557: 10996: 10451: 10431: 10407: 10383: 10249: 9985: 9982: 9957: 8867: 8755: 8739: 8402: 8350: 7312: 7092:

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

4807:

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

3982: 3523: 3308: 1531: 1422: 1411: 1072: 11298: 9884: 3711:

is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast,

12390: 12143: 12104: 12036: 11924: 11896: 11733: 11723: 11719: 11524: 11489: 11430: 11401: 11363: 11259: 11209: 11114: 11061: 11003: 10947: 10939: 10886: 10813: 10784: 10256: 10052: 9932: 9779:, are written all on one typographical line, but take 3 or more typographical spaces. 9011: 8747: 8390: 8386: 7003: 6996: 6990: 1947: 1728: 1296: 460: 11172:

Easterday, Kenneth E. (Winter 1982). "One-hundred fifty years of vulgar fractions".

10352:, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), 5981:

To multiply fractions, multiply the numerators and multiply the denominators. Thus:

12434: 12380: 12028: 11963: 11801: 11553: 11514: 11234: 10852: 10822: 10742: 10727: 10004: 9968: 9149:

of the denominator so that the denominator becomes a rational number. For example:

8896: 7710:{\displaystyle (a,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)} 3743:

in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.

3291: 1811:{\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.} 1286: 972: 603: 370:

is not zero; the set of all rational numbers is commonly represented by the symbol

162: 9763:: 1/2, so called because this notation was used for pre-decimal British currency ( 9753:

if they take only one typographical space, regardless of the direction of the bar.

5328:{\displaystyle {\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}} 12424: 12385: 12170: 11203: 11185: 11051: 10981: 10777: 10594: 10349: 10260: 10008: 9946: 8743: 8343: 7002:

A conventional way to indicate a repeating decimal is to place a bar (known as a

4277:(where the dot signifies multiplication and is an alternative symbol to ×). Then 2347: 1905: 1363: 1310: 1292: 1141:. The product of a fraction and its reciprocal is 1, hence the reciprocal is the 1013: 847: 762: 631: 623: 611: 562: 558: 554: 335: 11565: 11238: 10372:

Several states in the United States have adopted learning trajectories from the

4717:{\displaystyle {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}} 3831:

can be reduced to lowest terms by dividing the numerator and denominator by 21:

3042:

is a fraction of a fraction, or any number of fractions connected with the word

2331:{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}} 553:

amounts to eight parts, each of which is of the type named "fifth". In terms of

10770: 10670: 10518: 10345: 10316: 10268: 9965: 9953: 9851: in this History section. Unsourced material may be challenged and removed. 9007: 7367: 3656: 1390: 1084: 615: 607: 530: 518: 70: 10041: 9145:

denominators involves multiplying the top and the bottom of a fraction by the

8030:

of fractions. Each fraction from one equivalence class may be considered as a

12464: 12449: 12370: 12016: 11916: 11737: 10511: 10353: 10296: 9942: 8742:

of the integers, while the integers themselves are not a field but rather an

7048: 3416:. When the numerator and denominator are both multiplied by 2, the result is 1995: 1901: 988: 971:. The term was originally used to distinguish this type of fraction from the 12140:

The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook

12032: 11023: 5037:

into fifteenths by multiplying both the numerator and denominator by three:

3920:

gives a method for finding the greatest common divisor of any two integers.

12400: 12395: 12365: 11944:

Curtis, Lorenzo J. (1978). "Concept of the exponential law prior to 1900".

10913: 10720: 10284: 9961: 9706: 7271: 7044: 5151:

into fifteenths by multiplying both the numerator and denominator by five:

3661: 1058: 1018: 643: 12062:. Vol. 1. La Salle, Illinois: Open Court Publishing Company. p. 11441:

When you read a mixed number out loud, you say the whole number, the word

10899: 10180: 4801: 1818:

Conversely, an improper fraction can be converted to a mixed number using

598:). These marks are respectively known as the horizontal bar; the virgule, 11989: 11255:

Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation

10556: 10300: 10288: 10030: 8885: 7363: 6429: 3620:, which can be reduced by dividing both the numerator and denominator by 3283: 3279: 1273: 1148:

When the numerator and denominator of a fraction are equal (for example,

982:; though, unless irrational, they can be evaluated to a common fraction. 12023:. Boston Series in the Philosophy of Science. Vol. 238. Dordrecht: 11488:. American Mathematical Society. §14.3 Mixed Numbers, pp. 225–227. 9701:, simple fractions are sometimes printed as a single character, e.g. ½ ( 1098: 525:, "counter" or "numberer"), and the type or variety of the parts is the 12444: 12439: 12241: 11813: 10078: 10019: 9993: 9971: 9698: 8533: 8358: 5205:{\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}} 3162:{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}} 1386: 1382:= 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... 782: 619: 599: 11519: 7077:

If a non-repeating set of decimals precede the pattern (such as 0.1523

5091:{\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}} 10308: 1635:. Subtraction or negation is applied to the entire mixed numeral, so 11967: 11805: 9826: 5752:

subtracting the numerators of the original fractions. For instance,

12307: 10942:(2003). "CRC Concise Encyclopedia of Mathematics, Second Edition". 10496: 10486: 10361: 9989: 9702: 9049: 9037: 9027: 8408: 7359: 7213:

Continue the subtraction operation to clear the repeating decimal:

6918:

can be written exactly with two decimal digits, while the fraction

6030:{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}} 1430: 1426: 1402: 900: 745:

and "two fifths" is the same fraction understood as 2 instances of

401: 11299:"Ask Dr. Math: Can Negative Fractions Also Be Proper or Improper?" 8361:

in one indeterminate, with coefficients from some integral domain

7047:

precede the pattern, the nines are suffixed by the same number of

12407: 12303: 12286: 11229:

Perry, Owen; Perry, Joyce (1981). "Chapter 2: Common fractions".

10632: 10324: 10199: 9996:, who is said to have been executed for revealing this fact.) In 9928: 9710: 8323:

are often taken as uniquely determined representatives for their

8313: 7307: 6212: 5406:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}} 4728: 2169:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.} 1410:(ppt) has an implied denominator of 1000, while the more general 821:

as "six-millionths", "six millionths", or "six one-millionths").

703: 665:

The denominators of English fractions are generally expressed as

627: 537:, "thing that names or designates"). As an example, the fraction 6693:. To divide a number by a fraction, multiply that number by the 2257:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}} 513:

In a fraction, the number of equal parts being described is the

39: 11513:. OBP Series in Mathematics. Open Book Publishers. p. 89. 11362:(2nd ed., reprinted ed.). Cheltenham: Thornes. p. 5. 11286:

James Flesher, and are to be sold by Edward Dod. pp. 266–.

10276: 12138:

Berggren, J. Lennart (2007). "Mathematics in Medieval Islam".

9764: 8722:{\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}} 338:

is a number that can be represented by a fraction of the form

12276: 10906: 10295:

many centuries before Stevin and that the Persian astronomer

10179:

The horizontal fraction bar is first attested in the work of

6978: 6894:"), and stop when the desired accuracy is obtained, e.g., at 5976: 2455:{\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26} 1213: 205: 12189:. Common Core State Standards Initiative. 2010. p. 85. 10917:, although in fact both are synonyms for the standard slash. 9988:. (This is commonly though probably erroneously ascribed to 6820: 1950:

is the sum of distinct positive unit fractions, for example

10704: 10000: 6218: 2893:{\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}} 1897: 3270:

used in their every day meaning of "consisting of parts".

10712: 10195: 10051:). Their works form fractions by placing the numerators ( 6657:{\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}} 669:, in the plural if the numerator is not 1. (For example, 509:

Unicode subscripts and superscripts § Fraction slash

459:), and even do not represent any number (for example the 789:

powers of ten are often rendered in this fashion (e.g.,

11258:. Springer Science & Business Media. pp. 87–. 6428:

Alternately, mixed numbers can be treated as sums, and

1303:(·), a comma) depends on the locale (for examples, see 1222:". For example, if a car lot had 12 vehicles, of which 1057:

In Unicode, precomposed fraction characters are in the

12019:; Cohen, Robert S.; Renn, Jürgen; et al. (eds.). 11484:

Understanding Numbers in Elementary School Mathematics

11296: 11161:(1st ed.). Oxford: Oxford University Press. 1913. 11142:(1st ed.). Oxford: Oxford University Press. 1917. 8806: 8767: 8705: 8685: 8016:{\displaystyle (a,b)\sim (c,d)\quad \iff \quad ad=bc,} 7274:. Mathematicians define a fraction as an ordered pair 6952: 6796: 6765: 6750: 6735: 6720: 6705: 6672: 6643: 6620: 6591: 6556: 6285: 6270: 6255: 6240: 5944: 5919: 5867: 5823: 5808: 5793: 5778: 5763: 5454: 5425: 5191: 5176: 5161: 5077: 5062: 5047: 5016: 4769: 4740: 4703: 4685: 4670: 4655: 4608: 4593: 4520: 4475: 4443: 4414: 4382: 4367: 4338: 4309: 4240: 4195: 4166: 4137: 4076: 4056: 4034: 4014: 3954: 3939: 3892: 3857: 3842: 3719: 3690: 3497: 3393: 3362: 3333: 3223: 3192: 3177: 3148: 3133: 3118: 3089: 3060: 2918: 2879: 2854: 2683: 2592: 2501: 2487: 2424: 2365: 2317: 2302: 2287: 2272: 2243: 2228: 2213: 2184: 2152: 2137: 2122: 2093: 2064: 2035: 2006: 1973: 1958: 1851: 1830: 1794: 1779: 1764: 1749: 1694: 1649: 1591: 1556: 1472: 868: 829: 469: 435: 126: 97: 10678: 10640: 10602: 10564: 10526: 10454: 10434: 10410: 10386: 10356:, pie-shaped pieces, plastic rectangles, grid paper, 10291:. It is true that decimal fractions were used by the 10213: 9791: 9422: 9158: 9061: 8954: 8905: 8870: 8837: 8766: 8674: 8616: 8544: 8479: 8421: 8169: 8105: 8044: 7954: 7730: 7628: 7553: 7468: 7383: 7335: 7315: 7280: 6944: 6703: 6670: 6618: 6589: 6554: 6441: 6326: 6232: 6053: 5990: 5856: 5761: 5485: 5452: 5423: 5347: 5272: 5226: 5159: 5045: 5014: 4971: 4816: 4767: 4738: 4653: 4591: 4563: 4518: 4473: 4441: 4412: 4365: 4336: 4307: 4238: 4193: 4164: 4135: 4012: 3985: 3937: 3840: 3717: 3688: 3659:. One says that the fraction has been reduced to its 3577: 3548: 3526: 3495: 3391: 3360: 3331: 3311: 3221: 3206:{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}} 3175: 3116: 3087: 3058: 2995: 2951: 2906: 2852: 2816: 2476: 2412: 2363: 2270: 2211: 2182: 2120: 2091: 2062: 2033: 2004: 1956: 1828: 1741: 1718:{\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.} 1676: 1641: 1583: 1554: 1534: 1499: 1467: 866: 468: 434: 382: 124: 95: 11895:(6th ed.). Philadelphia: Saunders College Pub. 10968:

H. Wu, "The Mis-Education of Mathematics Teachers",

9949:, used for certain systems of weights and measures. 8289:

In the case of fractions of integers, the fractions

7265: 6985:

Decimal representation § Conversion to fraction

4622:{\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}} 1279: 646:, a piece of type bearing a complete fraction (e.g. 8274:{\displaystyle ((a,b)+(c,d))\sim ((a',b')+(c',d'))} 7100:Alternatively, algebra can be used, such as below: 3483: 1868:{\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.} 1017:is a common fraction in which the denominator is a 11056:(2nd ed.). Lausanne: AVA Publishing. p. 10995: 10690: 10652: 10614: 10576: 10538: 10460: 10440: 10416: 10392: 10238: 9804: 9671: 9407: 9130: 8996: 8938: 8876: 8856: 8823: 8786: 8721: 8652: 8586: 8515: 8463: 8273: 8153: 8091: 8015: 7930: 7709: 7613: 7538: 7453: 7350: 7321: 7298: 6968: 6809: 6685: 6656: 6604: 6575: 6532: 6417: 6298: 6200: 6029: 5960: 5836: 5739:The smallest possible denominator is given by the 5728: 5467: 5438: 5405: 5327: 5252: 5204: 5090: 5029: 4997: 4951: 4782: 4753: 4716: 4621: 4575: 4550:{\displaystyle {\tfrac {18\times 4}{18\times 17}}} 4549: 4505:{\displaystyle {\tfrac {5\times 17}{18\times 17}}} 4504: 4459: 4427: 4396:{\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}} 4395: 4351: 4322: 4269: 4224: 4179: 4150: 4114: 3991: 3968:{\displaystyle {\tfrac {3}{4}}>{\tfrac {2}{4}}} 3967: 3905: 3732: 3703: 3592: 3563: 3532: 3510: 3406: 3375: 3346: 3317: 3256: 3205: 3161: 3102: 3073: 3028: 2981: 2937: 2892: 2838: 2799: 2454: 2398: 2330: 2256: 2197: 2168: 2106: 2077: 2048: 2019: 1986: 1867: 1810: 1717: 1662: 1604: 1569: 1540: 1520: 1485: 1418:(ppm), means that the proportion is 75/1,000,000. 992:is a common fraction with a numerator of 1 (e.g., 881: 622:, fractions stacked vertically are also known as " 483: 451: 390: 139: 110: 11709: 10018:seems to have originated in India in the work of 8997:{\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}} 6969:{\displaystyle 12.3456={\tfrac {123456}{10000}}.} 2604: 2590: 2499: 2485: 2341: 1994:. This definition derives from the fact that the 452:{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}} 151:, displayed above a line (or before a slash like 12462: 12103:. New Jersey: Princeton University Press. 1986. 11921:Mathematical Puzzles: A Connoisseur's Collection 8349:, in which case a fraction is an element of the 7096:1523 / 10000 + 987 / 9990000 = 1522464 / 9990000 1064: 11990:"Earliest Uses of Various Mathematical Symbols" 11612:A new mathematical and philosophical dictionary 8946:can be decomposed as the sum of two fractions: 7152:Subtract the two equations from each other (if 6142: 6125: 6101: 6084: 1987:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}} 501:Numeral (linguistics) § Fractional numbers 12008: 11420: 11201: 10972:, Volume 58, Issue 03 (March 2011), 10060: 8653:{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} 8525:. Algebraic fractions are subject to the same 8516:{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} 4962:Consider adding the following two quantities: 2938:{\displaystyle 5{\big /}{\tfrac {10}{20}}=10.} 1548:representing the whole cakes and the fraction 1359:moves the decimal point 7 places to the left. 1335:Decimal fractions can also be expressed using 1299:, the appearance of which (e.g., a period, an 1079:In general, a common fraction is said to be a 505:English numerals § Fractions and decimals 12257: 12184:"Common Core State Standards for Mathematics" 11893:An introduction to the history of mathematics 11251: 11049: 10279:in 1585, together with a French translation, 9986:cannot be expressed as a fraction of integers 8365:, are themselves an integral domain, call it 8154:{\displaystyle \quad (c,d)\sim (c',d')\quad } 7006:) over the digits that repeat, for example 0. 5932: 5907: 5253:{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}} 4998:{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}} 4270:{\displaystyle {\tfrac {b\cdot c}{b\cdot d}}} 4225:{\displaystyle {\tfrac {a\cdot d}{b\cdot d}}} 2437: 2415: 1732: 1707: 1682: 12125:Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī 11710:Schoenborn, Barry; Simkins, Bradley (2010). 11245: 11197: 11195: 10970:Notices of the American Mathematical Society 10239:{\displaystyle {\frac {3\quad 1}{5\quad 3}}} 10112:and would be written in modern notation as 6 7234:Divide both sides by 9,990,000 to represent 6312: 5338:This method can be expressed algebraically: 3273: 903:. As with other fractions, the denominator ( 572:. The fraction bar may be horizontal (as in 12142:. Princeton University Press. p. 518. 11761:. Encyclopedia of Mathematics. 2012-04-06. 11359:New comprehensive mathematics for 'O' level 10993: 10348:, fractions have been demonstrated through 9686: 7874:multiplicative inverse fractions, for 5132:does not change the value of the fraction. 4794: 4281:is a common denominator and the numerators 1099:Reciprocals and the "invisible denominator" 846:, where vulgar is Latin for "common") is a 588:), oblique (as in 2/5), or diagonal (as in 12264: 12250: 11584: 11582: 11543: 11297:Laurel Brenner; Peterson (31 March 2004). 11272: 11228: 11104: 10987: 10682: 10644: 10606: 10568: 10530: 10287:(1548–1620), then settled in the Northern 10014:A modern expression of fractions known as 8327:fractions, which are considered to be the 7993: 7989: 6979:Converting repeating decimals to fractions 5977:Multiplying a fraction by another fraction 2603: 2589: 2498: 2484: 1183:, where 1 is sometimes referred to as the 204:Other uses for fractions are to represent 12014: 11657:"Complex fraction definition and meaning" 11604: 11602: 11518: 11233:. Palgrave Macmillan UK. pp. 13–25. 11192: 11171: 11098: 11045: 11043: 10938: 10684: 10646: 10608: 10570: 10532: 10367: 9911:Learn how and when to remove this message 8532:If the numerator and the denominator are 7535: 7450: 6821:Converting between decimals and fractions 3880: 3876: 3867: 3863: 3110:is a compound fraction, corresponding to 2846:could be plausibly interpreted as either 1506: 561:, and the denominator corresponds to the 484:{\displaystyle \textstyle {\frac {1}{x}}} 384: 12271: 12137: 11508: 10946:. Chapman & Hall/CRC. p. 1925. 9006:. This is useful for the computation of 8796:, are also rational fractions, as are a 8787:{\displaystyle \textstyle {\sqrt {2}}/2} 8587:{\displaystyle {\frac {3x}{x^{2}+2x-3}}} 8464:{\displaystyle {\frac {3x}{x^{2}+2x-3}}} 8286:and similarly for the other operations. 7614:{\displaystyle (a,b)\cdot (c,d)=(ac,bd)} 7539:{\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,} 7454:{\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,} 6219:Multiplying a fraction by a whole number 4727: 1727:Any mixed number can be converted to an 38: 12223:The Online Encyclopaedia of Mathematics 11983: 11981: 11979: 11977: 11915: 11588: 11579: 11511:Teaching Mathematics at Secondary Level 11146: 11113:: Hartley & Marks. pp. 81–82. 11002:. Mathematical Association of America. 10944:CRC Concise Encyclopedia of Mathematics 9046:Simplified form of a radical expression 8407:An algebraic fraction is the indicated 8092:{\displaystyle (a,b)\sim (a',b')\quad } 3300: 3278:Like whole numbers, fractions obey the 251:represents a half-dollar profit, then − 14: 12463: 12101:A Source Book in Mathematics 1200–1800 12087: 12055: 11987: 11943: 11783:Galen, Leslie Blackwell (March 2004). 11608: 11599: 11445:, then the fraction. The mixed number 11393: 11278: 11092: 11040: 10468:is a positive whole number. (The word 10374:Common Core State Standards Initiative 10334: 10065:), but without a bar between them. In 9772: 9021: 8396: 6576:{\displaystyle {\tfrac {10}{3}}\div 5} 3923: 3213:is equivalent to the complex fraction 12245: 11996:from the original on 20 February 2016 11782: 11776: 11397:The Complete Idiot's Guide to Algebra 11355: 8939:{\displaystyle {\frac {2x}{x^{2}-1}}} 8738:The field of rational numbers is the 8596:, the algebraic fraction is called a 1936: 1347:, which represents 0.0000006023. The 824: 614:); and the fraction bar, solidus, or 11974: 11890: 11884: 11309:from the original on 9 November 2014 11208:. Courier Corporation. p. 219. 11050:Ambrose, Gavin; et al. (2006). 10339: 9849:adding citations to reliable sources 9820: 8529:properties as arithmetic fractions. 3746:Using these rules, we can show that 3432:, which has the same value (0.5) as 3354:equals 1. Therefore, multiplying by 1941: 12059:A History of Mathematical Notations 12021:History of Science, History of Text 11127: 4557:. It is not necessary to calculate 3257:{\displaystyle {\tfrac {3/4}{7/5}}} 3048: 2399:{\displaystyle {\tfrac {1/2}{1/3}}} 907:) cannot be zero. Examples include 830:Simple, common, or vulgar fractions 557:, the numerator corresponds to the 24: 12163:"MacTutor's al-Uqlidisi biography" 11785:"Putting Fractions in Their Place" 11558:10.1111/j.1949-8594.2000.tb17254.x 11479: 11332:New England Compact Math Resources 11252:Jack Williams (19 November 2011). 10364:, counters and computer software. 9141:The process of rationalization of 630:fractions", and diagonal ones as " 25: 12482: 12211: 11988:Miller, Jeff (22 December 2014). 11107:The Elements of Typographic Style 11095:, "275. The solidus", pp. 312–314 10889:mistakenly distinguish the slash 10303:fractions with great ease in his 10204:Islamic inheritance jurisprudence 9709:for information on doing this in 8857:{\displaystyle {\sqrt {2}},\pi ,} 7266:Fractions in abstract mathematics 5971: 4460:{\displaystyle {\tfrac {4}{17}},} 3484:Simplifying (reducing) fractions 3029:{\displaystyle 5{\big /}(10/20).} 2198:{\displaystyle {\tfrac {13}{17}}} 1663:{\displaystyle -2{\tfrac {3}{4}}} 1605:{\displaystyle 2+{\tfrac {3}{4}}} 1339:with negative exponents, such as 1280:Decimal fractions and percentages 12316: 12285: 11425:Ready for Fractions and Decimals 11421:Wingard-Nelson, Rebecca (2014). 10892: 10105:1 1 −1 9956:used unit fractions and (later) 9825: 4428:{\displaystyle {\tfrac {5}{18}}} 4406:For the more laborious question 2982:{\displaystyle (5/10){\big /}20} 1486:{\displaystyle 2{\tfrac {3}{4}}} 1437: 1385:Another kind of fraction is the 140:{\displaystyle {\tfrac {17}{3}}} 12196:from the original on 2013-10-19 12176: 12156: 12131: 12117: 12093: 12081: 12070:from the original on 2014-04-14 12049: 11937: 11909: 11873:from the original on 2013-05-21 11859: 11848:from the original on 2013-05-26 11834: 11823:from the original on 2011-07-13 11765:from the original on 2014-10-21 11751: 11703: 11692:from the original on 2018-03-14 11678: 11667:from the original on 2017-12-01 11649: 11619: 11592:A complete system of arithmetic 11537: 11502: 11473: 11414: 11387: 11376:from the original on 2019-01-19 11349: 11320: 11290: 11222: 11165: 11074:from the original on 2016-03-04 10920: 10691:{\displaystyle :\;\mathbb {N} } 10653:{\displaystyle :\;\mathbb {Z} } 10615:{\displaystyle :\;\mathbb {Q} } 10577:{\displaystyle :\;\mathbb {R} } 10539:{\displaystyle :\;\mathbb {C} } 10283:, by the Flemish mathematician 10229: 10220: 10108:4 5 9 10102:6 1 2 10094:४ ५ ९ 10088:१ १ १ 10085:६ १ २ 9836:needs additional citations for 8150: 8106: 8088: 7994: 7988: 7918: as multiplicative unities 7686: 6686:{\displaystyle {\tfrac {2}{3}}} 6605:{\displaystyle {\tfrac {2}{3}}} 5468:{\displaystyle {\tfrac {5}{6}}} 5439:{\displaystyle {\tfrac {3}{4}}} 5030:{\displaystyle {\tfrac {3}{5}}} 4783:{\displaystyle {\tfrac {1}{4}}} 4754:{\displaystyle {\tfrac {1}{2}}} 4352:{\displaystyle {\tfrac {1}{2}}} 4323:{\displaystyle {\tfrac {2}{3}}} 4180:{\displaystyle {\tfrac {c}{d}}} 4151:{\displaystyle {\tfrac {a}{b}}} 3733:{\displaystyle {\tfrac {3}{8}}} 3704:{\displaystyle {\tfrac {3}{9}}} 3511:{\displaystyle {\tfrac {a}{b}}} 3407:{\displaystyle {\tfrac {1}{2}}} 3376:{\displaystyle {\tfrac {n}{n}}} 3347:{\displaystyle {\tfrac {n}{n}}} 3103:{\displaystyle {\tfrac {5}{7}}} 3074:{\displaystyle {\tfrac {3}{4}}} 2585: 2107:{\displaystyle {\tfrac {5}{7}}} 2078:{\displaystyle {\tfrac {3}{4}}} 2049:{\displaystyle {\tfrac {2}{3}}} 2020:{\displaystyle {\tfrac {1}{2}}} 1998:expressed all fractions except 1570:{\displaystyle {\tfrac {3}{4}}} 882:{\displaystyle {\tfrac {a}{b}}} 111:{\displaystyle {\tfrac {1}{2}}} 11546:School Science and Mathematics 11202:David E. Smith (1 June 1958). 11086: 11053:The Fundamentals of Typography 11016: 10962: 10879: 10331:as early as the 10th century. 9981: BC) discovered that the 9805:{\displaystyle {\frac {1}{2}}} 9621: 9602: 9581: 9567: 9547: 9528: 9357: 9338: 9317: 9303: 9283: 9264: 9016:partial fraction decomposition 8758:representing numbers, such as 8268: 8265: 8243: 8237: 8215: 8212: 8206: 8203: 8191: 8185: 8173: 8170: 8147: 8125: 8119: 8107: 8085: 8063: 8057: 8045: 7990: 7985: 7973: 7967: 7955: 7913: 7901: 7866: 7854: 7835: 7822: 7815: as additive unities, and 7810: 7798: 7775: 7760: 7750: 7738: 7704: 7687: 7683: 7665: 7659: 7647: 7641: 7629: 7608: 7590: 7584: 7572: 7566: 7554: 7532: 7505: 7499: 7487: 7481: 7469: 7447: 7420: 7414: 7402: 7396: 7384: 7293: 7281: 5899: 5881: 5746: 3542:, then they can be written as 3020: 3006: 2966: 2952: 2342:Complex and compound fractions 331:represents positive one-half. 13: 1: 11923:. A K Peters. pp. 2, 6. 11793:American Mathematical Monthly 10933: 10255:In discussing the origins of 10045: 10034: 10023: 9975: 9935: 9032:Rationalization (mathematics) 3979:, and the equal denominators 3624:to give the reduced fraction 2464: 1065:Proper and improper fractions 702:"hundredth"/"hundredths" or " 494: 11869:. allbusiness.com glossary. 11844:. allbusiness.com glossary. 10994:Schwartzman, Steven (1994). 10311:, early fifteenth century). 9923:The earliest fractions were 9036:A fraction may also contain 8666:contain a fraction, such as 8389:(also known as the field of 5116:equals 1, multiplication by 4761:of a cake is to be added to 1351:represents a denominator of 391:{\displaystyle \mathbb {Q} } 7: 12127:. Wiesbaden: Steiner. 1951. 11947:American Journal of Physics 11688:. Sosmath.com. 1996-02-05. 11394:Kelley, W. Michael (2004). 11239:10.1007/978-1-349-05230-1_2 10475: 10098:which is the equivalent of 7783:additive inverse fractions, 6543: 4638: 4576:{\displaystyle 18\times 17} 3600:, and the fraction becomes 3290:laws, and the rule against 10: 12487: 11716:Technical Math For Dummies 11661:Collins English Dictionary 11631:Collins English Dictionary 10885:Some typographers such as 9816: 9690: 9025: 8400: 6988: 6982: 2345: 498: 29: 12356: 12325: 12314: 12292: 12283: 11919:(2004). "Uses of fuses". 11159:Oxford English Dictionary 11140:Oxford English Dictionary 10059:) over the denominators ( 10056: 9697:In computer displays and 7692:with, additionally, 7358:for which the operations 7108:= the repeating decimal: 6313:Multiplying mixed numbers 5217:Now it can be seen that: 4187:, these are converted to 3464:) make up half the cake ( 3274:Arithmetic with fractions 1731:by applying the rules of 1276:that it would be yellow. 1206: 232:(three divided by four). 147:) consists of an integer 32:Fraction (disambiguation) 12219:"Fraction, arithmetical" 12056:Cajori, Florian (1928). 10872: 9687:Typographical variations 7374:are defined as follows: 7351:{\displaystyle b\neq 0,} 6697:of that fraction. Thus, 4795:Adding unlike quantities 2346:Not to be confused with 1733:adding unlike quantities 1612:cakes. The mixed number 1528:cakes, with the numeral 1521:{\displaystyle 2\ \,3/4} 12471:Fractions (mathematics) 12235:Encyclopædia Britannica 12173:. Retrieved 2011-11-22. 12033:10.1007/1-4020-2321-9_7 11712:"8. Fun with Fractions" 11589:Trotter, James (1853). 11509:Gardiner, Tony (2016). 11400:. Penguin. p. 25. 11279:Record, Robert (1654). 10903:and the fraction slash 10329:Abu'l-Hasan al-Uqlidisi 8750:with coefficients in a 8342:may be elements of any 7272:consistent and reliable 6430:multiplied as binomials 3653:greatest common divisor 2839:{\displaystyle 5/10/20} 1820:division with remainder 1400:The related concept of 642:square. In traditional 11609:Barlow, Peter (1814). 11429:. Enslow. p. 14. 11205:History of Mathematics 11174:Contemporary Education 10778:Dyadic (finite binary) 10692: 10654: 10616: 10578: 10540: 10462: 10448:is a whole number and 10442: 10418: 10394: 10368:Documents for teachers 10313: 10299:used both decimal and 10240: 10061: 9806: 9733:. An example would be 9705:). See the article on 9673: 9409: 9132: 8998: 8940: 8878: 8858: 8825: 8798:transcendental numbers 8788: 8723: 8654: 8588: 8517: 8465: 8275: 8155: 8093: 8017: 7932: 7711: 7615: 7540: 7455: 7352: 7323: 7300: 6970: 6811: 6687: 6658: 6606: 6577: 6534: 6419: 6300: 6202: 6031: 5962: 5838: 5730: 5469: 5440: 5407: 5329: 5254: 5206: 5092: 5031: 4999: 4953: 4791: 4784: 4755: 4718: 4623: 4577: 4551: 4506: 4461: 4429: 4397: 4353: 4324: 4271: 4226: 4181: 4152: 4116: 3993: 3969: 3907: 3734: 3705: 3594: 3565: 3534: 3512: 3408: 3377: 3348: 3319: 3258: 3207: 3163: 3104: 3075: 3030: 2983: 2939: 2894: 2840: 2801: 2456: 2400: 2332: 2258: 2199: 2170: 2108: 2079: 2050: 2021: 1988: 1933:means multiplication. 1869: 1812: 1719: 1664: 1606: 1571: 1542: 1522: 1487: 1394: 1143:multiplicative inverse 883: 534: 522: 485: 453: 392: 141: 112: 74: 62: 11891:Eves, Howard (1990). 11720:Wiley Publishing Inc. 11480:Wu, Hung-Hsi (2011). 10693: 10655: 10617: 10579: 10541: 10463: 10443: 10419: 10395: 10265: 10252:in the 13th century. 10241: 10202:, who specialized in 9834:This History section 9807: 9767:), as in "2/6" for a 9674: 9410: 9133: 8999: 8941: 8879: 8859: 8826: 8789: 8724: 8655: 8589: 8518: 8466: 8413:algebraic expressions 8276: 8156: 8094: 8018: 7933: 7712: 7616: 7541: 7456: 7353: 7324: 7301: 7299:{\displaystyle (a,b)} 6983:Further information: 6971: 6878:to a decimal, divide 6842:to a decimal, divide 6812: 6688: 6659: 6607: 6578: 6535: 6420: 6301: 6203: 6032: 5963: 5839: 5741:least common multiple 5731: 5470: 5441: 5408: 5330: 5255: 5207: 5093: 5032: 5000: 4954: 4802:invisible denominator 4785: 4756: 4731: 4719: 4624: 4578: 4552: 4507: 4462: 4430: 4398: 4354: 4325: 4297:, without evaluating 4272: 4227: 4182: 4153: 4117: 3994: 3970: 3908: 3735: 3706: 3595: 3566: 3535: 3513: 3409: 3378: 3349: 3320: 3259: 3208: 3164: 3105: 3076: 3049:§ Multiplication 3031: 2984: 2940: 2895: 2841: 2802: 2457: 2401: 2333: 2259: 2200: 2171: 2109: 2080: 2051: 2022: 1989: 1931:algebraic expressions 1896:hours or 5 3/16 1870: 1813: 1720: 1665: 1607: 1572: 1543: 1523: 1488: 1185:invisible denominator 884: 486: 454: 393: 142: 113: 42: 12025:Springer Netherlands 11686:"Compound Fractions" 10823:Algebraic irrational 10676: 10638: 10600: 10562: 10524: 10482:Cross multiplication 10452: 10432: 10408: 10384: 10211: 10192:Muslim mathematician 10144:, and 2 − 9845:improve this article 9789: 9420: 9156: 9059: 8952: 8903: 8868: 8835: 8824:{\textstyle \pi /2,} 8804: 8764: 8672: 8614: 8542: 8477: 8419: 8167: 8103: 8042: 8028:equivalence relation 7952: 7728: 7626: 7551: 7466: 7381: 7333: 7313: 7278: 6942: 6701: 6668: 6616: 6587: 6552: 6439: 6324: 6230: 6146: 6129: 6105: 6088: 6051: 5988: 5854: 5759: 5483: 5450: 5421: 5345: 5270: 5224: 5157: 5043: 5012: 4969: 4814: 4765: 4736: 4651: 4589: 4561: 4516: 4471: 4439: 4410: 4363: 4334: 4305: 4236: 4191: 4162: 4133: 4010: 3983: 3935: 3838: 3715: 3686: 3593:{\displaystyle b=ce} 3575: 3564:{\displaystyle a=cd} 3546: 3524: 3493: 3389: 3358: 3329: 3309: 3301:Equivalent fractions 3219: 3173: 3114: 3085: 3056: 2993: 2949: 2904: 2850: 2814: 2474: 2410: 2361: 2268: 2209: 2180: 2118: 2089: 2060: 2031: 2002: 1954: 1826: 1739: 1674: 1639: 1581: 1552: 1532: 1497: 1465: 973:sexagesimal fraction 864: 466: 432: 380: 212:. Thus the fraction 122: 93: 89:fraction (examples: 30:For other uses, see 27:Ratio of two numbers 11960:1978AmJPh..46..896C 10514: 10335:In formal education 10067:Sanskrit literature 9958:continued fractions 9022:Radical expressions 8756:radical expressions 8606:irrational fraction 8602:rational expression 8397:Algebraic fractions 7220:= 1,523,987 − 1,523 6664:, which reduces to 6432:. In this example, 4052: because 3924:Comparing fractions 3918:Euclidean algorithm 1337:scientific notation 1125:, for instance, is 975:used in astronomy. 638:square, or a wider 400:, which stands for 12293:Division and ratio 12169:2011-11-15 at the 11627:"complex fraction" 11466:two and one fourth 11356:Greer, A. (1986). 11028:www.mathsisfun.com 10980:2017-08-20 at the 10688: 10650: 10612: 10574: 10536: 10510: 10458: 10438: 10414: 10390: 10250:Leonardo Fibonacci 10236: 10003:mathematicians in 9983:square root of two 9933:Egyptian fractions 9802: 9783:Built-up fractions 9669: 9405: 9128: 9012:rational functions 8994: 8936: 8874: 8854: 8821: 8784: 8783: 8748:rational fractions 8740:field of fractions 8719: 8714: 8694: 8650: 8584: 8513: 8461: 8403:Algebraic fraction 8391:rational functions 8387:rational fractions 8383:field of fractions 8351:field of fractions 8271: 8151: 8089: 8013: 7928: 7926: 7707: 7611: 7536: 7451: 7348: 7319: 7296: 6966: 6961: 6807: 6805: 6790: 6759: 6744: 6729: 6714: 6683: 6681: 6654: 6652: 6637: 6602: 6600: 6573: 6565: 6530: 6415: 6296: 6294: 6279: 6264: 6249: 6211:A two is a common 6198: 6027: 5958: 5953: 5928: 5876: 5834: 5832: 5817: 5802: 5787: 5772: 5726: 5724: 5465: 5463: 5436: 5434: 5403: 5325: 5263:is equivalent to: 5250: 5202: 5200: 5185: 5170: 5088: 5086: 5071: 5056: 5027: 5025: 4995: 4949: 4792: 4780: 4778: 4751: 4749: 4714: 4712: 4694: 4679: 4664: 4619: 4617: 4602: 4573: 4547: 4545: 4502: 4500: 4457: 4452: 4425: 4423: 4393: 4391: 4376: 4349: 4347: 4320: 4318: 4301:, e.g., comparing 4267: 4265: 4222: 4220: 4177: 4175: 4148: 4146: 4112: 4090: 4070: 4048: 4028: 3989: 3965: 3963: 3948: 3903: 3901: 3886: 3851: 3730: 3728: 3701: 3699: 3590: 3561: 3530: 3508: 3506: 3404: 3402: 3373: 3371: 3344: 3342: 3315: 3254: 3252: 3203: 3201: 3186: 3159: 3157: 3142: 3127: 3100: 3098: 3071: 3069: 3026: 2979: 2935: 2927: 2890: 2888: 2863: 2836: 2797: 2795: 2692: 2601: 2510: 2496: 2452: 2433: 2396: 2394: 2328: 2326: 2311: 2296: 2281: 2254: 2252: 2237: 2222: 2195: 2193: 2166: 2161: 2146: 2131: 2114:can be written as 2104: 2102: 2075: 2073: 2046: 2044: 2017: 2015: 1984: 1982: 1967: 1937:Historical notions 1865: 1860: 1839: 1808: 1803: 1788: 1773: 1758: 1715: 1703: 1660: 1658: 1602: 1600: 1567: 1565: 1538: 1518: 1483: 1481: 1425:, it is easier to 1423:mental calculation 1412:parts-per notation 1408:parts per thousand 1295:to the right of a 1093:top-heavy fraction 1073:The Ground of Arts 879: 877: 825:Forms of fractions 481: 480: 449: 448: 388: 137: 135: 108: 106: 63: 12458: 12457: 12237:. 5 January 2024. 12149:978-0-691-11485-9 12110:978-0-691-02397-7 12042:978-1-4020-2320-0 11902:978-0-03-029558-4 11729:978-0-470-59874-0 11520:10.11647/OBP.0071 11495:978-0-8218-5260-6 11436:978-0-7660-4247-6 11369:978-0-85950-159-0 11328:"Proper Fraction" 11265:978-0-85729-862-1 11215:978-0-486-20430-7 11120:978-0-88179-206-5 11067:978-2-940411-76-4 11009:978-0-88385-511-9 10870: 10869: 10866: 10865: 10862: 10861: 10858: 10857: 10847: 10846: 10843: 10842: 10839: 10838: 10835: 10834: 10804: 10803: 10800: 10799: 10796: 10795: 10792: 10791: 10785:Repeating decimal 10752: 10751: 10748: 10747: 10743:Negative integers 10737: 10736: 10733: 10732: 10728:Composite numbers 10461:{\displaystyle b} 10441:{\displaystyle a} 10417:{\displaystyle b} 10393:{\displaystyle a} 10340:Pedagogical tools 10305:Key to arithmetic 10257:decimal fractions 10234: 10075:⟨+⟩ 10071:⟨०⟩ 9921: 9920: 9913: 9895: 9800: 9773:complex fractions 9761:solidus fractions 9721:Special fractions 9667: 9661: 9636: 9619: 9591: 9578: 9545: 9517: 9514: 9496: 9474: 9471: 9446: 9443: 9403: 9397: 9372: 9355: 9327: 9314: 9281: 9253: 9250: 9232: 9210: 9207: 9182: 9179: 9126: 9120: 9104: 9103: 9098: 9087: 9086: 9072: 9071: 8992: 8971: 8934: 8877:{\displaystyle 2} 8843: 8773: 8746:. Similarly, the 8717: 8713: 8693: 8648: 8629: 8598:rational fraction 8582: 8511: 8492: 8459: 7941:Furthermore, the 7919: 7899: 7875: 7816: 7796: 7784: 7693: 7322:{\displaystyle a} 6997:repeating decimal 6991:Repeating decimal 6960: 6804: 6789: 6758: 6743: 6728: 6713: 6680: 6651: 6636: 6599: 6564: 6525: 6509: 6490: 6459: 6413: 6397: 6384: 6357: 6344: 6293: 6278: 6263: 6248: 6196: 6183: 6170: 6157: 6151: 6134: 6116: 6110: 6093: 6075: 6062: 6025: 6012: 5999: 5952: 5927: 5875: 5831: 5816: 5801: 5786: 5771: 5720: 5705: 5692: 5679: 5650: 5614: 5599: 5586: 5573: 5544: 5511: 5498: 5462: 5433: 5401: 5369: 5356: 5323: 5307: 5294: 5281: 5248: 5235: 5199: 5184: 5169: 5085: 5070: 5055: 5024: 4993: 4980: 4944: 4931: 4919: 4915: 4902: 4874: 4870: 4841: 4829: 4825: 4800:number apply the 4777: 4748: 4711: 4693: 4678: 4663: 4616: 4601: 4544: 4499: 4451: 4422: 4390: 4375: 4346: 4317: 4264: 4219: 4174: 4145: 4095: 4089: 4069: 4053: 4047: 4027: 3992:{\displaystyle 4} 3962: 3947: 3900: 3885: 3850: 3727: 3698: 3680:in simplest terms 3647:If one takes for 3533:{\displaystyle c} 3518:are divisible by 3505: 3401: 3370: 3341: 3318:{\displaystyle n} 3251: 3200: 3185: 3156: 3141: 3126: 3097: 3068: 3040:compound fraction 2926: 2887: 2862: 2788: 2775: 2762: 2729: 2698: 2691: 2667: 2654: 2641: 2622: 2609: 2600: 2580: 2567: 2554: 2541: 2528: 2511: 2509: 2495: 2432: 2393: 2325: 2310: 2295: 2280: 2251: 2236: 2221: 2192: 2160: 2145: 2130: 2101: 2072: 2043: 2014: 1996:ancient Egyptians 1981: 1966: 1948:Egyptian fraction 1942:Egyptian fraction 1859: 1838: 1802: 1787: 1772: 1757: 1729:improper fraction 1702: 1657: 1599: 1564: 1541:{\displaystyle 2} 1505: 1480: 1416:parts per million 1306:Decimal separator 1297:decimal separator 1089:improper fraction 876: 838:(also known as a 478: 461:rational fraction 446: 442: 408:and the notation 366:are integers and 334:In mathematics a 134: 105: 16:(Redirected from 12478: 12435:Musical interval 12348: 12347: 12345: 12344: 12341: 12338: 12320: 12319: 12289: 12266: 12259: 12252: 12243: 12242: 12238: 12226: 12205: 12204: 12202: 12201: 12195: 12188: 12180: 12174: 12160: 12154: 12153: 12135: 12129: 12128: 12121: 12115: 12114: 12097: 12091: 12085: 12079: 12078: 12076: 12075: 12053: 12047: 12046: 12012: 12006: 12005: 12003: 12001: 11985: 11972: 11971: 11941: 11935: 11934: 11913: 11907: 11906: 11888: 11882: 11881: 11879: 11878: 11867:"piece fraction" 11863: 11857: 11856: 11854: 11853: 11842:"built fraction" 11838: 11832: 11831: 11829: 11828: 11822: 11789: 11780: 11774: 11773: 11771: 11770: 11755: 11749: 11748: 11746: 11744: 11707: 11701: 11700: 11698: 11697: 11682: 11676: 11675: 11673: 11672: 11653: 11647: 11646: 11644: 11642: 11633:. Archived from 11623: 11617: 11616: 11606: 11597: 11596: 11586: 11577: 11576: 11541: 11535: 11534: 11522: 11506: 11500: 11499: 11487: 11477: 11471: 11470: 11463: 11461: 11460: 11457: 11454: 11450: 11428: 11418: 11412: 11411: 11391: 11385: 11384: 11382: 11381: 11353: 11347: 11346: 11344: 11343: 11334:. Archived from 11324: 11318: 11317: 11315: 11314: 11294: 11288: 11287: 11276: 11270: 11269: 11249: 11243: 11242: 11226: 11220: 11219: 11199: 11190: 11189: 11169: 11163: 11162: 11150: 11144: 11143: 11131: 11125: 11124: 11109:(3rd ed.). 11102: 11096: 11090: 11084: 11082: 11080: 11079: 11047: 11038: 11037: 11035: 11034: 11020: 11014: 11013: 11001: 10991: 10985: 10966: 10957: 10927: 10924: 10918: 10910: 10896: 10883: 10819: 10818: 10810: 10809: 10767: 10766: 10758: 10757: 10701: 10700: 10697: 10695: 10694: 10689: 10687: 10667: 10666: 10663: 10662: 10659: 10657: 10656: 10651: 10649: 10629: 10628: 10625: 10624: 10621: 10619: 10618: 10613: 10611: 10591: 10590: 10587: 10586: 10583: 10581: 10580: 10575: 10573: 10553: 10552: 10549: 10548: 10545: 10543: 10542: 10537: 10535: 10515: 10509: 10506: 10505: 10502: 10501: 10467: 10465: 10464: 10459: 10447: 10445: 10444: 10439: 10427: 10425: 10424: 10423: 10421: 10420: 10415: 10403: 10400: 10399: 10397: 10396: 10391: 10321:Jamshīd al-Kāshī 10247: 10245: 10243: 10242: 10237: 10235: 10233: 10224: 10215: 10189: 10186: 10175: 10173: 10172: 10169: 10166: 10159: 10157: 10156: 10153: 10150: 10143: 10141: 10140: 10137: 10134: 10127: 10125: 10124: 10121: 10118: 10076: 10072: 10064: 10058: 10050: 10047: 10039: 10036: 10028: 10025: 9999: 9980: 9977: 9947:dyadic fractions 9940: 9937: 9916: 9909: 9905: 9902: 9896: 9894: 9853: 9829: 9821: 9811: 9809: 9808: 9803: 9801: 9793: 9748: 9746: 9745: 9742: 9739: 9693:Slash § Encoding 9678: 9676: 9675: 9670: 9668: 9663: 9662: 9657: 9645: 9637: 9635: 9624: 9620: 9615: 9597: 9592: 9590: 9589: 9588: 9579: 9574: 9563: 9562: 9557: 9550: 9546: 9541: 9523: 9518: 9516: 9515: 9510: 9498: 9497: 9492: 9480: 9475: 9473: 9472: 9467: 9452: 9447: 9445: 9444: 9439: 9424: 9414: 9412: 9411: 9406: 9404: 9399: 9398: 9393: 9381: 9373: 9371: 9360: 9356: 9351: 9333: 9328: 9326: 9325: 9324: 9315: 9310: 9299: 9298: 9293: 9286: 9282: 9277: 9259: 9254: 9252: 9251: 9246: 9234: 9233: 9228: 9216: 9211: 9209: 9208: 9203: 9188: 9183: 9181: 9180: 9175: 9160: 9137: 9135: 9134: 9129: 9127: 9122: 9121: 9116: 9110: 9105: 9099: 9094: 9093: 9088: 9082: 9078: 9073: 9067: 9063: 9005: 9003: 9001: 9000: 8995: 8993: 8991: 8977: 8972: 8970: 8956: 8945: 8943: 8942: 8937: 8935: 8933: 8926: 8925: 8915: 8907: 8897:partial fraction 8883: 8881: 8880: 8875: 8863: 8861: 8860: 8855: 8844: 8839: 8830: 8828: 8827: 8822: 8814: 8795: 8793: 8791: 8790: 8785: 8779: 8774: 8769: 8733:complex fraction 8730: 8728: 8726: 8725: 8720: 8718: 8716: 8715: 8706: 8696: 8695: 8686: 8676: 8661: 8659: 8657: 8656: 8651: 8649: 8647: 8640: 8639: 8619: 8618: 8595: 8593: 8591: 8590: 8585: 8583: 8581: 8565: 8564: 8554: 8546: 8524: 8522: 8520: 8519: 8514: 8512: 8510: 8503: 8502: 8482: 8481: 8470: 8468: 8467: 8462: 8460: 8458: 8442: 8441: 8431: 8423: 8385:is the field of 8381:, the generated 8334:More generally, 8322: 8312: 8308: 8304: 8302: 8301: 8298: 8295: 8280: 8278: 8277: 8272: 8264: 8253: 8236: 8225: 8160: 8158: 8157: 8152: 8146: 8135: 8098: 8096: 8095: 8090: 8084: 8073: 8022: 8020: 8019: 8014: 7937: 7935: 7934: 7929: 7927: 7920: 7917: 7900: 7897: 7894: 7893: 7892: 7876: 7873: 7870: 7846: 7845: 7817: 7814: 7797: 7794: 7791: 7790: 7789: 7785: 7782: 7779: 7716: 7714: 7713: 7708: 7694: 7691: 7620: 7618: 7617: 7612: 7545: 7543: 7542: 7537: 7460: 7458: 7457: 7452: 7357: 7355: 7354: 7349: 7328: 7326: 7325: 7320: 7305: 7303: 7302: 7297: 7259: 7257: 7256: 7253: 7250: 7228: 7208: 7204: 7189: 7171: 7161: 7147: 7132: 7117: 7088: 7080: 7072: 7065: 7058: 7038: 7031: 7024: 7017: 7009: 6975: 6973: 6972: 6967: 6962: 6953: 6937: 6933: 6931: 6930: 6927: 6924: 6917: 6915: 6914: 6911: 6908: 6901: 6897: 6893: 6889: 6885: 6881: 6877: 6875: 6874: 6871: 6868: 6861: 6857: 6853: 6849: 6845: 6841: 6839: 6838: 6835: 6832: 6816: 6814: 6813: 6808: 6806: 6797: 6791: 6788: 6777: 6766: 6760: 6751: 6745: 6736: 6730: 6721: 6715: 6706: 6692: 6690: 6689: 6684: 6682: 6673: 6663: 6661: 6660: 6655: 6653: 6644: 6638: 6635: 6621: 6612:and also equals 6611: 6609: 6608: 6603: 6601: 6592: 6582: 6580: 6579: 6574: 6566: 6557: 6539: 6537: 6536: 6531: 6526: 6518: 6510: 6502: 6491: 6483: 6460: 6452: 6424: 6422: 6421: 6416: 6414: 6406: 6398: 6390: 6385: 6380: 6363: 6358: 6350: 6345: 6337: 6305: 6303: 6302: 6297: 6295: 6286: 6280: 6271: 6265: 6256: 6250: 6241: 6207: 6205: 6204: 6199: 6197: 6189: 6184: 6176: 6171: 6163: 6158: 6156: 6155: 6149: 6147: 6139: 6138: 6132: 6130: 6122: 6117: 6115: 6114: 6108: 6106: 6098: 6097: 6091: 6089: 6081: 6076: 6068: 6063: 6055: 6036: 6034: 6033: 6028: 6026: 6018: 6013: 6005: 6000: 5992: 5967: 5965: 5964: 5959: 5954: 5945: 5936: 5935: 5929: 5920: 5911: 5910: 5877: 5868: 5843: 5841: 5840: 5835: 5833: 5824: 5818: 5809: 5803: 5794: 5788: 5779: 5773: 5764: 5735: 5733: 5732: 5727: 5725: 5721: 5713: 5706: 5698: 5693: 5685: 5680: 5678: 5667: 5656: 5651: 5649: 5638: 5627: 5619: 5615: 5607: 5600: 5592: 5587: 5579: 5574: 5572: 5561: 5550: 5545: 5543: 5532: 5521: 5512: 5504: 5499: 5491: 5474: 5472: 5471: 5466: 5464: 5455: 5445: 5443: 5442: 5437: 5435: 5426: 5412: 5410: 5409: 5404: 5402: 5400: 5392: 5375: 5370: 5362: 5357: 5349: 5334: 5332: 5331: 5326: 5324: 5316: 5308: 5300: 5295: 5287: 5282: 5274: 5259: 5257: 5256: 5251: 5249: 5241: 5236: 5228: 5213: 5211: 5209: 5208: 5203: 5201: 5192: 5186: 5177: 5171: 5162: 5150: 5148: 5147: 5144: 5141: 5135:Second, convert 5131: 5129: 5128: 5125: 5122: 5115: 5113: 5112: 5109: 5106: 5099: 5097: 5095: 5094: 5089: 5087: 5078: 5072: 5063: 5057: 5048: 5036: 5034: 5033: 5028: 5026: 5017: 5004: 5002: 5001: 4996: 4994: 4986: 4981: 4973: 4958: 4956: 4955: 4950: 4945: 4937: 4932: 4924: 4917: 4916: 4908: 4903: 4901: 4890: 4879: 4872: 4871: 4869: 4858: 4847: 4842: 4834: 4827: 4826: 4818: 4789: 4787: 4786: 4781: 4779: 4770: 4760: 4758: 4757: 4752: 4750: 4741: 4723: 4721: 4720: 4715: 4713: 4704: 4695: 4686: 4680: 4671: 4665: 4656: 4630: 4628: 4626: 4625: 4620: 4618: 4609: 4603: 4594: 4582: 4580: 4579: 4574: 4556: 4554: 4553: 4548: 4546: 4543: 4532: 4521: 4511: 4509: 4508: 4503: 4501: 4498: 4487: 4476: 4466: 4464: 4463: 4458: 4453: 4444: 4434: 4432: 4431: 4426: 4424: 4415: 4402: 4400: 4399: 4394: 4392: 4383: 4377: 4368: 4358: 4356: 4355: 4350: 4348: 4339: 4329: 4327: 4326: 4321: 4319: 4310: 4276: 4274: 4273: 4268: 4266: 4263: 4252: 4241: 4231: 4229: 4228: 4223: 4221: 4218: 4207: 4196: 4186: 4184: 4183: 4178: 4176: 4167: 4157: 4155: 4154: 4149: 4147: 4138: 4121: 4119: 4118: 4113: 4096: 4093: 4091: 4085: 4077: 4071: 4068: 4057: 4054: 4051: 4049: 4046: 4035: 4029: 4026: 4015: 3998: 3996: 3995: 3990: 3978: 3974: 3972: 3971: 3966: 3964: 3955: 3949: 3940: 3912: 3910: 3909: 3904: 3902: 3893: 3887: 3884: 3871: 3858: 3852: 3843: 3830: 3828: 3827: 3824: 3821: 3811: 3810: 3808: 3807: 3804: 3801: 3794: 3792: 3791: 3788: 3785: 3778: 3776: 3775: 3772: 3769: 3762: 3760: 3759: 3756: 3753: 3739: 3737: 3736: 3731: 3729: 3720: 3710: 3708: 3707: 3702: 3700: 3691: 3650: 3643: 3641: 3640: 3635: 3632: 3619: 3617: 3616: 3611: 3608: 3599: 3597: 3596: 3591: 3570: 3568: 3567: 3562: 3541: 3539: 3537: 3536: 3531: 3517: 3515: 3514: 3509: 3507: 3498: 3479: 3477: 3476: 3473: 3470: 3463: 3461: 3460: 3457: 3454: 3447: 3445: 3444: 3441: 3438: 3431: 3429: 3428: 3425: 3422: 3415: 3413: 3411: 3410: 3405: 3403: 3394: 3382: 3380: 3379: 3374: 3372: 3363: 3353: 3351: 3350: 3345: 3343: 3334: 3324: 3322: 3321: 3316: 3292:division by zero 3265: 3263: 3261: 3260: 3255: 3253: 3250: 3246: 3237: 3233: 3224: 3212: 3210: 3209: 3204: 3202: 3193: 3187: 3178: 3168: 3166: 3165: 3160: 3158: 3149: 3143: 3134: 3128: 3119: 3109: 3107: 3106: 3101: 3099: 3090: 3080: 3078: 3077: 3072: 3070: 3061: 3052:). For example, 3035: 3033: 3032: 3027: 3016: 3005: 3004: 2988: 2986: 2985: 2980: 2975: 2974: 2962: 2944: 2942: 2941: 2936: 2928: 2919: 2916: 2915: 2899: 2897: 2896: 2891: 2889: 2880: 2871: 2870: 2864: 2855: 2845: 2843: 2842: 2837: 2832: 2824: 2806: 2804: 2803: 2798: 2796: 2789: 2781: 2776: 2768: 2763: 2758: 2741: 2730: 2725: 2708: 2699: 2694: 2693: 2684: 2677: 2668: 2660: 2655: 2647: 2642: 2634: 2623: 2615: 2610: 2605: 2602: 2593: 2587: 2581: 2573: 2568: 2560: 2555: 2547: 2542: 2534: 2529: 2521: 2512: 2502: 2500: 2497: 2488: 2482: 2461: 2459: 2458: 2453: 2448: 2447: 2441: 2440: 2434: 2425: 2419: 2418: 2405: 2403: 2402: 2397: 2395: 2392: 2388: 2379: 2375: 2366: 2355:complex fraction 2337: 2335: 2334: 2329: 2327: 2318: 2312: 2303: 2297: 2288: 2282: 2273: 2263: 2261: 2260: 2255: 2253: 2244: 2238: 2229: 2223: 2214: 2204: 2202: 2201: 2196: 2194: 2185: 2175: 2173: 2172: 2167: 2162: 2153: 2147: 2138: 2132: 2123: 2113: 2111: 2110: 2105: 2103: 2094: 2084: 2082: 2081: 2076: 2074: 2065: 2055: 2053: 2052: 2047: 2045: 2036: 2026: 2024: 2023: 2018: 2016: 2007: 1993: 1991: 1990: 1985: 1983: 1974: 1968: 1959: 1927: 1925: 1924: 1919: 1916: 1895: 1893: 1892: 1889: 1886: 1882: 1874: 1872: 1871: 1866: 1861: 1852: 1840: 1831: 1817: 1815: 1814: 1809: 1804: 1795: 1789: 1780: 1774: 1765: 1759: 1750: 1724: 1722: 1721: 1716: 1711: 1710: 1704: 1695: 1686: 1685: 1669: 1667: 1666: 1661: 1659: 1650: 1630: 1628: 1627: 1624: 1621: 1617: 1611: 1609: 1608: 1603: 1601: 1592: 1576: 1574: 1573: 1568: 1566: 1557: 1547: 1545: 1544: 1539: 1527: 1525: 1524: 1519: 1514: 1503: 1492: 1490: 1489: 1484: 1482: 1473: 1381: 1379: 1378: 1375: 1372: 1358: 1354: 1350: 1346: 1344: 1331: 1329: 1328: 1325: 1322: 1318: 1287:decimal fraction 1271: 1269: 1268: 1265: 1262: 1255: 1253: 1252: 1249: 1246: 1202: 1200: 1199: 1196: 1193: 1182: 1180: 1179: 1176: 1173: 1163: 1161: 1160: 1157: 1154: 1140: 1138: 1137: 1134: 1131: 1124: 1122: 1121: 1118: 1115: 1052: 1050: 1049: 1046: 1043: 1036: 1034: 1033: 1030: 1027: 1007: 1005: 1004: 1001: 998: 980:common fractions 970: 968: 967: 964: 961: 954: 952: 951: 948: 945: 938: 936: 935: 932: 929: 922: 920: 919: 916: 913: 890: 888: 886: 885: 880: 878: 869: 820: 818: 817: 814: 811: 804: 802: 801: 798: 795: 780: 778: 777: 774: 771: 765:. (For example, 760: 758: 757: 754: 751: 744: 742: 741: 738: 735: 724: 722: 721: 718: 715: 700: 698: 697: 694: 691: 684: 682: 681: 678: 675: 661: 659: 658: 655: 652: 597: 596: 592: 587: 585: 584: 581: 578: 552: 550: 549: 546: 543: 490: 488: 487: 482: 479: 471: 458: 456: 455: 450: 447: 438: 437: 427: 425: 424: 419: 416: 399: 397: 395: 394: 389: 387: 357: 355: 354: 349: 346: 330: 328: 327: 324: 321: 314: 312: 311: 308: 305: 298: 296: 295: 292: 289: 282: 280: 279: 276: 273: 266: 264: 263: 260: 257: 250: 248: 247: 244: 241: 231: 227: 225: 224: 221: 218: 200: 198: 197: 194: 191: 184: 182: 181: 178: 175: 160: 159: 155: 146: 144: 143: 138: 136: 127: 117: 115: 114: 109: 107: 98: 61: 60: 58: 57: 54: 51: 21: 12486: 12485: 12481: 12480: 12479: 12477: 12476: 12475: 12461: 12460: 12459: 12454: 12425:Just intonation 12352: 12342: 12339: 12336: 12335: 12333: 12332: 12321: 12317: 12312: 12290: 12279: 12270: 12229: 12217: 12214: 12209: 12208: 12199: 12197: 12193: 12186: 12182: 12181: 12177: 12171:Wayback Machine 12161: 12157: 12150: 12136: 12132: 12123: 12122: 12118: 12111: 12099: 12098: 12094: 12086: 12082: 12073: 12071: 12054: 12050: 12043: 12027:. p. 152. 12013: 12009: 11999: 11997: 11986: 11975: 11968:10.1119/1.11512 11942: 11938: 11931: 11914: 11910: 11903: 11889: 11885: 11876: 11874: 11865: 11864: 11860: 11851: 11849: 11840: 11839: 11835: 11826: 11824: 11820: 11806:10.2307/4145131 11787: 11781: 11777: 11768: 11766: 11757: 11756: 11752: 11742: 11740: 11730: 11708: 11704: 11695: 11693: 11684: 11683: 11679: 11670: 11668: 11655: 11654: 11650: 11640: 11638: 11625: 11624: 11620: 11607: 11600: 11587: 11580: 11542: 11538: 11531: 11507: 11503: 11496: 11478: 11474: 11458: 11455: 11452: 11451: 11448: 11446: 11437: 11419: 11415: 11408: 11392: 11388: 11379: 11377: 11370: 11354: 11350: 11341: 11339: 11326: 11325: 11321: 11312: 11310: 11295: 11291: 11277: 11273: 11266: 11250: 11246: 11227: 11223: 11216: 11200: 11193: 11170: 11166: 11152: 11151: 11147: 11133: 11132: 11128: 11121: 11103: 11099: 11091: 11087: 11077: 11075: 11068: 11048: 11041: 11032: 11030: 11022: 11021: 11017: 11010: 10992: 10988: 10982:Wayback Machine 10967: 10963: 10954: 10940:Weisstein, Eric 10936: 10931: 10930: 10925: 10921: 10904: 10890: 10884: 10880: 10875: 10683: 10677: 10674: 10673: 10645: 10639: 10636: 10635: 10607: 10601: 10598: 10597: 10569: 10563: 10560: 10559: 10531: 10525: 10522: 10521: 10478: 10453: 10450: 10449: 10433: 10430: 10429: 10409: 10406: 10405: 10404: 10401: 10385: 10382: 10381: 10380: 10379: 10377: 10370: 10350:Cuisenaire rods 10346:primary schools 10342: 10337: 10275:, published at 10261:Dirk Jan Struik 10225: 10216: 10214: 10212: 10209: 10208: 10207: 10187: 10170: 10167: 10164: 10163: 10161: 10154: 10151: 10148: 10147: 10145: 10138: 10135: 10132: 10131: 10129: 10122: 10119: 10116: 10115: 10113: 10091: 10074: 10070: 10048: 10037: 10026: 10009:Sthananga Sutra 9997: 9978: 9938: 9917: 9906: 9900: 9897: 9854: 9852: 9842: 9830: 9819: 9792: 9790: 9787: 9786: 9777:piece fractions 9743: 9740: 9737: 9736: 9734: 9695: 9689: 9656: 9646: 9644: 9625: 9614: 9598: 9596: 9584: 9580: 9573: 9558: 9553: 9552: 9551: 9540: 9524: 9522: 9509: 9499: 9491: 9481: 9479: 9466: 9456: 9451: 9438: 9428: 9423: 9421: 9418: 9417: 9392: 9382: 9380: 9361: 9350: 9334: 9332: 9320: 9316: 9309: 9294: 9289: 9288: 9287: 9276: 9260: 9258: 9245: 9235: 9227: 9217: 9215: 9202: 9192: 9187: 9174: 9164: 9159: 9157: 9154: 9153: 9115: 9111: 9109: 9092: 9077: 9062: 9060: 9057: 9056: 9034: 9026:Main articles: 9024: 9008:antiderivatives 8981: 8976: 8960: 8955: 8953: 8950: 8949: 8947: 8921: 8917: 8916: 8908: 8906: 8904: 8901: 8900: 8869: 8866: 8865: 8838: 8836: 8833: 8832: 8810: 8805: 8802: 8801: 8775: 8768: 8765: 8762: 8761: 8759: 8744:integral domain 8704: 8697: 8684: 8677: 8675: 8673: 8670: 8669: 8667: 8635: 8631: 8630: 8617: 8615: 8612: 8611: 8609: 8560: 8556: 8555: 8547: 8545: 8543: 8540: 8539: 8537: 8498: 8494: 8493: 8480: 8478: 8475: 8474: 8472: 8437: 8433: 8432: 8424: 8422: 8420: 8417: 8416: 8405: 8399: 8357:. For example, 8344:integral domain 8317: 8310: 8306: 8299: 8296: 8293: 8292: 8290: 8257: 8246: 8229: 8218: 8168: 8165: 8164: 8139: 8128: 8104: 8101: 8100: 8077: 8066: 8043: 8040: 8039: 7953: 7950: 7949: 7945:, specified as 7925: 7924: 7916: 7896: 7890: 7889: 7872: 7869: 7847: 7838: 7834: 7819: 7818: 7813: 7793: 7787: 7786: 7781: 7778: 7753: 7731: 7729: 7726: 7725: 7690: 7627: 7624: 7623: 7552: 7549: 7548: 7467: 7464: 7463: 7382: 7379: 7378: 7334: 7331: 7330: 7314: 7311: 7310: 7279: 7276: 7275: 7268: 7254: 7251: 7248: 7247: 7245: 7223: 7206: 7202: 7173: 7163: 7153: 7145: 7130: 7115: 7086: 7085:0.1523 + 0.0000 7078: 7070: 7063: 7056: 7036: 7029: 7022: 7015: 7007: 6993: 6987: 6981: 6951: 6943: 6940: 6939: 6935: 6928: 6925: 6922: 6921: 6919: 6912: 6909: 6906: 6905: 6903: 6902:. The fraction 6899: 6895: 6891: 6887: 6883: 6879: 6872: 6869: 6866: 6865: 6863: 6859: 6855: 6851: 6847: 6843: 6836: 6833: 6830: 6829: 6827: 6823: 6795: 6778: 6767: 6764: 6749: 6734: 6719: 6704: 6702: 6699: 6698: 6671: 6669: 6666: 6665: 6642: 6625: 6619: 6617: 6614: 6613: 6590: 6588: 6585: 6584: 6555: 6553: 6550: 6549: 6546: 6517: 6501: 6482: 6451: 6440: 6437: 6436: 6405: 6389: 6364: 6362: 6349: 6336: 6325: 6322: 6321: 6315: 6284: 6269: 6254: 6239: 6231: 6228: 6227: 6221: 6188: 6175: 6162: 6148: 6141: 6140: 6131: 6124: 6123: 6121: 6107: 6100: 6099: 6090: 6083: 6082: 6080: 6067: 6054: 6052: 6049: 6048: 6017: 6004: 5991: 5989: 5986: 5985: 5979: 5974: 5943: 5931: 5930: 5918: 5906: 5905: 5866: 5855: 5852: 5851: 5822: 5807: 5792: 5777: 5762: 5760: 5757: 5756: 5749: 5723: 5722: 5712: 5707: 5697: 5684: 5668: 5657: 5655: 5639: 5628: 5626: 5617: 5616: 5606: 5601: 5591: 5578: 5562: 5551: 5549: 5533: 5522: 5520: 5513: 5503: 5490: 5486: 5484: 5481: 5480: 5453: 5451: 5448: 5447: 5424: 5422: 5419: 5418: 5393: 5376: 5374: 5361: 5348: 5346: 5343: 5342: 5315: 5299: 5286: 5273: 5271: 5268: 5267: 5240: 5227: 5225: 5222: 5221: 5190: 5175: 5160: 5158: 5155: 5154: 5152: 5145: 5142: 5139: 5138: 5136: 5126: 5123: 5120: 5119: 5117: 5110: 5107: 5104: 5103: 5101: 5076: 5061: 5046: 5044: 5041: 5040: 5038: 5015: 5013: 5010: 5009: 5008:First, convert 4985: 4972: 4970: 4967: 4966: 4936: 4923: 4907: 4891: 4880: 4878: 4859: 4848: 4846: 4833: 4817: 4815: 4812: 4811: 4797: 4768: 4766: 4763: 4762: 4739: 4737: 4734: 4733: 4702: 4684: 4669: 4654: 4652: 4649: 4648: 4641: 4607: 4592: 4590: 4587: 4586: 4584: 4562: 4559: 4558: 4533: 4522: 4519: 4517: 4514: 4513: 4488: 4477: 4474: 4472: 4469: 4468: 4442: 4440: 4437: 4436: 4413: 4411: 4408: 4407: 4381: 4366: 4364: 4361: 4360: 4337: 4335: 4332: 4331: 4308: 4306: 4303: 4302: 4253: 4242: 4239: 4237: 4234: 4233: 4208: 4197: 4194: 4192: 4189: 4188: 4165: 4163: 4160: 4159: 4136: 4134: 4131: 4130: 4094: and 4092: 4078: 4075: 4061: 4055: 4050: 4039: 4033: 4019: 4013: 4011: 4008: 4007: 3984: 3981: 3980: 3976: 3953: 3938: 3936: 3933: 3932: 3926: 3891: 3872: 3859: 3856: 3841: 3839: 3836: 3835: 3825: 3822: 3819: 3818: 3816: 3812:, for example. 3805: 3802: 3799: 3798: 3796: 3789: 3786: 3783: 3782: 3780: 3773: 3770: 3767: 3766: 3764: 3757: 3754: 3751: 3750: 3748: 3747: 3718: 3716: 3713: 3712: 3689: 3687: 3684: 3683: 3682:. For example, 3657:absolute values 3648: 3636: 3633: 3628: 3627: 3625: 3612: 3609: 3604: 3603: 3601: 3576: 3573: 3572: 3547: 3544: 3543: 3525: 3522: 3521: 3519: 3496: 3494: 3491: 3490: 3486: 3474: 3471: 3468: 3467: 3465: 3458: 3455: 3452: 3451: 3449: 3442: 3439: 3436: 3435: 3433: 3426: 3423: 3420: 3419: 3417: 3392: 3390: 3387: 3386: 3384: 3361: 3359: 3356: 3355: 3332: 3330: 3327: 3326: 3325:, the fraction 3310: 3307: 3306: 3303: 3276: 3242: 3238: 3229: 3225: 3222: 3220: 3217: 3216: 3214: 3191: 3176: 3174: 3171: 3170: 3147: 3132: 3117: 3115: 3112: 3111: 3088: 3086: 3083: 3082: 3059: 3057: 3054: 3053: 3012: 3000: 2999: 2994: 2991: 2990: 2970: 2969: 2958: 2950: 2947: 2946: 2917: 2911: 2910: 2905: 2902: 2901: 2878: 2866: 2865: 2853: 2851: 2848: 2847: 2828: 2820: 2815: 2812: 2811: 2794: 2793: 2780: 2767: 2742: 2740: 2709: 2707: 2700: 2682: 2678: 2676: 2673: 2672: 2659: 2646: 2633: 2614: 2591: 2588: 2586: 2572: 2559: 2546: 2533: 2520: 2513: 2486: 2483: 2481: 2477: 2475: 2472: 2471: 2467:. For example: 2465:§ Division 2443: 2442: 2436: 2435: 2423: 2414: 2413: 2411: 2408: 2407: 2384: 2380: 2371: 2367: 2364: 2362: 2359: 2358: 2351: 2348:Complex numbers 2344: 2316: 2301: 2286: 2271: 2269: 2266: 2265: 2242: 2227: 2212: 2210: 2207: 2206: 2183: 2181: 2178: 2177: 2151: 2136: 2121: 2119: 2116: 2115: 2092: 2090: 2087: 2086: 2063: 2061: 2058: 2057: 2034: 2032: 2029: 2028: 2005: 2003: 2000: 1999: 1972: 1957: 1955: 1952: 1951: 1944: 1939: 1920: 1917: 1912: 1911: 1909: 1908:, the quotient 1906:rational number 1890: 1887: 1884: 1883: 1880: 1878: 1850: 1829: 1827: 1824: 1823: 1793: 1778: 1763: 1748: 1740: 1737: 1736: 1735:. For example, 1706: 1705: 1693: 1681: 1680: 1675: 1672: 1671: 1648: 1640: 1637: 1636: 1625: 1622: 1619: 1618: 1615: 1613: 1590: 1582: 1579: 1578: 1555: 1553: 1550: 1549: 1533: 1530: 1529: 1510: 1498: 1495: 1494: 1471: 1466: 1463: 1462: 1446:(also called a 1440: 1376: 1373: 1370: 1369: 1367: 1366:. For example, 1364:infinite series 1356: 1352: 1348: 1342: 1340: 1326: 1323: 1320: 1319: 1316: 1314: 1311:fractional part 1282: 1266: 1263: 1260: 1259: 1257: 1256:of the cars or 1250: 1247: 1244: 1243: 1241: 1209: 1197: 1194: 1191: 1190: 1188: 1177: 1174: 1171: 1170: 1168: 1158: 1155: 1152: 1151: 1149: 1135: 1132: 1129: 1128: 1126: 1119: 1116: 1113: 1112: 1110: 1101: 1091:, or sometimes 1081:proper fraction 1067: 1047: 1044: 1041: 1040: 1038: 1031: 1028: 1025: 1024: 1022: 1014:dyadic fraction 1002: 999: 996: 995: 993: 965: 962: 959: 958: 956: 949: 946: 943: 942: 940: 933: 930: 927: 926: 924: 917: 914: 911: 910: 908: 867: 865: 862: 861: 859: 848:rational number 844:vulgar fraction 840:common fraction 836:simple fraction 832: 827: 815: 812: 809: 808: 806: 799: 796: 793: 792: 790: 775: 772: 769: 768: 766: 763:cardinal number 755: 752: 749: 748: 746: 739: 736: 733: 732: 730: 719: 716: 713: 712: 710: 695: 692: 689: 688: 686: 679: 676: 673: 672: 670: 667:ordinal numbers 656: 653: 650: 649: 647: 594: 590: 589: 582: 579: 576: 575: 573: 547: 544: 541: 540: 538: 511: 497: 470: 467: 464: 463: 436: 433: 430: 429: 420: 417: 412: 411: 409: 383: 381: 378: 377: 375: 350: 347: 342: 341: 339: 336:rational number 325: 322: 319: 318: 316: 309: 306: 303: 302: 300: 293: 290: 287: 286: 284: 277: 274: 271: 270: 268: 261: 258: 255: 254: 252: 245: 242: 239: 238: 236: 229: 222: 219: 216: 215: 213: 195: 192: 189: 188: 186: 179: 176: 173: 172: 170: 157: 153: 152: 125: 123: 120: 119: 96: 94: 91: 90: 55: 52: 49: 48: 46: 44: 35: 28: 23: 22: 15: 12: 11: 5: 12484: 12474: 12473: 12456: 12455: 12453: 12452: 12447: 12442: 12437: 12432: 12427: 12422: 12421: 12420: 12410: 12405: 12404: 12403: 12393: 12388: 12383: 12378: 12373: 12368: 12363: 12357: 12354: 12353: 12351: 12350: 12329: 12327: 12323: 12322: 12315: 12313: 12311: 12310: 12296: 12294: 12291: 12284: 12281: 12280: 12269: 12268: 12261: 12254: 12246: 12240: 12239: 12227: 12213: 12212:External links 12210: 12207: 12206: 12175: 12155: 12148: 12130: 12116: 12109: 12092: 12080: 12048: 12041: 12017:Chemla, Karine 12007: 11973: 11954:(9): 896–906. 11936: 11929: 11917:Winkler, Peter 11908: 11901: 11883: 11858: 11833: 11800:(3): 238–242. 11775: 11750: 11728: 11702: 11677: 11663:. 2018-03-09. 11648: 11618: 11598: 11578: 11552:(4): 173–180. 11536: 11529: 11501: 11494: 11472: 11435: 11413: 11406: 11386: 11368: 11348: 11319: 11289: 11271: 11264: 11244: 11221: 11214: 11191: 11164: 11145: 11126: 11119: 11097: 11085: 11066: 11039: 11015: 11008: 10986: 10960: 10959: 10952: 10935: 10932: 10929: 10928: 10919: 10877: 10876: 10874: 10871: 10868: 10867: 10864: 10863: 10860: 10859: 10856: 10855: 10849: 10848: 10845: 10844: 10841: 10840: 10837: 10836: 10833: 10832: 10830:Transcendental 10826: 10825: 10816: 10806: 10805: 10802: 10801: 10798: 10797: 10794: 10793: 10790: 10789: 10787: 10781: 10780: 10774: 10773: 10771:Finite decimal 10764: 10754: 10753: 10750: 10749: 10746: 10745: 10739: 10738: 10735: 10734: 10731: 10730: 10724: 10723: 10717: 10716: 10709: 10708: 10698: 10686: 10681: 10660: 10648: 10643: 10622: 10610: 10605: 10584: 10572: 10567: 10546: 10534: 10529: 10512:Number systems 10500: 10499: 10494: 10489: 10484: 10477: 10474: 10457: 10437: 10413: 10389: 10369: 10366: 10354:pattern blocks 10341: 10338: 10336: 10333: 10327:mathematician 10319:mathematician 10232: 10228: 10223: 10219: 10110: 10109: 10106: 10103: 10096: 10095: 10092: 10089: 10086: 9943:unit fractions 9919: 9918: 9833: 9831: 9824: 9818: 9815: 9814: 9813: 9799: 9796: 9780: 9754: 9751:case fractions 9727:Case fractions 9724: 9688: 9685: 9680: 9679: 9666: 9660: 9655: 9652: 9649: 9643: 9640: 9634: 9631: 9628: 9623: 9618: 9613: 9610: 9607: 9604: 9601: 9595: 9587: 9583: 9577: 9572: 9569: 9566: 9561: 9556: 9549: 9544: 9539: 9536: 9533: 9530: 9527: 9521: 9513: 9508: 9505: 9502: 9495: 9490: 9487: 9484: 9478: 9470: 9465: 9462: 9459: 9455: 9450: 9442: 9437: 9434: 9431: 9427: 9415: 9402: 9396: 9391: 9388: 9385: 9379: 9376: 9370: 9367: 9364: 9359: 9354: 9349: 9346: 9343: 9340: 9337: 9331: 9323: 9319: 9313: 9308: 9305: 9302: 9297: 9292: 9285: 9280: 9275: 9272: 9269: 9266: 9263: 9257: 9249: 9244: 9241: 9238: 9231: 9226: 9223: 9220: 9214: 9206: 9201: 9198: 9195: 9191: 9186: 9178: 9173: 9170: 9167: 9163: 9139: 9138: 9125: 9119: 9114: 9108: 9102: 9097: 9091: 9085: 9081: 9076: 9070: 9066: 9023: 9020: 8990: 8987: 8984: 8980: 8975: 8969: 8966: 8963: 8959: 8932: 8929: 8924: 8920: 8914: 8911: 8892:coefficients. 8873: 8853: 8850: 8847: 8842: 8820: 8817: 8813: 8809: 8782: 8778: 8772: 8731:, is called a 8712: 8709: 8703: 8700: 8692: 8689: 8683: 8680: 8646: 8643: 8638: 8634: 8628: 8625: 8622: 8580: 8577: 8574: 8571: 8568: 8563: 8559: 8553: 8550: 8509: 8506: 8501: 8497: 8491: 8488: 8485: 8457: 8454: 8451: 8448: 8445: 8440: 8436: 8430: 8427: 8401:Main article: 8398: 8395: 8284: 8283: 8282: 8281: 8270: 8267: 8263: 8260: 8256: 8252: 8249: 8245: 8242: 8239: 8235: 8232: 8228: 8224: 8221: 8217: 8214: 8211: 8208: 8205: 8202: 8199: 8196: 8193: 8190: 8187: 8184: 8181: 8178: 8175: 8172: 8149: 8145: 8142: 8138: 8134: 8131: 8127: 8124: 8121: 8118: 8115: 8112: 8109: 8087: 8083: 8080: 8076: 8072: 8069: 8065: 8062: 8059: 8056: 8053: 8050: 8047: 8032:representative 8024: 8023: 8012: 8009: 8006: 8003: 8000: 7997: 7992: 7987: 7984: 7981: 7978: 7975: 7972: 7969: 7966: 7963: 7960: 7957: 7939: 7938: 7923: 7915: 7912: 7909: 7906: 7903: 7895: 7891: 7888: 7885: 7882: 7879: 7871: 7868: 7865: 7862: 7859: 7856: 7853: 7850: 7848: 7844: 7841: 7837: 7833: 7830: 7827: 7824: 7821: 7820: 7812: 7809: 7806: 7803: 7800: 7792: 7788: 7780: 7777: 7774: 7771: 7768: 7765: 7762: 7759: 7756: 7754: 7752: 7749: 7746: 7743: 7740: 7737: 7734: 7733: 7718: 7717: 7706: 7703: 7700: 7697: 7689: 7685: 7682: 7679: 7676: 7673: 7670: 7667: 7664: 7661: 7658: 7655: 7652: 7649: 7646: 7643: 7640: 7637: 7634: 7631: 7621: 7610: 7607: 7604: 7601: 7598: 7595: 7592: 7589: 7586: 7583: 7580: 7577: 7574: 7571: 7568: 7565: 7562: 7559: 7556: 7546: 7534: 7531: 7528: 7525: 7522: 7519: 7516: 7513: 7510: 7507: 7504: 7501: 7498: 7495: 7492: 7489: 7486: 7483: 7480: 7477: 7474: 7471: 7461: 7449: 7446: 7443: 7440: 7437: 7434: 7431: 7428: 7425: 7422: 7419: 7416: 7413: 7410: 7407: 7404: 7401: 7398: 7395: 7392: 7389: 7386: 7368:multiplication 7347: 7344: 7341: 7338: 7318: 7295: 7292: 7289: 7286: 7283: 7267: 7264: 7263: 7262: 7261: 7260: 7238:as a fraction 7232: 7231: 7230: 7221: 7211: 7210: 7209: 7150: 7149: 7148: 7135: 7134: 7133: 7120: 7119: 7118: 7098: 7097: 7090: 7089: 7075: 7074: 7067: 7060: 7049:trailing zeros 7041: 7040: 7033: 7026: 7019: 6980: 6977: 6965: 6959: 6956: 6950: 6947: 6898:decimals with 6858:"), to obtain 6822: 6819: 6803: 6800: 6794: 6787: 6784: 6781: 6776: 6773: 6770: 6763: 6757: 6754: 6748: 6742: 6739: 6733: 6727: 6724: 6718: 6712: 6709: 6679: 6676: 6650: 6647: 6641: 6634: 6631: 6628: 6624: 6598: 6595: 6572: 6569: 6563: 6560: 6545: 6542: 6541: 6540: 6529: 6524: 6521: 6516: 6513: 6508: 6505: 6500: 6497: 6494: 6489: 6486: 6481: 6478: 6475: 6472: 6469: 6466: 6463: 6458: 6455: 6450: 6447: 6444: 6426: 6425: 6412: 6409: 6404: 6401: 6396: 6393: 6388: 6383: 6379: 6376: 6373: 6370: 6367: 6361: 6356: 6353: 6348: 6343: 6340: 6335: 6332: 6329: 6314: 6311: 6307: 6306: 6292: 6289: 6283: 6277: 6274: 6268: 6262: 6259: 6253: 6247: 6244: 6238: 6235: 6220: 6217: 6209: 6208: 6195: 6192: 6187: 6182: 6179: 6174: 6169: 6166: 6161: 6154: 6145: 6137: 6128: 6120: 6113: 6104: 6096: 6087: 6079: 6074: 6071: 6066: 6061: 6058: 6038: 6037: 6024: 6021: 6016: 6011: 6008: 6003: 5998: 5995: 5978: 5975: 5973: 5972:Multiplication 5970: 5969: 5968: 5957: 5951: 5948: 5942: 5939: 5934: 5926: 5923: 5917: 5914: 5909: 5904: 5901: 5898: 5895: 5892: 5889: 5886: 5883: 5880: 5874: 5871: 5865: 5862: 5859: 5845: 5844: 5830: 5827: 5821: 5815: 5812: 5806: 5800: 5797: 5791: 5785: 5782: 5776: 5770: 5767: 5748: 5745: 5737: 5736: 5719: 5716: 5711: 5708: 5704: 5701: 5696: 5691: 5688: 5683: 5677: 5674: 5671: 5666: 5663: 5660: 5654: 5648: 5645: 5642: 5637: 5634: 5631: 5625: 5622: 5620: 5618: 5613: 5610: 5605: 5602: 5598: 5595: 5590: 5585: 5582: 5577: 5571: 5568: 5565: 5560: 5557: 5554: 5548: 5542: 5539: 5536: 5531: 5528: 5525: 5519: 5516: 5514: 5510: 5507: 5502: 5497: 5494: 5489: 5488: 5461: 5458: 5432: 5429: 5414: 5413: 5399: 5396: 5391: 5388: 5385: 5382: 5379: 5373: 5368: 5365: 5360: 5355: 5352: 5336: 5335: 5322: 5319: 5314: 5311: 5306: 5303: 5298: 5293: 5290: 5285: 5280: 5277: 5261: 5260: 5247: 5244: 5239: 5234: 5231: 5198: 5195: 5189: 5183: 5180: 5174: 5168: 5165: 5084: 5081: 5075: 5069: 5066: 5060: 5054: 5051: 5023: 5020: 5006: 5005: 4992: 4989: 4984: 4979: 4976: 4960: 4959: 4948: 4943: 4940: 4935: 4930: 4927: 4922: 4914: 4911: 4906: 4900: 4897: 4894: 4889: 4886: 4883: 4877: 4868: 4865: 4862: 4857: 4854: 4851: 4845: 4840: 4837: 4832: 4824: 4821: 4796: 4793: 4776: 4773: 4747: 4744: 4726: 4725: 4710: 4707: 4701: 4698: 4692: 4689: 4683: 4677: 4674: 4668: 4662: 4659: 4640: 4637: 4615: 4612: 4606: 4600: 4597: 4572: 4569: 4566: 4542: 4539: 4536: 4531: 4528: 4525: 4497: 4494: 4491: 4486: 4483: 4480: 4456: 4450: 4447: 4421: 4418: 4389: 4386: 4380: 4374: 4371: 4345: 4342: 4316: 4313: 4262: 4259: 4256: 4251: 4248: 4245: 4217: 4214: 4211: 4206: 4203: 4200: 4173: 4170: 4144: 4141: 4123: 4122: 4111: 4108: 4105: 4102: 4099: 4088: 4084: 4081: 4074: 4067: 4064: 4060: 4045: 4042: 4038: 4032: 4025: 4022: 4018: 4001: 4000: 3988: 3961: 3958: 3952: 3946: 3943: 3925: 3922: 3914: 3913: 3899: 3896: 3890: 3883: 3879: 3875: 3870: 3866: 3862: 3855: 3849: 3846: 3726: 3723: 3697: 3694: 3589: 3586: 3583: 3580: 3560: 3557: 3554: 3551: 3529: 3504: 3501: 3485: 3482: 3400: 3397: 3369: 3366: 3340: 3337: 3314: 3302: 3299: 3275: 3272: 3249: 3245: 3241: 3236: 3232: 3228: 3199: 3196: 3190: 3184: 3181: 3155: 3152: 3146: 3140: 3137: 3131: 3125: 3122: 3096: 3093: 3067: 3064: 3025: 3022: 3019: 3015: 3011: 3008: 3003: 2998: 2978: 2973: 2968: 2965: 2961: 2957: 2954: 2934: 2931: 2925: 2922: 2914: 2909: 2886: 2883: 2877: 2874: 2869: 2861: 2858: 2835: 2831: 2827: 2823: 2819: 2808: 2807: 2792: 2787: 2784: 2779: 2774: 2771: 2766: 2761: 2757: 2754: 2751: 2748: 2745: 2739: 2736: 2733: 2728: 2724: 2721: 2718: 2715: 2712: 2706: 2703: 2701: 2697: 2690: 2687: 2681: 2675: 2674: 2671: 2666: 2663: 2658: 2653: 2650: 2645: 2640: 2637: 2632: 2629: 2626: 2621: 2618: 2613: 2608: 2599: 2596: 2584: 2579: 2576: 2571: 2566: 2563: 2558: 2553: 2550: 2545: 2540: 2537: 2532: 2527: 2524: 2519: 2516: 2514: 2508: 2505: 2494: 2491: 2480: 2479: 2451: 2446: 2439: 2431: 2428: 2422: 2417: 2391: 2387: 2383: 2378: 2374: 2370: 2343: 2340: 2324: 2321: 2315: 2309: 2306: 2300: 2294: 2291: 2285: 2279: 2276: 2250: 2247: 2241: 2235: 2232: 2226: 2220: 2217: 2191: 2188: 2165: 2159: 2156: 2150: 2144: 2141: 2135: 2129: 2126: 2100: 2097: 2071: 2068: 2042: 2039: 2013: 2010: 1980: 1977: 1971: 1965: 1962: 1943: 1940: 1938: 1935: 1864: 1858: 1855: 1849: 1846: 1843: 1837: 1834: 1807: 1801: 1798: 1792: 1786: 1783: 1777: 1771: 1768: 1762: 1756: 1753: 1747: 1744: 1714: 1709: 1701: 1698: 1692: 1689: 1684: 1679: 1656: 1653: 1647: 1644: 1598: 1595: 1589: 1586: 1563: 1560: 1537: 1517: 1513: 1509: 1502: 1479: 1476: 1470: 1448:mixed fraction 1439: 1436: 1355:. Dividing by 1281: 1278: 1234: 1233: 1230: 1229:6 are red, and 1227: 1208: 1205: 1100: 1097: 1085:absolute value 1066: 1063: 1055: 1054: 1009: 875: 872: 831: 828: 826: 823: 616:fraction slash 496: 493: 477: 474: 445: 441: 386: 133: 130: 104: 101: 26: 9: 6: 4: 3: 2: 12483: 12472: 12469: 12468: 12466: 12451: 12448: 12446: 12443: 12441: 12438: 12436: 12433: 12431: 12428: 12426: 12423: 12419: 12416: 12415: 12414: 12411: 12409: 12406: 12402: 12399: 12398: 12397: 12394: 12392: 12389: 12387: 12384: 12382: 12379: 12377: 12374: 12372: 12369: 12367: 12364: 12362: 12359: 12358: 12355: 12331: 12330: 12328: 12324: 12309: 12305: 12301: 12298: 12297: 12295: 12288: 12282: 12278: 12274: 12267: 12262: 12260: 12255: 12253: 12248: 12247: 12244: 12236: 12232: 12228: 12224: 12220: 12216: 12215: 12192: 12185: 12179: 12172: 12168: 12164: 12159: 12151: 12145: 12141: 12134: 12126: 12120: 12112: 12106: 12102: 12096: 12089: 12088:Cajori (1928) 12084: 12069: 12065: 12061: 12060: 12052: 12044: 12038: 12034: 12030: 12026: 12022: 12018: 12011: 11995: 11991: 11984: 11982: 11980: 11978: 11969: 11965: 11961: 11957: 11953: 11949: 11948: 11940: 11932: 11930:1-56881-201-9 11926: 11922: 11918: 11912: 11904: 11898: 11894: 11887: 11872: 11868: 11862: 11847: 11843: 11837: 11819: 11815: 11811: 11807: 11803: 11799: 11795: 11794: 11786: 11779: 11764: 11760: 11754: 11739: 11735: 11731: 11725: 11722:p. 120. 11721: 11717: 11713: 11706: 11691: 11687: 11681: 11666: 11662: 11658: 11652: 11637:on 2017-12-01 11636: 11632: 11628: 11622: 11614: 11613: 11605: 11603: 11595:. p. 65. 11594: 11593: 11585: 11583: 11575: 11573: 11567: 11563: 11559: 11555: 11551: 11547: 11540: 11532: 11530:9781783741373 11526: 11521: 11516: 11512: 11505: 11497: 11491: 11486: 11485: 11476: 11469: 11467: 11444: 11438: 11432: 11427: 11426: 11417: 11409: 11407:9781592571611 11403: 11399: 11398: 11390: 11375: 11371: 11365: 11361: 11360: 11352: 11338:on 2012-04-15 11337: 11333: 11329: 11323: 11308: 11304: 11300: 11293: 11285: 11284: 11275: 11267: 11261: 11257: 11256: 11248: 11240: 11236: 11232: 11231:Mathematics I 11225: 11217: 11211: 11207: 11206: 11198: 11196: 11187: 11183: 11179: 11175: 11168: 11160: 11156: 11149: 11141: 11137: 11130: 11122: 11116: 11112: 11111:Point Roberts 11108: 11101: 11094: 11093:Cajori (1928) 11089: 11073: 11069: 11063: 11059: 11055: 11054: 11046: 11044: 11029: 11025: 11019: 11011: 11005: 11000: 10999: 10990: 10983: 10979: 10975: 10971: 10965: 10961: 10958: 10955: 10953:1-58488-347-2 10949: 10945: 10941: 10923: 10916: 10915: 10908: 10902: 10901: 10894: 10888: 10882: 10878: 10854: 10851: 10850: 10831: 10828: 10827: 10824: 10821: 10820: 10817: 10815: 10812: 10811: 10808: 10807: 10788: 10786: 10783: 10782: 10779: 10776: 10775: 10772: 10769: 10768: 10765: 10763: 10760: 10759: 10756: 10755: 10744: 10741: 10740: 10729: 10726: 10725: 10722: 10721:Prime numbers 10719: 10718: 10714: 10711: 10710: 10706: 10703: 10702: 10699: 10679: 10672: 10669: 10668: 10665: 10664: 10661: 10641: 10634: 10631: 10630: 10627: 10626: 10623: 10603: 10596: 10593: 10592: 10589: 10588: 10585: 10565: 10558: 10555: 10554: 10551: 10550: 10547: 10527: 10520: 10517: 10516: 10513: 10508: 10507: 10504: 10503: 10498: 10495: 10493: 10490: 10488: 10485: 10483: 10480: 10479: 10473: 10471: 10455: 10435: 10411: 10387: 10375: 10365: 10363: 10359: 10355: 10351: 10347: 10332: 10330: 10326: 10322: 10318: 10312: 10310: 10306: 10302: 10298: 10294: 10290: 10286: 10282: 10278: 10274: 10270: 10264: 10262: 10258: 10253: 10251: 10230: 10226: 10221: 10217: 10205: 10201: 10197: 10193: 10182: 10177: 10107: 10104: 10101: 10100: 10099: 10093: 10087: 10084: 10083: 10082: 10080: 10068: 10063: 10054: 10043: 10032: 10027: AD 500 10021: 10017: 10012: 10010: 10006: 10002: 9995: 9991: 9987: 9984: 9973: 9970: 9967: 9963: 9959: 9955: 9950: 9948: 9944: 9934: 9930: 9926: 9915: 9912: 9904: 9893: 9890: 9886: 9883: 9879: 9876: 9872: 9869: 9865: 9862: – 9861: 9857: 9856:Find sources: 9850: 9846: 9840: 9839: 9832: 9828: 9823: 9822: 9797: 9794: 9784: 9781: 9778: 9774: 9770: 9766: 9762: 9758: 9755: 9752: 9732: 9728: 9725: 9722: 9719: 9718: 9717: 9714: 9712: 9708: 9704: 9700: 9694: 9684: 9664: 9658: 9653: 9650: 9647: 9641: 9638: 9632: 9629: 9626: 9616: 9611: 9608: 9605: 9599: 9593: 9585: 9575: 9570: 9564: 9559: 9554: 9542: 9537: 9534: 9531: 9525: 9519: 9511: 9506: 9503: 9500: 9493: 9488: 9485: 9482: 9476: 9468: 9463: 9460: 9457: 9453: 9448: 9440: 9435: 9432: 9429: 9425: 9416: 9400: 9394: 9389: 9386: 9383: 9377: 9374: 9368: 9365: 9362: 9352: 9347: 9344: 9341: 9335: 9329: 9321: 9311: 9306: 9300: 9295: 9290: 9278: 9273: 9270: 9267: 9261: 9255: 9247: 9242: 9239: 9236: 9229: 9224: 9221: 9218: 9212: 9204: 9199: 9196: 9193: 9189: 9184: 9176: 9171: 9168: 9165: 9161: 9152: 9151: 9150: 9148: 9144: 9123: 9117: 9112: 9106: 9100: 9095: 9089: 9083: 9079: 9074: 9068: 9064: 9055: 9054: 9053: 9051: 9047: 9043: 9039: 9033: 9029: 9019: 9017: 9013: 9009: 8988: 8985: 8982: 8978: 8973: 8967: 8964: 8961: 8957: 8930: 8927: 8922: 8918: 8912: 8909: 8898: 8893: 8891: 8887: 8871: 8851: 8848: 8845: 8840: 8831:since all of 8818: 8815: 8811: 8807: 8799: 8780: 8776: 8770: 8757: 8753: 8749: 8745: 8741: 8736: 8734: 8710: 8707: 8701: 8698: 8690: 8687: 8681: 8678: 8663: 8644: 8641: 8636: 8632: 8626: 8623: 8620: 8607: 8603: 8599: 8578: 8575: 8572: 8569: 8566: 8561: 8557: 8551: 8548: 8535: 8530: 8528: 8507: 8504: 8499: 8495: 8489: 8486: 8483: 8455: 8452: 8449: 8446: 8443: 8438: 8434: 8428: 8425: 8414: 8410: 8404: 8394: 8392: 8388: 8384: 8380: 8376: 8372: 8368: 8364: 8360: 8356: 8352: 8348: 8345: 8341: 8337: 8332: 8330: 8326: 8320: 8315: 8287: 8261: 8258: 8254: 8250: 8247: 8240: 8233: 8230: 8226: 8222: 8219: 8209: 8200: 8197: 8194: 8188: 8182: 8179: 8176: 8163: 8162: 8143: 8140: 8136: 8132: 8129: 8122: 8116: 8113: 8110: 8081: 8078: 8074: 8070: 8067: 8060: 8054: 8051: 8048: 8038: 8037: 8036: 8033: 8029: 8010: 8007: 8004: 8001: 7998: 7995: 7982: 7979: 7976: 7970: 7964: 7961: 7958: 7948: 7947: 7946: 7944: 7921: 7910: 7907: 7904: 7886: 7883: 7880: 7877: 7863: 7860: 7857: 7851: 7849: 7842: 7839: 7831: 7828: 7825: 7807: 7804: 7801: 7772: 7769: 7766: 7763: 7757: 7755: 7747: 7744: 7741: 7735: 7724: 7723: 7722: 7701: 7698: 7695: 7680: 7677: 7674: 7671: 7668: 7662: 7656: 7653: 7650: 7644: 7638: 7635: 7632: 7622: 7605: 7602: 7599: 7596: 7593: 7587: 7581: 7578: 7575: 7569: 7563: 7560: 7557: 7547: 7529: 7526: 7523: 7520: 7517: 7514: 7511: 7508: 7502: 7496: 7493: 7490: 7484: 7478: 7475: 7472: 7462: 7444: 7441: 7438: 7435: 7432: 7429: 7426: 7423: 7417: 7411: 7408: 7405: 7399: 7393: 7390: 7387: 7377: 7376: 7375: 7373: 7369: 7365: 7361: 7345: 7342: 7339: 7336: 7316: 7309: 7290: 7287: 7284: 7273: 7243: 7240: 7239: 7237: 7233: 7227: 7222: 7219: 7215: 7214: 7212: 7200: 7196: 7192: 7191: 7188: 7184: 7180: 7176: 7170: 7166: 7160: 7156: 7151: 7143: 7139: 7138: 7136: 7128: 7124: 7123: 7121: 7113: 7110: 7109: 7107: 7103: 7102: 7101: 7095: 7094: 7093: 7084: 7083: 7082: 7068: 7061: 7054: 7053: 7052: 7050: 7046: 7045:leading zeros 7034: 7027: 7020: 7013: 7012: 7011: 7005: 7000: 6998: 6992: 6986: 6976: 6963: 6957: 6954: 6948: 6945: 6818: 6801: 6798: 6792: 6785: 6782: 6779: 6774: 6771: 6768: 6761: 6755: 6752: 6746: 6740: 6737: 6731: 6725: 6722: 6716: 6710: 6707: 6696: 6677: 6674: 6648: 6645: 6639: 6632: 6629: 6626: 6622: 6596: 6593: 6570: 6567: 6561: 6558: 6527: 6522: 6519: 6514: 6511: 6506: 6503: 6498: 6495: 6492: 6487: 6484: 6479: 6476: 6473: 6470: 6467: 6464: 6461: 6456: 6453: 6448: 6445: 6442: 6435: 6434: 6433: 6431: 6410: 6407: 6402: 6399: 6394: 6391: 6386: 6381: 6377: 6374: 6371: 6368: 6365: 6359: 6354: 6351: 6346: 6341: 6338: 6333: 6330: 6327: 6320: 6319: 6318: 6310: 6290: 6287: 6281: 6275: 6272: 6266: 6260: 6257: 6251: 6245: 6242: 6236: 6233: 6226: 6225: 6224: 6216: 6214: 6193: 6190: 6185: 6180: 6177: 6172: 6167: 6164: 6159: 6152: 6143: 6135: 6126: 6118: 6111: 6102: 6094: 6085: 6077: 6072: 6069: 6064: 6059: 6056: 6047: 6046: 6045: 6042: 6022: 6019: 6014: 6009: 6006: 6001: 5996: 5993: 5984: 5983: 5982: 5955: 5949: 5946: 5940: 5937: 5924: 5921: 5915: 5912: 5902: 5896: 5893: 5890: 5887: 5884: 5878: 5872: 5869: 5863: 5860: 5857: 5850: 5849: 5848: 5828: 5825: 5819: 5813: 5810: 5804: 5798: 5795: 5789: 5783: 5780: 5774: 5768: 5765: 5755: 5754: 5753: 5744: 5742: 5717: 5714: 5709: 5702: 5699: 5694: 5689: 5686: 5681: 5675: 5672: 5669: 5664: 5661: 5658: 5652: 5646: 5643: 5640: 5635: 5632: 5629: 5623: 5621: 5611: 5608: 5603: 5596: 5593: 5588: 5583: 5580: 5575: 5569: 5566: 5563: 5558: 5555: 5552: 5546: 5540: 5537: 5534: 5529: 5526: 5523: 5517: 5515: 5508: 5505: 5500: 5495: 5492: 5479: 5478: 5477: 5459: 5456: 5430: 5427: 5397: 5394: 5389: 5386: 5383: 5380: 5377: 5371: 5366: 5363: 5358: 5353: 5350: 5341: 5340: 5339: 5320: 5317: 5312: 5309: 5304: 5301: 5296: 5291: 5288: 5283: 5278: 5275: 5266: 5265: 5264: 5245: 5242: 5237: 5232: 5229: 5220: 5219: 5218: 5215: 5196: 5193: 5187: 5181: 5178: 5172: 5166: 5163: 5133: 5082: 5079: 5073: 5067: 5064: 5058: 5052: 5049: 5021: 5018: 4990: 4987: 4982: 4977: 4974: 4965: 4964: 4963: 4946: 4941: 4938: 4933: 4928: 4925: 4920: 4912: 4909: 4904: 4898: 4895: 4892: 4887: 4884: 4881: 4875: 4866: 4863: 4860: 4855: 4852: 4849: 4843: 4838: 4835: 4830: 4822: 4819: 4810: 4809: 4808: 4805: 4803: 4774: 4771: 4745: 4742: 4730: 4708: 4705: 4699: 4696: 4690: 4687: 4681: 4675: 4672: 4666: 4660: 4657: 4647: 4646: 4645: 4636: 4632: 4613: 4610: 4604: 4598: 4595: 4570: 4567: 4564: 4540: 4537: 4534: 4529: 4526: 4523: 4495: 4492: 4489: 4484: 4481: 4478: 4454: 4448: 4445: 4419: 4416: 4404: 4387: 4384: 4378: 4372: 4369: 4343: 4340: 4314: 4311: 4300: 4296: 4292: 4288: 4284: 4280: 4260: 4257: 4254: 4249: 4246: 4243: 4215: 4212: 4209: 4204: 4201: 4198: 4171: 4168: 4142: 4139: 4127: 4109: 4106: 4103: 4100: 4097: 4086: 4082: 4079: 4072: 4065: 4062: 4058: 4043: 4040: 4036: 4030: 4023: 4020: 4016: 4006: 4005: 4004: 3999:are positive. 3986: 3959: 3956: 3950: 3944: 3941: 3931: 3930: 3929: 3921: 3919: 3897: 3894: 3888: 3881: 3877: 3873: 3868: 3864: 3860: 3853: 3847: 3844: 3834: 3833: 3832: 3813: 3744: 3742: 3724: 3721: 3695: 3692: 3681: 3677: 3673: 3672: 3666: 3664: 3663: 3658: 3654: 3645: 3639: 3631: 3623: 3615: 3607: 3587: 3584: 3581: 3578: 3558: 3555: 3552: 3549: 3527: 3502: 3499: 3481: 3398: 3395: 3367: 3364: 3338: 3335: 3312: 3298: 3295: 3293: 3289: 3285: 3281: 3271: 3267: 3247: 3243: 3239: 3234: 3230: 3226: 3197: 3194: 3188: 3182: 3179: 3153: 3150: 3144: 3138: 3135: 3129: 3123: 3120: 3094: 3091: 3065: 3062: 3051: 3050: 3045: 3041: 3036: 3023: 3017: 3013: 3009: 2996: 2976: 2963: 2959: 2955: 2932: 2929: 2923: 2920: 2907: 2884: 2881: 2875: 2872: 2859: 2856: 2833: 2829: 2825: 2821: 2817: 2790: 2785: 2782: 2777: 2772: 2769: 2764: 2759: 2755: 2752: 2749: 2746: 2743: 2737: 2734: 2731: 2726: 2722: 2719: 2716: 2713: 2710: 2704: 2702: 2695: 2688: 2685: 2679: 2669: 2664: 2661: 2656: 2651: 2648: 2643: 2638: 2635: 2630: 2627: 2624: 2619: 2616: 2611: 2606: 2597: 2594: 2582: 2577: 2574: 2569: 2564: 2561: 2556: 2551: 2548: 2543: 2538: 2535: 2530: 2525: 2522: 2517: 2515: 2506: 2503: 2492: 2489: 2470: 2469: 2468: 2466: 2449: 2429: 2426: 2420: 2389: 2385: 2381: 2376: 2372: 2368: 2356: 2349: 2339: 2322: 2319: 2313: 2307: 2304: 2298: 2292: 2289: 2283: 2277: 2274: 2248: 2245: 2239: 2233: 2230: 2224: 2218: 2215: 2189: 2186: 2163: 2157: 2154: 2148: 2142: 2139: 2133: 2127: 2124: 2098: 2095: 2069: 2066: 2040: 2037: 2011: 2008: 1997: 1978: 1975: 1969: 1963: 1960: 1949: 1934: 1932: 1923: 1915: 1907: 1903: 1902:metric system 1899: 1875: 1862: 1856: 1853: 1847: 1844: 1841: 1835: 1832: 1821: 1805: 1799: 1796: 1790: 1784: 1781: 1775: 1769: 1766: 1760: 1754: 1751: 1745: 1742: 1734: 1730: 1725: 1712: 1699: 1696: 1690: 1687: 1677: 1654: 1651: 1645: 1642: 1634: 1596: 1593: 1587: 1584: 1561: 1558: 1535: 1515: 1511: 1507: 1500: 1477: 1474: 1468: 1459: 1457: 1456:concatenation 1453: 1452:mixed numeral 1449: 1445: 1438:Mixed numbers 1435: 1432: 1428: 1424: 1419: 1417: 1413: 1409: 1405: 1404: 1398: 1396: 1392: 1388: 1383: 1365: 1360: 1338: 1333: 1312: 1308: 1307: 1302: 1298: 1294: 1289: 1288: 1277: 1275: 1237: 1232:4 are yellow, 1231: 1228: 1225: 1224: 1223: 1221: 1216: 1215: 1204: 1186: 1165: 1146: 1144: 1108: 1107: 1096: 1094: 1090: 1086: 1082: 1077: 1075: 1074: 1062: 1060: 1020: 1016: 1015: 1010: 991: 990: 989:unit fraction 985: 984: 983: 981: 976: 974: 906: 902: 898: 894: 873: 870: 857: 853: 849: 845: 841: 837: 822: 788: 784: 764: 726: 707: 705: 668: 663: 645: 641: 637: 633: 629: 625: 621: 617: 613: 609: 605: 601: 571: 566: 564: 560: 556: 536: 532: 528: 524: 520: 516: 510: 506: 502: 492: 475: 472: 462: 443: 439: 423: 415: 407: 403: 373: 369: 365: 361: 353: 345: 337: 332: 233: 211: 207: 202: 168: 164: 150: 131: 128: 102: 99: 88: 84: 80: 76: 72: 68: 41: 37: 33: 19: 12272: 12234: 12222: 12198:. Retrieved 12178: 12158: 12139: 12133: 12124: 12119: 12100: 12095: 12083: 12072:. Retrieved 12058: 12051: 12020: 12010: 11998:. Retrieved 11951: 11945: 11939: 11920: 11911: 11892: 11886: 11875:. Retrieved 11861: 11850:. Retrieved 11836: 11825:. Retrieved 11797: 11791: 11778: 11767:. Retrieved 11753: 11743:28 September 11741:. Retrieved 11715: 11705: 11694:. Retrieved 11680: 11669:. Retrieved 11660: 11651: 11639:. Retrieved 11635:the original 11630: 11621: 11611: 11591: 11571: 11569: 11549: 11545: 11539: 11510: 11504: 11483: 11475: 11465: 11442: 11440: 11424: 11416: 11396: 11389: 11378:. Retrieved 11358: 11351: 11340:. Retrieved 11336:the original 11331: 11322: 11311:. Retrieved 11302: 11292: 11281: 11274: 11254: 11247: 11230: 11224: 11204: 11180:(2): 83–88. 11177: 11173: 11167: 11158: 11154: 11148: 11139: 11135: 11129: 11106: 11100: 11088: 11076:. Retrieved 11052: 11031:. Retrieved 11027: 11018: 10997: 10989: 10969: 10964: 10943: 10937: 10922: 10912: 10898: 10881: 10761: 10469: 10371: 10343: 10314: 10304: 10285:Simon Stevin 10280: 10272: 10266: 10254: 10178: 10111: 10097: 10015: 10013: 9951: 9922: 9907: 9898: 9888: 9881: 9874: 9867: 9855: 9843:Please help 9838:verification 9835: 9782: 9776: 9760: 9756: 9750: 9730: 9726: 9720: 9715: 9707:Number Forms 9696: 9681: 9140: 9044:it (compare 9035: 8894: 8889: 8886:real numbers 8737: 8732: 8664: 8605: 8601: 8597: 8531: 8406: 8382: 8378: 8377:elements of 8374: 8370: 8366: 8362: 8354: 8346: 8339: 8335: 8333: 8328: 8324: 8318: 8288: 8285: 8025: 7940: 7719: 7269: 7241: 7235: 7225: 7217: 7201:= 1,523,987. 7198: 7194: 7186: 7182: 7178: 7174: 7168: 7164: 7158: 7154: 7144:= 1,523,987. 7141: 7126: 7111: 7105: 7099: 7091: 7076: 7066:= 392/999000 7042: 7001: 6994: 6862:. To change 6824: 6547: 6427: 6316: 6308: 6222: 6210: 6043: 6039: 5980: 5846: 5750: 5738: 5415: 5337: 5262: 5216: 5134: 5007: 4961: 4806: 4798: 4642: 4633: 4405: 4298: 4294: 4290: 4286: 4282: 4278: 4128: 4124: 4002: 3927: 3915: 3814: 3745: 3740: 3679: 3675: 3669: 3667: 3662:lowest terms 3660: 3646: 3637: 3629: 3621: 3613: 3605: 3487: 3304: 3296: 3288:distributive 3277: 3268: 3047: 3043: 3039: 3037: 2809: 2354: 2352: 1945: 1921: 1913: 1876: 1726: 1632: 1460: 1455: 1451: 1447: 1444:mixed number 1443: 1441: 1420: 1415: 1407: 1401: 1399: 1384: 1361: 1334: 1304: 1285: 1283: 1238: 1235: 1226:2 are white, 1219: 1212: 1210: 1184: 1166: 1147: 1104: 1102: 1092: 1088: 1080: 1078: 1071: 1068: 1059:Number Forms 1056: 1019:power of two 1012: 987: 979: 977: 904: 896: 892: 855: 851: 843: 839: 835: 833: 786: 727: 708: 664: 644:typefounding 639: 635: 570:fraction bar 569: 567: 526: 514: 512: 421: 413: 405: 371: 367: 363: 359: 351: 343: 333: 234: 203: 166: 148: 86: 82: 78: 66: 64: 36: 12413:Irreducible 12343:Denominator 12000:15 February 11718:. Hoboken: 11464:is read as 11024:"Fractions" 10301:sexagesimal 10289:Netherlands 10188: 1200 10049: 1150 10031:Brahmagupta 10007:wrote the " 9969:philosopher 9939: 1000 9925:reciprocals 9042:rationalize 9018:for more). 8534:polynomials 8359:polynomials 7364:subtraction 7229:= 1,522,464 7039:= 6291/9999 5747:Subtraction 4635:fractions. 3671:irreducible 3284:associative 3280:commutative 1414:, as in 75 1274:probability 850:written as 535:dēnōminātor 527:denominator 404:. The term 201:of a cake. 167:denominator 12445:Percentage 12440:Paper size 12349:= Quotient 12231:"Fraction" 12200:2013-10-10 12074:2017-08-30 11877:2013-06-18 11852:2013-06-18 11827:2010-01-27 11769:2012-08-15 11759:"Fraction" 11696:2018-03-13 11671:2018-03-13 11380:2014-07-29 11342:2011-12-31 11313:2014-10-30 11303:Math Forum 11186:1291644250 11153:"solidus, 11134:"virgule, 11078:2016-02-20 11033:2020-08-27 10934:References 10887:Bringhurst 10814:Irrational 10315:While the 10273:De Thiende 10079:Bhaskara I 10038: 628 10020:Aryabhatta 10016:bhinnarasi 9994:Metapontum 9979: 530 9972:Pythagoras 9871:newspapers 9860:"Fraction" 9769:half crown 9699:typography 9691:See also: 8325:equivalent 8035:fractions 7898:with 7795:with 7193:10,000,000 7140:10,000,000 6989:See also: 6695:reciprocal 1395:per centum 1387:percentage 1301:interpunct 1106:reciprocal 783:slash mark 620:typography 499:See also: 495:Vocabulary 12418:Reduction 12376:Continued 12361:Algebraic 12337:Numerator 12273:Fractions 11738:719886424 11641:29 August 11566:195210281 10853:Imaginary 10362:geoboards 10358:dot paper 10309:Samarkand 10271:pamphlet 10181:Al-Hassār 10073:or cross 9962:Followers 9901:June 2023 9651:− 9642:− 9630:− 9609:− 9565:− 9535:− 9504:− 9486:− 9477:⋅ 9378:− 9366:− 9301:− 9213:⋅ 9197:− 9169:− 9147:conjugate 9090:⋅ 8986:− 8928:− 8895:The term 8849:π 8808:π 8702:− 8642:− 8576:− 8505:− 8453:− 8369:. So for 8210:∼ 8123:∼ 8061:∼ 7991:⟺ 7971:∼ 7881:≠ 7840:− 7764:− 7736:− 7699:≠ 7645:÷ 7570:⋅ 7515:− 7485:− 7340:≠ 7224:9,990,000 7216:9,990,000 7073:= 12/9900 7032:= 264/999 6783:⋅ 6772:⋅ 6747:× 6717:÷ 6630:⋅ 6568:÷ 6480:× 6468:× 6446:× 6369:× 6360:× 6331:× 6267:× 6237:× 6173:× 6119:× 6065:× 6002:× 5916:− 5894:− 5888:− 5861:− 5805:− 5775:− 5673:⋅ 5662:⋅ 5644:⋅ 5633:⋅ 5567:⋅ 5556:⋅ 5538:⋅ 5527:⋅ 5173:× 5059:× 4896:× 4885:× 4864:× 4853:× 4568:× 4538:× 4527:× 4493:× 4482:× 4258:⋅ 4247:⋅ 4213:⋅ 4202:⋅ 4107:− 4098:− 4080:− 4063:− 4041:− 4021:− 3878:÷ 3865:÷ 3189:× 3130:× 2765:× 2747:× 2732:÷ 2714:× 2644:× 2625:÷ 2557:× 2531:÷ 1678:− 1643:− 1493:cakes or 1083:, if the 899:are both 523:numerātor 515:numerator 161:), and a 149:numerator 45:fraction 18:Fractions 12465:Category 12391:Egyptian 12326:Fraction 12308:Quotient 12300:Dividend 12191:Archived 12167:Archived 12068:Archived 11994:Archived 11871:Archived 11846:Archived 11818:Archived 11763:Archived 11690:Archived 11665:Archived 11562:ProQuest 11447:⁠2 11374:Archived 11307:Archived 11182:ProQuest 11072:Archived 10978:Archived 10909:⟩ 10905:⟨ 10895:⟩ 10891:⟨ 10762:Fraction 10595:Rational 10497:FRACTRAN 10492:Multiple 10487:0.999... 10476:See also 10470:fraction 10325:Baghdadi 10297:Al-Kāshī 10281:La Disme 10263:states: 10160:(i.e., 1 10081:writes: 10053:Sanskrit 10042:Bhaskara 9990:Hippasus 9929:integers 9757:Shilling 9703:one half 9143:binomial 9050:monomial 9038:radicals 9028:Nth root 8800:such as 8536:, as in 8409:quotient 8262:′ 8251:′ 8234:′ 8223:′ 8144:′ 8133:′ 8082:′ 8071:′ 7943:relation 7372:division 7360:addition 7308:integers 7205:− 1,523. 7197:− 10,000 7129:= 1,523. 7114:= 0.1523 7004:vinculum 6892:1.000... 6880:1.000... 6544:Division 5100:. Since 4639:Addition 4512: ? 4435: ? 4330: ? 3977:3 > 2 3975:because 1879:⁠2 1614:⁠2 1431:accurate 1427:multiply 1403:permille 1315:⁠3 901:integers 891:, where 559:dividend 555:division 406:fraction 402:quotient 358:, where 210:division 165:integer 163:non-zero 67:fraction 12408:Integer 12381:Decimal 12346:⁠ 12334:⁠ 12304:Divisor 12090:, p. 89 11956:Bibcode 11814:4145131 11462:⁠ 10914:solidus 10911:as the 10900:virgule 10897:as the 10671:Natural 10633:Integer 10519:Complex 10426:⁠ 10378:⁠ 10317:Persian 10293:Chinese 10269:Flemish 10200:Morocco 10174:⁠ 10162:⁠ 10158:⁠ 10146:⁠ 10142:⁠ 10130:⁠ 10126:⁠ 10114:⁠ 10040:), and 9964:of the 9885:scholar 9817:History 9747:⁠ 9735:⁠ 9731:upright 9711:Unicode 9004:⁠ 8948:⁠ 8890:integer 8794:⁠ 8760:⁠ 8729:⁠ 8668:⁠ 8660:⁠ 8610:⁠ 8594:⁠ 8538:⁠ 8523:⁠ 8473:⁠ 8411:of two 8314:coprime 8303:⁠ 8291:⁠ 7258:⁠ 7255:9990000 7249:1522464 7246:⁠ 7172:, then 7025:= 62/99 6946:12.3456 6932:⁠ 6920:⁠ 6916:⁠ 6904:⁠ 6876:⁠ 6864:⁠ 6840:⁠ 6828:⁠ 6583:equals 5212:⁠ 5153:⁠ 5149:⁠ 5137:⁠ 5130:⁠ 5118:⁠ 5114:⁠ 5102:⁠ 5098:⁠ 5039:⁠ 4629:⁠ 4585:⁠ 3829:⁠ 3817:⁠ 3809:⁠ 3797:⁠ 3793:⁠ 3781:⁠ 3777:⁠ 3765:⁠ 3761:⁠ 3749:⁠ 3676:reduced 3642:⁠ 3626:⁠ 3618:⁠ 3602:⁠ 3540:⁠ 3520:⁠ 3478:⁠ 3466:⁠ 3462:⁠ 3450:⁠ 3446:⁠ 3434:⁠ 3430:⁠ 3418:⁠ 3414:⁠ 3385:⁠ 3264:⁠ 3215:⁠ 1926:⁠ 1910:⁠ 1894:⁠ 1629:⁠ 1380:⁠ 1368:⁠ 1330:⁠ 1270:⁠ 1258:⁠ 1254:⁠ 1242:⁠ 1201:⁠ 1189:⁠ 1181:⁠ 1169:⁠ 1162:⁠ 1150:⁠ 1139:⁠ 1127:⁠ 1123:⁠ 1111:⁠ 1061:block. 1051:⁠ 1039:⁠ 1035:⁠ 1023:⁠ 1021:, e.g. 1006:⁠ 994:⁠ 969:⁠ 957:⁠ 953:⁠ 941:⁠ 937:⁠ 925:⁠ 921:⁠ 909:⁠ 889:⁠ 860:⁠ 819:⁠ 816:1000000 807:⁠ 803:⁠ 791:⁠ 779:⁠ 767:⁠ 759:⁠ 747:⁠ 743:⁠ 731:⁠ 723:⁠ 711:⁠ 704:percent 699:⁠ 687:⁠ 683:⁠ 671:⁠ 660:⁠ 648:⁠ 593:⁄ 586:⁠ 574:⁠ 563:divisor 551:⁠ 539:⁠ 426:⁠ 410:⁠ 398:⁠ 376:⁠ 356:⁠ 340:⁠ 329:⁠ 317:⁠ 313:⁠ 301:⁠ 297:⁠ 285:⁠ 281:⁠ 269:⁠ 265:⁠ 253:⁠ 249:⁠ 237:⁠ 226:⁠ 214:⁠ 199:⁠ 187:⁠ 183:⁠ 171:⁠ 156:⁄ 75:fractus 59:⁠ 47:⁠ 12401:Silver 12396:Golden 12386:Dyadic 12371:Binary 12366:Aspect 12277:ratios 12146: 12107: 12039: 11927: 11899: 11812: 11736: 11726: 11564: 11527: 11492: 11433: 11404: 11366: 11262: 11212: 11184: 11117: 11064: 11006: 10974:p. 374 10950: 10428:where 10277:Leyden 9998:150 BC 9954:Greeks 9887: 9880: 9873: 9866: 9858: 8604:). An 8321:> 0 8161:imply 8026:is an 7370:, and 7125:10,000 7059:= 5/90 6955:123456 6900:0.3333 6213:factor 6150: 6133: 6109: 6092: 4918: 4873: 4828: 4359:gives 3286:, and 2900:or as 1898:inches 1670:means 1504: 1389:(from 1293:digits 1207:Ratios 955:, and 626:" or " 608:stroke 606:), or 529:(from 517:(from 507:, and 206:ratios 87:simple 83:vulgar 79:common 69:(from 12194:(PDF) 12187:(PDF) 11821:(PDF) 11810:JSTOR 11788:(PDF) 10873:Notes 10194:from 10190:), a 10062:cheda 10005:India 9966:Greek 9892:JSTOR 9878:books 9014:(see 8752:field 8527:field 8305:with 7062:0.000 7018:= 5/9 6958:10000 6890:into 6854:into 3678:, or 2353:In a 1391:Latin 1341:6.023 1214:ratio 618:. In 600:slash 531:Latin 519:Latin 230:3 ÷ 4 85:, or 71:Latin 12450:Unit 12275:and 12144:ISBN 12105:ISBN 12037:ISBN 12002:2016 11925:ISBN 11897:ISBN 11745:2020 11734:OCLC 11724:ISBN 11643:2022 11525:ISBN 11490:ISBN 11431:ISBN 11402:ISBN 11364:ISBN 11260:ISBN 11210:ISBN 11115:ISBN 11062:ISBN 11004:ISBN 10948:ISBN 10715:: 1 10707:: 0 10705:Zero 10557:Real 10057:amsa 10001:Jain 9952:The 9864:news 9030:and 8884:are 8864:and 8600:(or 8471:and 8373:and 8338:and 8329:same 8316:and 8309:and 8099:and 7329:and 7162:and 7104:Let 7069:0.00 7037:6291 6860:0.25 6856:1.00 6844:1.00 5446:and 4605:> 4379:> 4293:and 4285:and 4232:and 4158:and 4104:< 4031:< 3951:> 3916:The 3651:the 2406:and 2264:and 2205:are 2056:and 1103:The 895:and 685:and 362:and 299:and 208:and 118:and 12430:LCD 12064:269 12029:doi 11964:doi 11802:doi 11798:111 11554:doi 11550:100 11515:doi 11443:and 11235:doi 11157:". 11138:". 10713:One 10344:In 10196:Fez 10185:fl. 10176:). 10128:, 1 10029:), 9992:of 9927:of 9847:by 9765:£sd 9759:or 9010:of 8393:). 8353:of 7306:of 7207:987 7203:987 7190:): 7146:987 7131:987 7116:987 7087:987 7079:987 7064:392 7055:0.0 7043:If 7030:264 7008:789 6882:by 6846:by 4804:1. 4732:If 3874:462 3848:462 3826:462 3806:100 3480:). 3266:.) 3081:of 2989:or 2933:10. 2786:104 1946:An 1633:and 1450:or 1406:or 1327:100 923:, − 858:or 842:or 800:117 787:not 706:". 628:nut 491:). 374:or 12467:: 12306:= 12302:÷ 12233:. 12221:. 12165:. 12066:. 12035:. 11992:. 11976:^ 11962:. 11952:46 11950:. 11816:. 11808:. 11796:. 11790:. 11732:. 11714:. 11659:. 11629:. 11601:^ 11581:^ 11568:. 11560:. 11548:. 11523:. 11439:. 11372:. 11330:. 11305:. 11301:. 11194:^ 11178:53 11176:. 11155:n. 11136:n. 11070:. 11060:. 11058:74 11042:^ 11026:. 10976:. 10360:, 10259:, 10246:". 10198:, 10055:: 10046:c. 10035:c. 10024:c. 9976:c. 9960:. 9936:c. 9785:: 9713:. 9665:11 9633:20 9401:11 9369:20 8735:. 8662:. 7366:, 7362:, 7244:= 7185:− 7181:= 7177:− 7167:= 7157:= 7071:12 7051:: 7035:0. 7028:0. 7023:62 7021:0. 7014:0. 6886:(" 6850:(" 6817:. 6649:15 6646:10 6623:10 6559:10 6392:33 6288:18 6023:12 5718:12 5715:19 5703:12 5700:10 5690:12 5612:12 5609:19 5597:24 5594:20 5584:24 5581:18 5321:15 5305:15 5302:19 5292:15 5289:10 5279:15 5214:. 5197:15 5194:10 5083:15 4942:12 4929:12 4913:12 4631:. 4614:17 4599:18 4571:17 4565:18 4541:17 4535:18 4524:18 4496:17 4490:18 4485:17 4449:17 4420:18 4403:. 4299:bd 4295:bc 4291:ad 4287:bc 4283:ad 4279:bd 4110:2. 3898:22 3882:21 3869:21 3861:63 3845:63 3820:63 3800:50 3795:= 3790:20 3784:10 3779:= 3763:= 3758:10 3741:is 3674:, 3665:. 3644:. 3614:ce 3606:cd 3571:, 3294:. 3282:, 3154:28 3151:15 3044:of 3038:A 3018:20 3010:10 2977:20 2964:10 2924:20 2921:10 2885:40 2873:20 2860:10 2834:20 2826:10 2783:51 2773:26 2744:12 2735:26 2711:12 2696:26 2680:12 2665:10 2450:26 2421:12 2338:. 2323:68 2249:68 2190:17 2187:13 2158:21 2027:, 1833:11 1797:11 1442:A 1393:: 1357:10 1353:10 1349:10 1345:10 1332:. 1321:75 1284:A 1251:12 1211:A 1203:. 1198:17 1172:17 1076:. 1037:= 1011:A 986:A 966:−5 944:−8 939:, 834:A 640:em 636:en 632:em 624:en 612:UK 604:US 565:. 533:: 521:: 503:, 326:−2 320:−1 310:−2 288:−1 283:, 129:17 81:, 73:: 65:A 12340:/ 12265:e 12258:t 12251:v 12225:. 12203:. 12152:. 12113:. 12077:. 12045:. 12031:: 12004:. 11970:. 11966:: 11958:: 11933:. 11905:. 11880:. 11855:. 11830:. 11804:: 11772:. 11747:. 11699:. 11674:. 11645:. 11615:. 11572:x 11556:: 11533:. 11517:: 11498:. 11468:. 11459:4 11456:/ 11453:1 11449:+ 11410:. 11383:. 11345:. 11316:. 11268:. 11241:. 11237:: 11218:. 11188:. 11123:. 11083:. 11081:. 11036:. 11012:. 10984:. 10956:. 10907:⁄ 10893:/ 10685:N 10680:: 10647:Z 10642:: 10609:Q 10604:: 10571:R 10566:: 10533:C 10528:: 10456:b 10436:a 10412:b 10402:/ 10388:a 10307:( 10231:3 10227:5 10222:1 10218:3 10183:( 10171:9 10168:/ 10165:8 10155:9 10152:/ 10149:1 10139:5 10136:/ 10133:1 10123:4 10120:/ 10117:1 10090:० 10044:( 10033:( 10022:( 9974:( 9914:) 9908:( 9903:) 9899:( 9889:· 9882:· 9875:· 9868:· 9841:. 9798:2 9795:1 9744:2 9741:/ 9738:1 9659:5 9654:6 9648:9 9639:= 9627:9 9622:) 9617:5 9612:2 9606:3 9603:( 9600:3 9594:= 9586:2 9582:) 9576:5 9571:2 9568:( 9560:2 9555:3 9548:) 9543:5 9538:2 9532:3 9529:( 9526:3 9520:= 9512:5 9507:2 9501:3 9494:5 9489:2 9483:3 9469:5 9464:2 9461:+ 9458:3 9454:3 9449:= 9441:5 9436:2 9433:+ 9430:3 9426:3 9395:5 9390:6 9387:+ 9384:9 9375:= 9363:9 9358:) 9353:5 9348:2 9345:+ 9342:3 9339:( 9336:3 9330:= 9322:2 9318:) 9312:5 9307:2 9304:( 9296:2 9291:3 9284:) 9279:5 9274:2 9271:+ 9268:3 9265:( 9262:3 9256:= 9248:5 9243:2 9240:+ 9237:3 9230:5 9225:2 9222:+ 9219:3 9205:5 9200:2 9194:3 9190:3 9185:= 9177:5 9172:2 9166:3 9162:3 9124:7 9118:7 9113:3 9107:= 9101:7 9096:7 9084:7 9080:3 9075:= 9069:7 9065:3 8989:1 8983:x 8979:1 8974:+ 8968:1 8965:+ 8962:x 8958:1 8931:1 8923:2 8919:x 8913:x 8910:2 8872:2 8852:, 8846:, 8841:2 8819:, 8816:2 8812:/ 8781:2 8777:/ 8771:2 8711:x 8708:1 8699:1 8691:x 8688:1 8682:+ 8679:1 8645:3 8637:2 8633:x 8627:2 8624:+ 8621:x 8579:3 8573:x 8570:2 8567:+ 8562:2 8558:x 8552:x 8549:3 8508:3 8500:2 8496:x 8490:2 8487:+ 8484:x 8456:3 8450:x 8447:2 8444:+ 8439:2 8435:x 8429:x 8426:3 8379:P 8375:b 8371:a 8367:P 8363:D 8355:R 8347:R 8340:b 8336:a 8319:b 8311:b 8307:a 8300:b 8297:/ 8294:a 8269:) 8266:) 8259:d 8255:, 8248:c 8244:( 8241:+ 8238:) 8231:b 8227:, 8220:a 8216:( 8213:( 8207:) 8204:) 8201:d 8198:, 8195:c 8192:( 8189:+ 8186:) 8183:b 8180:, 8177:a 8174:( 8171:( 8148:) 8141:d 8137:, 8130:c 8126:( 8120:) 8117:d 8114:, 8111:c 8108:( 8086:) 8079:b 8075:, 8068:a 8064:( 8058:) 8055:b 8052:, 8049:a 8046:( 8011:, 8008:c 8005:b 8002:= 7999:d 7996:a 7986:) 7983:d 7980:, 7977:c 7974:( 7968:) 7965:b 7962:, 7959:a 7956:( 7922:. 7914:) 7911:b 7908:, 7905:b 7902:( 7887:, 7884:0 7878:a 7867:) 7864:a 7861:, 7858:b 7855:( 7852:= 7843:1 7836:) 7832:b 7829:, 7826:a 7823:( 7811:) 7808:b 7805:, 7802:0 7799:( 7776:) 7773:b 7770:, 7767:a 7761:( 7758:= 7751:) 7748:b 7745:, 7742:a 7739:( 7705:) 7702:0 7696:c 7688:( 7684:) 7681:c 7678:b 7675:, 7672:d 7669:a 7666:( 7663:= 7660:) 7657:d 7654:, 7651:c 7648:( 7642:) 7639:b 7636:, 7633:a 7630:( 7609:) 7606:d 7603:b 7600:, 7597:c 7594:a 7591:( 7588:= 7585:) 7582:d 7579:, 7576:c 7573:( 7567:) 7564:b 7561:, 7558:a 7555:( 7533:) 7530:d 7527:b 7524:, 7521:c 7518:b 7512:d 7509:a 7506:( 7503:= 7500:) 7497:d 7494:, 7491:c 7488:( 7482:) 7479:b 7476:, 7473:a 7470:( 7448:) 7445:d 7442:b 7439:, 7436:c 7433:b 7430:+ 7427:d 7424:a 7421:( 7418:= 7415:) 7412:d 7409:, 7406:c 7403:( 7400:+ 7397:) 7394:b 7391:, 7388:a 7385:( 7346:, 7343:0 7337:b 7317:a 7294:) 7291:b 7288:, 7285:a 7282:( 7252:/ 7242:x 7236:x 7226:x 7218:x 7199:x 7195:x 7187:d 7183:b 7179:c 7175:a 7169:d 7165:c 7159:b 7155:a 7142:x 7127:x 7112:x 7106:x 7057:5 7016:5 6964:. 6949:= 6936:1 6929:3 6926:/ 6923:1 6913:4 6910:/ 6907:1 6896:4 6888:3 6884:3 6873:3 6870:/ 6867:1 6852:4 6848:4 6837:4 6834:/ 6831:1 6802:3 6799:2 6793:= 6786:3 6780:2 6775:4 6769:1 6762:= 6756:3 6753:4 6741:2 6738:1 6732:= 6726:4 6723:3 6711:2 6708:1 6678:3 6675:2 6640:= 6633:5 6627:3 6597:3 6594:2 6571:5 6562:3 6528:. 6523:4 6520:1 6515:8 6512:= 6507:4 6504:9 6499:+ 6496:6 6493:= 6488:4 6485:3 6477:3 6474:+ 6471:2 6465:3 6462:= 6457:4 6454:3 6449:2 6443:3 6411:4 6408:1 6403:8 6400:= 6395:4 6387:= 6382:4 6378:3 6375:+ 6372:4 6366:2 6355:1 6352:3 6347:= 6342:4 6339:3 6334:2 6328:3 6291:4 6282:= 6276:4 6273:3 6261:1 6258:6 6252:= 6246:4 6243:3 6234:6 6194:2 6191:1 6186:= 6181:2 6178:1 6168:1 6165:1 6160:= 6153:2 6144:4 6136:1 6127:3 6112:1 6103:3 6095:1 6086:2 6078:= 6073:4 6070:3 6060:3 6057:2 6020:6 6015:= 6010:4 6007:3 5997:3 5994:2 5956:. 5950:4 5947:1 5941:1 5938:= 5933:) 5925:4 5922:3 5913:1 5908:( 5903:+ 5900:) 5897:1 5891:2 5885:4 5882:( 5879:= 5873:4 5870:3 5864:2 5858:4 5829:6 5826:1 5820:= 5814:6 5811:3 5799:6 5796:4 5790:= 5784:2 5781:1 5769:3 5766:2 5710:= 5695:+ 5687:9 5682:= 5676:6 5670:2 5665:5 5659:2 5653:+ 5647:3 5641:4 5636:3 5630:3 5624:= 5604:= 5589:+ 5576:= 5570:6 5564:4 5559:5 5553:4 5547:+ 5541:6 5535:4 5530:6 5524:3 5518:= 5509:6 5506:5 5501:+ 5496:4 5493:3 5460:6 5457:5 5431:4 5428:3 5398:d 5395:b 5390:b 5387:c 5384:+ 5381:d 5378:a 5372:= 5367:d 5364:c 5359:+ 5354:b 5351:a 5318:4 5313:1 5310:= 5297:= 5284:+ 5276:9 5246:3 5243:2 5238:+ 5233:5 5230:3 5188:= 5182:5 5179:5 5167:3 5164:2 5146:3 5143:/ 5140:2 5127:3 5124:/ 5121:3 5111:3 5108:/ 5105:3 5080:9 5074:= 5068:3 5065:3 5053:5 5050:3 5022:5 5019:3 4991:3 4988:2 4983:+ 4978:5 4975:3 4947:. 4939:7 4934:= 4926:4 4921:+ 4910:3 4905:= 4899:4 4893:3 4888:4 4882:1 4876:+ 4867:3 4861:4 4856:3 4850:1 4844:= 4839:3 4836:1 4831:+ 4823:4 4820:1 4775:4 4772:1 4746:2 4743:1 4724:. 4709:4 4706:1 4700:1 4697:= 4691:4 4688:5 4682:= 4676:4 4673:3 4667:+ 4661:4 4658:2 4611:4 4596:5 4530:4 4479:5 4455:, 4446:4 4417:5 4388:6 4385:3 4373:6 4370:4 4344:2 4341:1 4315:3 4312:2 4261:d 4255:b 4250:c 4244:b 4216:d 4210:b 4205:d 4199:a 4172:d 4169:c 4143:b 4140:a 4101:3 4087:b 4083:a 4073:= 4066:b 4059:a 4044:4 4037:2 4024:4 4017:3 3987:4 3960:4 3957:2 3945:4 3942:3 3895:3 3889:= 3854:= 3823:/ 3803:/ 3787:/ 3774:2 3771:/ 3768:1 3755:/ 3752:5 3725:8 3722:3 3696:9 3693:3 3649:c 3638:e 3634:/ 3630:d 3622:c 3610:/ 3588:e 3585:c 3582:= 3579:b 3559:d 3556:c 3553:= 3550:a 3528:c 3503:b 3500:a 3475:2 3472:/ 3469:1 3459:4 3456:/ 3453:2 3443:2 3440:/ 3437:1 3427:4 3424:/ 3421:2 3399:2 3396:1 3368:n 3365:n 3339:n 3336:n 3313:n 3248:5 3244:/ 3240:7 3235:4 3231:/ 3227:3 3198:7 3195:5 3183:4 3180:3 3145:= 3139:7 3136:5 3124:4 3121:3 3095:7 3092:5 3066:4 3063:3 3024:. 3021:) 3014:/ 3007:( 3002:/ 2997:5 2972:/ 2967:) 2960:/ 2956:5 2953:( 2930:= 2913:/ 2908:5 2882:1 2876:= 2868:/ 2857:5 2830:/ 2822:/ 2818:5 2791:. 2778:= 2770:1 2760:4 2756:3 2753:+ 2750:4 2738:= 2727:4 2723:3 2720:+ 2717:4 2705:= 2689:4 2686:3 2670:, 2662:3 2657:= 2652:5 2649:1 2639:2 2636:3 2631:= 2628:5 2620:2 2617:3 2612:= 2607:5 2598:2 2595:3 2583:, 2578:2 2575:3 2570:= 2565:1 2562:3 2552:2 2549:1 2544:= 2539:3 2536:1 2526:2 2523:1 2518:= 2507:3 2504:1 2493:2 2490:1 2445:/ 2438:) 2430:4 2427:3 2416:( 2390:3 2386:/ 2382:1 2377:2 2373:/ 2369:1 2350:. 2320:1 2314:+ 2308:6 2305:1 2299:+ 2293:4 2290:1 2284:+ 2278:3 2275:1 2246:1 2240:+ 2234:4 2231:1 2225:+ 2219:2 2216:1 2164:. 2155:1 2149:+ 2143:6 2140:1 2134:+ 2128:2 2125:1 2099:7 2096:5 2070:4 2067:3 2041:3 2038:2 2012:2 2009:1 1979:3 1976:1 1970:+ 1964:2 1961:1 1922:q 1918:/ 1914:p 1891:2 1888:/ 1885:1 1881:+ 1863:. 1857:4 1854:3 1848:+ 1845:2 1842:= 1836:4 1806:. 1800:4 1791:= 1785:4 1782:3 1776:+ 1770:4 1767:8 1761:= 1755:4 1752:3 1746:+ 1743:2 1713:. 1708:) 1700:4 1697:3 1691:+ 1688:2 1683:( 1655:4 1652:3 1646:2 1626:4 1623:/ 1620:3 1616:+ 1597:4 1594:3 1588:+ 1585:2 1562:4 1559:3 1536:2 1516:4 1512:/ 1508:3 1501:2 1478:4 1475:3 1469:2 1377:3 1374:/ 1371:1 1343:× 1324:/ 1317:+ 1267:3 1264:/ 1261:1 1248:/ 1245:4 1220:n 1195:/ 1192:1 1178:1 1175:/ 1159:7 1156:/ 1153:7 1136:3 1133:/ 1130:7 1120:7 1117:/ 1114:3 1053:. 1048:2 1045:/ 1042:1 1032:8 1029:/ 1026:1 1003:7 1000:/ 997:1 963:/ 960:8 950:5 947:/ 934:5 931:/ 928:8 918:2 915:/ 912:1 905:b 897:b 893:a 874:b 871:a 856:b 854:/ 852:a 813:/ 810:6 797:/ 794:1 776:1 773:/ 770:3 756:5 753:/ 750:1 740:5 737:/ 734:2 720:1 717:/ 714:3 696:5 693:/ 690:3 680:5 677:/ 674:2 657:2 654:/ 651:1 610:( 602:( 595:9 591:4 583:3 580:/ 577:1 548:5 545:/ 542:8 476:x 473:1 444:2 440:2 422:b 418:/ 414:a 385:Q 372:Q 368:b 364:b 360:a 352:b 348:/ 344:a 323:/ 307:/ 304:1 294:2 291:/ 278:2 275:/ 272:1 262:2 259:/ 256:1 246:2 243:/ 240:1 223:4 220:/ 217:3 196:4 193:/ 190:3 180:4 177:/ 174:3 158:2 154:1 132:3 103:2 100:1 56:4 53:/ 50:1 34:. 20:)

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