Fraction - Knowledge

2794: 12307: 2462: 7925: 5723: 2789:{\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}}} 9816: 4718: 12276: 29: 7716: 5471: 9666: 9402: 9712:: 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. 6030:

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.

6195: 7920:{\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}}} 5718:{\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}}} 9408: 9144: 4946: 3477:

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

9760:, 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 ( 8023:

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

4788:

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

4623:

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

3258:

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

1422:

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

8654:

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,

11271:

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

6029:

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

1058:

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

4632:

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

5740:

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

4114:

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

690:

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

6814:

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

6804: 6039: 1206:

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

5405:

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

4109: 7709:

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:

9125: 6412: 1386:, 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. 1917:

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

9801:. 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. 2451:

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

6527: 11563:, 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. 66:, "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 5955: 5831: 9661:{\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}}} 9397:{\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}}} 6293: 158:, 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 3900: 1279:

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

1228:

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

4802: 9671:

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.

5464:

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.

1298:). 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 7704: 6999:= 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: 1805: 5322: 4711: 2325: 717:

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

10058:, 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 10365:'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 8268: 5199: 3156: 5085: 6024: 1423:

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.

1893:. 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 5400: 2163: 2251: 10915:

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.

6190:{\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}}} 8716: 6689: 2467: 2449: 750:.) 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 9037:), 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 2887: 6204:

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.

6651: 1447:) 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. 1866:

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

8148: 3294:

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

10312:

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

8010: 3998: 3200: 1712: 4616: 1134:

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.

7721: 5476: 1862: 9047: 4544: 4499: 4390: 3962: 8991: 8086: 6963: 446: 11272:

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 ...

1981: 8647: 8510: 2932: 1302:

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,

5247: 4992: 4264: 4219: 10233: 6927:

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

6312: 1084:, 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. 6537:

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,

478: 8781: 8581: 8458: 7608: 7533: 7448: 2799:

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

967:

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

6570: 8933: 3251: 2393: 1410:

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

8851: 6427: 4454: 3023: 2192: 1657: 1599: 4422: 2976: 1480: 134: 10685: 10647: 10609: 10571: 10533: 6680: 6599: 5462: 5433: 5024: 4777: 4748: 4346: 4317: 4174: 4145: 3727: 3698: 3505: 3401: 3370: 3341: 3097: 3068: 2101: 2072: 2043: 2014: 1564: 876: 105: 9799: 385: 174:, 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 4570: 5732:

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.

7345: 3286:

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.

1515: 6984:

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

5842: 2833: 11559:

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

5747: 7293: 6218: 8818: 3587: 3558: 4941:{\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}}.} 3826: 3158:. 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 10455: 10435: 10411: 10387: 8871: 7316: 3986: 3527: 3312: 1535: 1059:

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

698:

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,

11316: 3372:

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

770:

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

7111:

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:

11599: 7614: 7259:

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

1727: 256:

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), −

5258: 3657:

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

11060: 6033:

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

4639: 2256: 4633:

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

11653: 11615: 1225:

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.

12155: 11859: 11834: 12252: 11806: 5145: 3102: 794:

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

12179: 5031: 3035:, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see 2346:, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, 10173: 1811:, with the proper fraction consisting of the remainder divided by the divisor. For example, since 4 goes into 11 twice, with 3 left over, 714:

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").

557:

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

10966: 10346: 9930: BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with 5976: 2934:

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

497: 11362: 6799:{\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}}} 5333: 2106: 623:" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow 304:

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

11678: 11412: 8743:

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

2197: 997:). 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. 6923:

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

9718:: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them 8660: 1261:

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

12056: 2398: 11751: 8404:. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are 4278:

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

11295: 2838: 10362: 6604: 4572:– only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is 1153:), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper. 4104:{\displaystyle {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-2.} 12136: 12097: 12029: 11889: 11716: 11482: 11423: 11356: 11252: 11202: 11107: 11054: 10996: 7940: 4115:

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

9120:{\displaystyle {\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}} 3161: 1662: 1076:

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

11579: 4577: 8155: 4456:

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

1814: 8597:

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

4504: 4459: 4351: 3923: 12245: 10195:. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, 9880: 8940: 6930: 3644:

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

774:. (For example, 1/2 may be read "one-half", "one half", or "one over two".) Fractions with large denominators that are 420: 32:

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

11533:

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

11471: 6973: 3992:

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

655: 11917: 11517: 11394: 10940: 10192: 9899: 9852: 6407:{\displaystyle 3\times 2{\frac {3}{4}}={\frac {3}{1}}\times {\frac {2\times 4+3}{4}}={\frac {33}{4}}=8{\frac {1}{4}}} 1942: 11982: 10461:

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

8602: 8465: 6212:

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

4118:

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

2892: 12459: 12013: 9920:: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians used 9041:

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

8091: 5212: 4957: 4224: 4179: 11324: 11094:

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

10199: 2165:

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

12004:

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

1294: 9859: 6988:

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

1176:. Therefore, every fraction or integer, except for zero, has a reciprocal. For example, the reciprocal of 17 is 651:) was known as a "case fraction", while those representing only part of fraction were called "piece fractions". 489: 9837: 9004: 8888:

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

7126:

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

1156:

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

454: 8752: 8530: 8407: 7539: 7454: 7369: 12238: 11781: 10180: 9030: 9020: 8030: 4779:

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

9738:, but rendered with the same height as other characters. Some sources include all rendering of fractions as 6540: 3478:

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

1418:

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

9034: 8891: 8020: 3917:

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

1889:, and remain widespread in daily life and in trades, especially in regions that do not use the decimalized 224:

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

10237:

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

9866: 6522:{\displaystyle 3\times 2{\frac {3}{4}}=3\times 2+3\times {\frac {3}{4}}=6+{\frac {9}{4}}=8{\frac {1}{4}}.} 1566:

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

493: 11935: 11700: 10066:, it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, 10000:", which contains work on the theory of numbers, arithmetical operations, and operations with fractions. 3207: 2349: 11645: 11623: 9934:. Their methods gave the same answer as modern methods. The Egyptians also had a different notation for 12151: 11855: 11830: 10317: 8823: 4427: 2981: 2168: 1627: 1569: 6306:

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

4398: 2937: 1453: 1443:) is the sum of a non-zero integer and a proper fraction, conventionally written by juxtaposition (or 110: 12418: 11099: 10664: 10626: 10588: 10550: 10512: 9848: 9833: 9757: 6656: 6575: 5438: 5409: 5000: 4753: 4724: 4322: 4293: 4150: 4121: 3703: 3674: 3481: 3377: 3346: 3317: 3073: 3044: 2077: 2074:

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

2048: 2019: 1990: 1620:

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

1540: 852: 81: 20: 9777: 8877:, and thus considered as coefficients. These same numbers, however, are not rational fractions with 11773: 1094: 417:

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

217:

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

12052: 10309: 10256:

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

9681: 1289: 1098:

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

368: 12406: 12172: 9826: 9705:

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

4549: 3641: 3437:. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ( 1808: 1351:

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

9029:

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

6298:

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

5950:{\displaystyle 4-2{\tfrac {3}{4}}=(4-2-1)+{\bigl (}1-{\tfrac {3}{4}}{\bigr )}=1{\tfrac {1}{4}}.} 12288: 10962: 10480: 9913: 8320:

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

7931: 7360: 7070:), one may write the number as the sum of the non-repeating and repeating parts, respectively: 6683: 5826:{\displaystyle {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}} 1131: 198: 11269: 7321: 6288:{\displaystyle 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}} 1485: 12219: 11550: 11384: 11242: 11170: 11046: 10818: 9131: 8786: 5729: 3276: 2802: 11470: 11411: 11346: 3895:{\displaystyle {\tfrac {63}{462}}={\tfrac {63\,\div \,21}{462\,\div \,21}}={\tfrac {3}{22}}} 1450:

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

12401: 11944: 11674: 10470: 8401: 8016: 7266: 3659: 2990: 2960: 2901: 2856: 2433: 1919: 10987:

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

3563: 3534: 8: 10281: 10055: 9135: 8792: 8740: 8515: 3906: 1325: 12046: 11948: 9764:) or within exponents to increase legibility. Fractions written this way, also known as 5836:

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

3804:

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

12364: 12349: 12207: 11798: 11747: 11546: 10985: 10440: 10420: 10396: 10372: 10238: 9974: 9971: 9946: 8856: 8744: 8728: 8391: 8339: 7301: 7081:

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

4796:

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

3971: 3512: 3297: 1520: 1411: 1400: 1061: 11287: 9873: 3700:

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

12379: 12132: 12093: 12025: 11913: 11885: 11722: 11712: 11708: 11513: 11478: 11419: 11390: 11352: 11248: 11198: 11103: 11050: 10992: 10936: 10928: 10875: 10802: 10773: 10245: 10041: 9921: 9768:, are written all on one typographical line, but take 3 or more typographical spaces. 9000: 8736: 8379: 8375: 6992: 6985: 6979: 1936: 1717: 1285: 449: 11161:

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

10341:, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), 5970:

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

12423: 12369: 12017: 11952: 11790: 11542: 11503: 11223: 10841: 10811: 10731: 10716: 9993: 9957: 9138:

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

8885: 7699:{\displaystyle (a,b)\div (c,d)=(ad,bc)\quad ({\text{with, additionally, }}c\neq 0)} 3732:

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

3280: 1800:{\displaystyle 2+{\tfrac {3}{4}}={\tfrac {8}{4}}+{\tfrac {3}{4}}={\tfrac {11}{4}}.} 1275: 961: 592: 359:

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

151: 9752:: 1/2, so called because this notation was used for pre-decimal British currency ( 9742:

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

5317:{\displaystyle {\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}} 12413: 12374: 12159: 11192: 11174: 11040: 10970: 10766: 10583: 10338: 10249: 9997: 9935: 8732: 8332: 6991:

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

4266:(where the dot signifies multiplication and is an alternative symbol to ×). Then 2336: 1894: 1352: 1299: 1281: 1130:. The product of a fraction and its reciprocal is 1, hence the reciprocal is the 1002: 836: 751: 620: 612: 600: 551: 547: 543: 324: 11554: 11227: 10361:

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

4706:{\displaystyle {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}} 3820:

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

3031:

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

2320:{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}} 542:

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

10759: 10659: 10507: 10334: 10305: 10257: 9954: 9942: 9840: in this History section. Unsourced material may be challenged and removed. 8996: 7356: 3645: 1379: 1073: 604: 596: 519: 507: 59: 10030: 9134:

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

8019:

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

12453: 12438: 12359: 12005: 11905: 11726: 10500: 10342: 10285: 9931: 8731:

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

7037: 3405:. When the numerator and denominator are both multiplied by 2, the result is 1984: 1890: 977: 960:. The term was originally used to distinguish this type of fraction from the 12129:

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

12021: 11012: 5026:

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

3909:

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

12389: 12384: 12354: 11933:

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

10902: 10709: 10273: 9950: 9695: 7260: 7033: 5140:

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

3650: 1047: 1007: 632: 12051:. Vol. 1. La Salle, Illinois: Open Court Publishing Company. p. 11430:

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

10888: 10169: 4790: 1807:

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

587:). These marks are respectively known as the horizontal bar; the virgule, 11978: 11244:

Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation

10545: 10289: 10277: 10019: 8874: 7352: 6418: 3609:, which can be reduced by dividing both the numerator and denominator by 3272: 3268: 1262: 1137:

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

971:; though, unless irrational, they can be evaluated to a common fraction. 12012:. Boston Series in the Philosophy of Science. Vol. 238. Dordrecht: 11477:. American Mathematical Society. §14.3 Mixed Numbers, pp. 225–227. 9690:, simple fractions are sometimes printed as a single character, e.g. ½ ( 1087: 514:, "counter" or "numberer"), and the type or variety of the parts is the 12433: 12428: 12230: 11802: 10067: 10008: 9982: 9960: 9687: 8522: 8347: 5194:{\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}} 3151:{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}} 1375: 1371:= 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + .... 771: 608: 588: 11508: 7066:

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

5080:{\displaystyle {\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}} 10297: 1624:. Subtraction or negation is applied to the entire mixed numeral, so 11956: 11794: 9815: 5741:

subtracting the numerators of the original fractions. For instance,

12296: 10931:(2003). "CRC Concise Encyclopedia of Mathematics, Second Edition". 10485: 10475: 10350: 9978: 9691: 9038: 9026: 9016: 8397: 7348: 7202:

Continue the subtraction operation to clear the repeating decimal:

6907:

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

6019:{\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}} 1419: 1415: 1391: 889: 734:

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

390: 11288:"Ask Dr. Math: Can Negative Fractions Also Be Proper or Improper?" 8350:

in one indeterminate, with coefficients from some integral domain

7036:

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

12396: 12292: 12275: 11218:

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

10621: 10313: 10188: 9985:, who is said to have been executed for revealing this fact.) In 9917: 9699: 8312:

are often taken as uniquely determined representatives for their

8302: 7296: 6201: 5395:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}} 4717: 2158:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.} 1399:(ppt) has an implied denominator of 1000, while the more general 810:

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

692: 654:

The denominators of English fractions are generally expressed as

616: 526:, "thing that names or designates"). As an example, the fraction 6682:. To divide a number by a fraction, multiply that number by the 2246:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}} 502:

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

28: 11502:. OBP Series in Mathematics. Open Book Publishers. p. 89. 11351:(2nd ed., reprinted ed.). Cheltenham: Thornes. p. 5. 11275:

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

10265: 12127:

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

9753: 8711:{\displaystyle {\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}} 327:

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

12265: 10895: 10284:

many centuries before Stevin and that the Persian astronomer

10168:

The horizontal fraction bar is first attested in the work of

6967: 6883:"), and stop when the desired accuracy is obtained, e.g., at 5965: 2444:{\displaystyle {\bigl (}12{\tfrac {3}{4}}{\bigr )}{\big /}26} 1202: 194: 12178:. Common Core State Standards Initiative. 2010. p. 85. 10906:, although in fact both are synonyms for the standard slash. 9977:. (This is commonly though probably erroneously ascribed to 6809: 1939:

is the sum of distinct positive unit fractions, for example

10693: 9989: 6207: 2882:{\displaystyle {\tfrac {5}{10}}{\big /}20={\tfrac {1}{40}}} 1886: 3259:

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

10701: 10184: 10040:). Their works form fractions by placing the numerators ( 6646:{\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}} 658:, in the plural if the numerator is not 1. (For example, 498:

Unicode subscripts and superscripts § Fraction slash

448:), and even do not represent any number (for example the 778:

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

11247:. Springer Science & Business Media. pp. 87–. 6417:

Alternately, mixed numbers can be treated as sums, and

1292:(·), a comma) depends on the locale (for examples, see 1211:". For example, if a car lot had 12 vehicles, of which 1046:

In Unicode, precomposed fraction characters are in the

12008:; Cohen, Robert S.; Renn, Jürgen; et al. (eds.). 11473:

Understanding Numbers in Elementary School Mathematics

11285: 11150:(1st ed.). Oxford: Oxford University Press. 1913. 11131:(1st ed.). Oxford: Oxford University Press. 1917. 8795: 8756: 8694: 8674: 8005:{\displaystyle (a,b)\sim (c,d)\quad \iff \quad ad=bc,} 7263:. Mathematicians define a fraction as an ordered pair 6941: 6785: 6754: 6739: 6724: 6709: 6694: 6661: 6632: 6609: 6580: 6545: 6274: 6259: 6244: 6229: 5933: 5908: 5856: 5812: 5797: 5782: 5767: 5752: 5443: 5414: 5180: 5165: 5150: 5066: 5051: 5036: 5005: 4758: 4729: 4692: 4674: 4659: 4644: 4597: 4582: 4509: 4464: 4432: 4403: 4371: 4356: 4327: 4298: 4229: 4184: 4155: 4126: 4065: 4045: 4023: 4003: 3943: 3928: 3881: 3846: 3831: 3708: 3679: 3486: 3382: 3351: 3322: 3212: 3181: 3166: 3137: 3122: 3107: 3078: 3049: 2907: 2868: 2843: 2672: 2581: 2490: 2476: 2413: 2354: 2306: 2291: 2276: 2261: 2232: 2217: 2202: 2173: 2141: 2126: 2111: 2082: 2053: 2024: 1995: 1962: 1947: 1840: 1819: 1783: 1768: 1753: 1738: 1683: 1638: 1580: 1545: 1461: 857: 818: 458: 424: 115: 86: 10667: 10629: 10591: 10553: 10515: 10443: 10423: 10399: 10375: 10345:, pie-shaped pieces, plastic rectangles, grid paper, 10280:. It is true that decimal fractions were used by the 10202: 9780: 9411: 9147: 9050: 8943: 8894: 8859: 8826: 8755: 8663: 8605: 8533: 8468: 8410: 8158: 8094: 8033: 7943: 7719: 7617: 7542: 7457: 7372: 7324: 7304: 7269: 6933: 6692: 6659: 6607: 6578: 6543: 6430: 6315: 6221: 6042: 5979: 5845: 5750: 5474: 5441: 5412: 5336: 5261: 5215: 5148: 5034: 5003: 4960: 4805: 4756: 4727: 4642: 4580: 4552: 4507: 4462: 4430: 4401: 4354: 4325: 4296: 4227: 4182: 4153: 4124: 4001: 3974: 3926: 3829: 3706: 3677: 3648:. One says that the fraction has been reduced to its 3566: 3537: 3515: 3484: 3380: 3349: 3320: 3300: 3210: 3195:{\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}} 3164: 3105: 3076: 3047: 2984: 2940: 2895: 2841: 2805: 2465: 2401: 2352: 2259: 2200: 2171: 2109: 2080: 2051: 2022: 1993: 1945: 1817: 1730: 1707:{\displaystyle -{\bigl (}2+{\tfrac {3}{4}}{\bigr )}.} 1665: 1630: 1572: 1543: 1523: 1488: 1456: 855: 457: 423: 371: 113: 84: 11884:(6th ed.). Philadelphia: Saunders College Pub. 10957:

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

9938:, used for certain systems of weights and measures. 8278:

In the case of fractions of integers, the fractions

7254: 6974:

Decimal representation § Conversion to fraction

4611:{\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}} 1268: 635:, a piece of type bearing a complete fraction (e.g. 8263:{\displaystyle ((a,b)+(c,d))\sim ((a',b')+(c',d'))} 7089:Alternatively, algebra can be used, such as below: 3472: 1857:{\displaystyle {\tfrac {11}{4}}=2+{\tfrac {3}{4}}.} 1006:is a common fraction in which the denominator is a 11045:(2nd ed.). Lausanne: AVA Publishing. p. 10984: 10679: 10641: 10603: 10565: 10527: 10449: 10429: 10405: 10381: 10227: 9793: 9660: 9396: 9119: 8985: 8927: 8865: 8845: 8812: 8775: 8710: 8641: 8575: 8504: 8452: 8262: 8142: 8080: 8004: 7919: 7698: 7602: 7527: 7442: 7339: 7310: 7287: 6957: 6798: 6674: 6645: 6593: 6564: 6521: 6406: 6287: 6189: 6018: 5949: 5825: 5728:The smallest possible denominator is given by the 5717: 5456: 5427: 5394: 5316: 5241: 5193: 5079: 5018: 4986: 4940: 4771: 4742: 4705: 4610: 4564: 4539:{\displaystyle {\tfrac {18\times 4}{18\times 17}}} 4538: 4494:{\displaystyle {\tfrac {5\times 17}{18\times 17}}} 4493: 4448: 4416: 4385:{\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}} 4384: 4340: 4311: 4258: 4213: 4168: 4139: 4103: 3980: 3957:{\displaystyle {\tfrac {3}{4}}>{\tfrac {2}{4}}} 3956: 3894: 3721: 3692: 3581: 3552: 3521: 3499: 3395: 3364: 3335: 3306: 3245: 3194: 3150: 3091: 3062: 3017: 2970: 2926: 2881: 2827: 2788: 2443: 2387: 2319: 2245: 2186: 2157: 2095: 2066: 2037: 2008: 1975: 1856: 1799: 1706: 1651: 1593: 1558: 1529: 1509: 1474: 1407:(ppm), means that the proportion is 75/1,000,000. 981:is a common fraction with a numerator of 1 (e.g., 870: 611:, fractions stacked vertically are also known as " 472: 440: 379: 128: 99: 11698: 10007:seems to have originated in India in the work of 8986:{\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-1}}} 6958:{\displaystyle 12.3456={\tfrac {123456}{10000}}.} 2593: 2579: 2488: 2474: 2330: 1983:. This definition derives from the fact that the 441:{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}} 140:, displayed above a line (or before a slash like 12451: 12092:. New Jersey: Princeton University Press. 1986. 11910:Mathematical Puzzles: A Connoisseur's Collection 8338:, in which case a fraction is an element of the 7085:1523 / 10000 + 987 / 9990000 = 1522464 / 9990000 1053: 11979:"Earliest Uses of Various Mathematical Symbols" 11601:A new mathematical and philosophical dictionary 8935:can be decomposed as the sum of two fractions: 7141:Subtract the two equations from each other (if 6131: 6114: 6090: 6073: 1976:{\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}} 490:Numeral (linguistics) § Fractional numbers 11997: 11409: 11190: 10961:, Volume 58, Issue 03 (March 2011), 10049: 8642:{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} 8514:. Algebraic fractions are subject to the same 8505:{\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} 4951:Consider adding the following two quantities: 2927:{\displaystyle 5{\big /}{\tfrac {10}{20}}=10.} 1537:representing the whole cakes and the fraction 1348:moves the decimal point 7 places to the left. 1324:Decimal fractions can also be expressed using 1288:, the appearance of which (e.g., a period, an 1068:In general, a common fraction is said to be a 494:English numerals § Fractions and decimals 12246: 12173:"Common Core State Standards for Mathematics" 11882:An introduction to the history of mathematics 11240: 11038: 10268:in 1585, together with a French translation, 9975:cannot be expressed as a fraction of integers 8354:, are themselves an integral domain, call it 8143:{\displaystyle \quad (c,d)\sim (c',d')\quad } 6995:) over the digits that repeat, for example 0. 5921: 5896: 5242:{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}} 4987:{\displaystyle {\frac {3}{5}}+{\frac {2}{3}}} 4259:{\displaystyle {\tfrac {b\cdot c}{b\cdot d}}} 4214:{\displaystyle {\tfrac {a\cdot d}{b\cdot d}}} 2426: 2404: 1721: 1696: 1671: 12114:Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī 11699:Schoenborn, Barry; Simkins, Bradley (2010). 11234: 11186: 11184: 10959:Notices of the American Mathematical Society 10228:{\displaystyle {\frac {3\quad 1}{5\quad 3}}} 10101:and would be written in modern notation as 6 7223:Divide both sides by 9,990,000 to represent 6301: 5327:This method can be expressed algebraically: 3262: 892:. As with other fractions, the denominator ( 561:. The fraction bar may be horizontal (as in 12131:. Princeton University Press. p. 518. 11750:. Encyclopedia of Mathematics. 2012-04-06. 11348:New comprehensive mathematics for 'O' level 10982: 10337:, fractions have been demonstrated through 9675: 7863:multiplicative inverse fractions, for 5121:does not change the value of the fraction. 4783: 4270:is a common denominator and the numerators 1088:Reciprocals and the "invisible denominator" 835:, where vulgar is Latin for "common") is a 577:), oblique (as in 2/5), or diagonal (as in 12253: 12239: 11573: 11571: 11532: 11286:Laurel Brenner; Peterson (31 March 2004). 11261: 11217: 11093: 10976: 10671: 10633: 10595: 10557: 10519: 10276:(1548–1620), then settled in the Northern 10003:A modern expression of fractions known as 8316:fractions, which are considered to be the 7982: 7978: 6968:Converting repeating decimals to fractions 5966:Multiplying a fraction by another fraction 2592: 2578: 2487: 2473: 1172:, where 1 is sometimes referred to as the 193:Other uses for fractions are to represent 12003: 11646:"Complex fraction definition and meaning" 11593: 11591: 11507: 11222:. Palgrave Macmillan UK. pp. 13–25. 11181: 11160: 11087: 11034: 11032: 10927: 10673: 10635: 10597: 10559: 10521: 10356: 9900:Learn how and when to remove this message 8521:If the numerator and the denominator are 7524: 7439: 6810:Converting between decimals and fractions 3869: 3865: 3856: 3852: 3099:is a compound fraction, corresponding to 2835:could be plausibly interpreted as either 1495: 550:, and the denominator corresponds to the 473:{\displaystyle \textstyle {\frac {1}{x}}} 373: 12260: 12126: 11497: 10935:. Chapman & Hall/CRC. p. 1925. 8995:. This is useful for the computation of 8785:, are also rational fractions, as are a 8776:{\displaystyle \textstyle {\sqrt {2}}/2} 8576:{\displaystyle {\frac {3x}{x^{2}+2x-3}}} 8453:{\displaystyle {\frac {3x}{x^{2}+2x-3}}} 8275:and similarly for the other operations. 7603:{\displaystyle (a,b)\cdot (c,d)=(ac,bd)} 7528:{\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,} 7443:{\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,} 6208:Multiplying a fraction by a whole number 4716: 1716:Any mixed number can be converted to an 27: 12212:The Online Encyclopaedia of Mathematics 11972: 11970: 11968: 11966: 11904: 11577: 11568: 11500:Teaching Mathematics at Secondary Level 11135: 11102:: Hartley & Marks. pp. 81–82. 10991:. Mathematical Association of America. 10933:CRC Concise Encyclopedia of Mathematics 9035:Simplified form of a radical expression 8396:An algebraic fraction is the indicated 8081:{\displaystyle (a,b)\sim (a',b')\quad } 3289: 3267:Like whole numbers, fractions obey the 240:represents a half-dollar profit, then − 12452: 12090:A Source Book in Mathematics 1200–1800 12076: 12044: 11976: 11932: 11772:Galen, Leslie Blackwell (March 2004). 11597: 11588: 11434:, then the fraction. The mixed number 11382: 11267: 11081: 11029: 10457:is a positive whole number. (The word 10363:Common Core State Standards Initiative 10323: 10054:), but without a bar between them. In 9761: 9010: 8385: 6565:{\displaystyle {\tfrac {10}{3}}\div 5} 3912: 3202:is equivalent to the complex fraction 12234: 11985:from the original on 20 February 2016 11771: 11765: 11386:The Complete Idiot's Guide to Algebra 11344: 8928:{\displaystyle {\frac {2x}{x^{2}-1}}} 8727:The field of rational numbers is the 8585:, the algebraic fraction is called a 1925: 1336:, which represents 0.0000006023. The 813: 603:); and the fraction bar, solidus, or 11963: 11879: 11873: 11298:from the original on 9 November 2014 11197:. Courier Corporation. p. 219. 11039:Ambrose, Gavin; et al. (2006). 10328: 9838:adding citations to reliable sources 9809: 8518:properties as arithmetic fractions. 3735:Using these rules, we can show that 3421:, which has the same value (0.5) as 3343:equals 1. Therefore, multiplying by 1930: 12048:A History of Mathematical Notations 12010:History of Science, History of Text 11116: 4546:. It is not necessary to calculate 3246:{\displaystyle {\tfrac {3/4}{7/5}}} 3037: 2388:{\displaystyle {\tfrac {1/2}{1/3}}} 896:) cannot be zero. Examples include 819:Simple, common, or vulgar fractions 546:, the numerator corresponds to the 13: 12152:"MacTutor's al-Uqlidisi biography" 11774:"Putting Fractions in Their Place" 11547:10.1111/j.1949-8594.2000.tb17254.x 11468: 11321:New England Compact Math Resources 11241:Jack Williams (19 November 2011). 10353:, counters and computer software. 9130:The process of rationalization of 619:fractions", and diagonal ones as " 14: 12471: 12200: 11977:Miller, Jeff (22 December 2014). 11096:The Elements of Typographic Style 11084:, "275. The solidus", pp. 312–314 10878:mistakenly distinguish the slash 10292:fractions with great ease in his 10193:Islamic inheritance jurisprudence 9698:for information on doing this in 8846:{\displaystyle {\sqrt {2}},\pi ,} 7255:Fractions in abstract mathematics 5960: 4449:{\displaystyle {\tfrac {4}{17}},} 3473:Simplifying (reducing) fractions 3018:{\displaystyle 5{\big /}(10/20).} 2187:{\displaystyle {\tfrac {13}{17}}} 1652:{\displaystyle -2{\tfrac {3}{4}}} 1594:{\displaystyle 2+{\tfrac {3}{4}}} 1328:with negative exponents, such as 1269:Decimal fractions and percentages 12305: 12274: 11414:Ready for Fractions and Decimals 11410:Wingard-Nelson, Rebecca (2014). 10881: 10094:1 1 −1 9945:used unit fractions and (later) 9814: 4417:{\displaystyle {\tfrac {5}{18}}} 4395:For the more laborious question 2971:{\displaystyle (5/10){\big /}20} 1475:{\displaystyle 2{\tfrac {3}{4}}} 1426: 1374:Another kind of fraction is the 129:{\displaystyle {\tfrac {17}{3}}} 12185:from the original on 2013-10-19 12165: 12145: 12120: 12106: 12082: 12070: 12059:from the original on 2014-04-14 12038: 11926: 11898: 11862:from the original on 2013-05-21 11848: 11837:from the original on 2013-05-26 11823: 11812:from the original on 2011-07-13 11754:from the original on 2014-10-21 11740: 11692: 11681:from the original on 2018-03-14 11667: 11656:from the original on 2017-12-01 11638: 11608: 11581:A complete system of arithmetic 11526: 11491: 11462: 11403: 11376: 11365:from the original on 2019-01-19 11338: 11309: 11279: 11211: 11154: 11063:from the original on 2016-03-04 10909: 10680:{\displaystyle :\;\mathbb {N} } 10642:{\displaystyle :\;\mathbb {Z} } 10604:{\displaystyle :\;\mathbb {Q} } 10566:{\displaystyle :\;\mathbb {R} } 10528:{\displaystyle :\;\mathbb {C} } 10272:, by the Flemish mathematician 10218: 10209: 10097:4 5 9 10091:6 1 2 10083:४ ५ ९ 10077:१ १ १ 10074:६ १ २ 9825:needs additional citations for 8139: 8095: 8077: 7983: 7977: 7907: as multiplicative unities 7675: 6675:{\displaystyle {\tfrac {2}{3}}} 6594:{\displaystyle {\tfrac {2}{3}}} 5457:{\displaystyle {\tfrac {5}{6}}} 5428:{\displaystyle {\tfrac {3}{4}}} 5019:{\displaystyle {\tfrac {3}{5}}} 4772:{\displaystyle {\tfrac {1}{4}}} 4743:{\displaystyle {\tfrac {1}{2}}} 4341:{\displaystyle {\tfrac {1}{2}}} 4312:{\displaystyle {\tfrac {2}{3}}} 4169:{\displaystyle {\tfrac {c}{d}}} 4140:{\displaystyle {\tfrac {a}{b}}} 3722:{\displaystyle {\tfrac {3}{8}}} 3693:{\displaystyle {\tfrac {3}{9}}} 3500:{\displaystyle {\tfrac {a}{b}}} 3396:{\displaystyle {\tfrac {1}{2}}} 3365:{\displaystyle {\tfrac {n}{n}}} 3336:{\displaystyle {\tfrac {n}{n}}} 3092:{\displaystyle {\tfrac {5}{7}}} 3063:{\displaystyle {\tfrac {3}{4}}} 2574: 2096:{\displaystyle {\tfrac {5}{7}}} 2067:{\displaystyle {\tfrac {3}{4}}} 2038:{\displaystyle {\tfrac {2}{3}}} 2009:{\displaystyle {\tfrac {1}{2}}} 1987:expressed all fractions except 1559:{\displaystyle {\tfrac {3}{4}}} 871:{\displaystyle {\tfrac {a}{b}}} 100:{\displaystyle {\tfrac {1}{2}}} 11535:School Science and Mathematics 11191:David E. Smith (1 June 1958). 11075: 11042:The Fundamentals of Typography 11005: 10951: 10868: 10320:as early as the 10th century. 9970: BC) discovered that the 9794:{\displaystyle {\frac {1}{2}}} 9610: 9591: 9570: 9556: 9536: 9517: 9346: 9327: 9306: 9292: 9272: 9253: 9005:partial fraction decomposition 8747:representing numbers, such as 8257: 8254: 8232: 8226: 8204: 8201: 8195: 8192: 8180: 8174: 8162: 8159: 8136: 8114: 8108: 8096: 8074: 8052: 8046: 8034: 7979: 7974: 7962: 7956: 7944: 7902: 7890: 7855: 7843: 7824: 7811: 7804: as additive unities, and 7799: 7787: 7764: 7749: 7739: 7727: 7693: 7676: 7672: 7654: 7648: 7636: 7630: 7618: 7597: 7579: 7573: 7561: 7555: 7543: 7521: 7494: 7488: 7476: 7470: 7458: 7436: 7409: 7403: 7391: 7385: 7373: 7282: 7270: 5888: 5870: 5735: 3531:, then they can be written as 3009: 2995: 2955: 2941: 2331:Complex and compound fractions 320:represents positive one-half. 1: 11912:. A K Peters. pp. 2, 6. 11782:American Mathematical Monthly 10922: 10244:In discussing the origins of 10034: 10023: 10012: 9964: 9924: 9021:Rationalization (mathematics) 3968:, and the equal denominators 3613:to give the reduced fraction 2453: 1054:Proper and improper fractions 691:"hundredth"/"hundredths" or " 483: 11858:. allbusiness.com glossary. 11833:. allbusiness.com glossary. 10983:Schwartzman, Steven (1994). 10300:, early fifteenth century). 9912:The earliest fractions were 9025:A fraction may also contain 8655:contain a fraction, such as 8378:(also known as the field of 5105:equals 1, multiplication by 4750:of a cake is to be added to 1340:represents a denominator of 380:{\displaystyle \mathbb {Q} } 7: 12116:. Wiesbaden: Steiner. 1951. 11936:American Journal of Physics 11677:. Sosmath.com. 1996-02-05. 11383:Kelley, W. Michael (2004). 11228:10.1007/978-1-349-05230-1_2 10464: 10087:which is the equivalent of 7772:additive inverse fractions, 6532: 4627: 4565:{\displaystyle 18\times 17} 3589:, and the fraction becomes 3279:laws, and the rule against 10: 12476: 11705:Technical Math For Dummies 11650:Collins English Dictionary 11620:Collins English Dictionary 10874:Some typographers such as 9805: 9679: 9014: 8389: 6977: 6971: 2334: 487: 18: 12345: 12314: 12303: 12281: 12272: 11908:(2004). "Uses of fuses". 11148:Oxford English Dictionary 11129:Oxford English Dictionary 10048:) over the denominators ( 10045: 9686:In computer displays and 7681:with, additionally, 7347:for which the operations 7097:= the repeating decimal: 6302:Multiplying mixed numbers 5206:Now it can be seen that: 4176:, these are converted to 3453:) make up half the cake ( 3263:Arithmetic with fractions 1720:by applying the rules of 1265:that it would be yellow. 1195: 221:(three divided by four). 136:) consists of an integer 21:Fraction (disambiguation) 12208:"Fraction, arithmetical" 12045:Cajori, Florian (1928). 10861: 9676:Typographical variations 7363:are defined as follows: 7340:{\displaystyle b\neq 0,} 6686:of that fraction. Thus, 4784:Adding unlike quantities 2335:Not to be confused with 1722:adding unlike quantities 1601:cakes. The mixed number 1517:cakes, with the numeral 1510:{\displaystyle 2\ \,3/4} 12460:Fractions (mathematics) 12224:Encyclopædia Britannica 12162:. Retrieved 2011-11-22. 12022:10.1007/1-4020-2321-9_7 11701:"8. Fun with Fractions" 11578:Trotter, James (1853). 11498:Gardiner, Tony (2016). 11389:. Penguin. p. 25. 11268:Record, Robert (1654). 10892:and the fraction slash 10318:Abu'l-Hasan al-Uqlidisi 8739:with coefficients in a 8331:may be elements of any 7261:consistent and reliable 6419:multiplied as binomials 3642:greatest common divisor 2828:{\displaystyle 5/10/20} 1809:division with remainder 1389:The related concept of 631:square. In traditional 11598:Barlow, Peter (1814). 11418:. Enslow. p. 14. 11194:History of Mathematics 11163:Contemporary Education 10767:Dyadic (finite binary) 10681: 10643: 10605: 10567: 10529: 10451: 10437:is a whole number and 10431: 10407: 10383: 10357:Documents for teachers 10302: 10288:used both decimal and 10229: 10050: 9795: 9722:. An example would be 9694:). See the article on 9662: 9398: 9121: 8987: 8929: 8867: 8847: 8814: 8787:transcendental numbers 8777: 8712: 8643: 8577: 8506: 8454: 8264: 8144: 8082: 8006: 7921: 7700: 7604: 7529: 7444: 7341: 7312: 7289: 6959: 6800: 6676: 6647: 6595: 6566: 6523: 6408: 6289: 6191: 6020: 5951: 5827: 5719: 5458: 5429: 5396: 5318: 5243: 5195: 5081: 5020: 4988: 4942: 4780: 4773: 4744: 4707: 4612: 4566: 4540: 4495: 4450: 4418: 4386: 4342: 4313: 4260: 4215: 4170: 4141: 4105: 3982: 3958: 3896: 3723: 3694: 3583: 3554: 3523: 3501: 3397: 3366: 3337: 3308: 3247: 3196: 3152: 3093: 3064: 3019: 2972: 2928: 2883: 2829: 2790: 2445: 2389: 2321: 2247: 2188: 2159: 2097: 2068: 2039: 2010: 1977: 1922:means multiplication. 1858: 1801: 1708: 1653: 1595: 1560: 1531: 1511: 1476: 1383: 1132:multiplicative inverse 872: 523: 511: 474: 442: 381: 130: 101: 63: 51: 11880:Eves, Howard (1990). 11709:Wiley Publishing Inc. 11469:Wu, Hung-Hsi (2011). 10682: 10644: 10606: 10568: 10530: 10452: 10432: 10408: 10384: 10254: 10241:in the 13th century. 10230: 10191:, who specialized in 9823:This History section 9796: 9756:), as in "2/6" for a 9663: 9399: 9122: 8988: 8930: 8868: 8848: 8815: 8778: 8713: 8644: 8578: 8507: 8455: 8402:algebraic expressions 8265: 8145: 8083: 8007: 7922: 7701: 7605: 7530: 7445: 7342: 7313: 7290: 7288:{\displaystyle (a,b)} 6972:Further information: 6960: 6867:to a decimal, divide 6831:to a decimal, divide 6801: 6677: 6648: 6596: 6567: 6524: 6409: 6290: 6192: 6021: 5952: 5828: 5730:least common multiple 5720: 5459: 5430: 5397: 5319: 5244: 5196: 5082: 5021: 4989: 4943: 4791:invisible denominator 4774: 4745: 4720: 4708: 4613: 4567: 4541: 4496: 4451: 4419: 4387: 4343: 4314: 4286:, without evaluating 4261: 4216: 4171: 4142: 4106: 3983: 3959: 3897: 3724: 3695: 3584: 3555: 3524: 3502: 3398: 3367: 3338: 3309: 3248: 3197: 3153: 3094: 3065: 3038:§ Multiplication 3020: 2973: 2929: 2884: 2830: 2791: 2446: 2390: 2322: 2248: 2189: 2160: 2098: 2069: 2040: 2011: 1978: 1920:algebraic expressions 1885:hours or 5 3/16 1859: 1802: 1709: 1654: 1596: 1561: 1532: 1512: 1477: 1174:invisible denominator 873: 475: 443: 382: 131: 102: 31: 12014:Springer Netherlands 11675:"Compound Fractions" 10812:Algebraic irrational 10665: 10627: 10589: 10551: 10513: 10471:Cross multiplication 10441: 10421: 10397: 10373: 10200: 10181:Muslim mathematician 10133:, and 2 − 9834:improve this article 9778: 9409: 9145: 9048: 8941: 8892: 8857: 8824: 8813:{\textstyle \pi /2,} 8793: 8753: 8661: 8603: 8531: 8466: 8408: 8156: 8092: 8031: 8017:equivalence relation 7941: 7717: 7615: 7540: 7455: 7370: 7322: 7302: 7267: 6931: 6690: 6657: 6605: 6576: 6541: 6428: 6313: 6219: 6135: 6118: 6094: 6077: 6040: 5977: 5843: 5748: 5472: 5439: 5410: 5334: 5259: 5213: 5146: 5032: 5001: 4958: 4803: 4754: 4725: 4640: 4578: 4550: 4505: 4460: 4428: 4399: 4352: 4323: 4294: 4225: 4180: 4151: 4122: 3999: 3972: 3924: 3827: 3704: 3675: 3582:{\displaystyle b=ce} 3564: 3553:{\displaystyle a=cd} 3535: 3513: 3482: 3378: 3347: 3318: 3298: 3290:Equivalent fractions 3208: 3162: 3103: 3074: 3045: 2982: 2938: 2893: 2839: 2803: 2463: 2399: 2350: 2257: 2198: 2169: 2107: 2078: 2049: 2020: 1991: 1943: 1815: 1728: 1663: 1628: 1570: 1541: 1521: 1486: 1454: 962:sexagesimal fraction 853: 455: 421: 369: 201:. Thus the fraction 111: 82: 78:fraction (examples: 19:For other uses, see 16:Ratio of two numbers 11949:1978AmJPh..46..896C 10503: 10324:In formal education 10056:Sanskrit literature 9947:continued fractions 9011:Radical expressions 8745:radical expressions 8595:irrational fraction 8591:rational expression 8386:Algebraic fractions 7209:= 1,523,987 − 1,523 6653:, which reduces to 6421:. In this example, 4041: because 3913:Comparing fractions 3907:Euclidean algorithm 1326:scientific notation 1114:, for instance, is 964:used in astronomy. 627:square, or a wider 389:, which stands for 12282:Division and ratio 12158:2011-11-15 at the 11616:"complex fraction" 11455:two and one fourth 11345:Greer, A. (1986). 11017:www.mathsisfun.com 10969:2017-08-20 at the 10677: 10639: 10601: 10563: 10525: 10499: 10447: 10427: 10403: 10379: 10239:Leonardo Fibonacci 10225: 9992:mathematicians in 9972:square root of two 9922:Egyptian fractions 9791: 9772:Built-up fractions 9658: 9394: 9117: 9001:rational functions 8983: 8925: 8863: 8843: 8810: 8773: 8772: 8737:rational fractions 8729:field of fractions 8708: 8703: 8683: 8639: 8573: 8502: 8450: 8392:Algebraic fraction 8380:rational functions 8376:rational fractions 8372:field of fractions 8340:field of fractions 8260: 8140: 8078: 8002: 7917: 7915: 7696: 7600: 7525: 7440: 7337: 7308: 7285: 6955: 6950: 6796: 6794: 6779: 6748: 6733: 6718: 6703: 6672: 6670: 6643: 6641: 6626: 6591: 6589: 6562: 6554: 6519: 6404: 6285: 6283: 6268: 6253: 6238: 6200:A two is a common 6187: 6016: 5947: 5942: 5917: 5865: 5823: 5821: 5806: 5791: 5776: 5761: 5715: 5713: 5454: 5452: 5425: 5423: 5392: 5314: 5252:is equivalent to: 5239: 5191: 5189: 5174: 5159: 5077: 5075: 5060: 5045: 5016: 5014: 4984: 4938: 4781: 4769: 4767: 4740: 4738: 4703: 4701: 4683: 4668: 4653: 4608: 4606: 4591: 4562: 4536: 4534: 4491: 4489: 4446: 4441: 4414: 4412: 4382: 4380: 4365: 4338: 4336: 4309: 4307: 4290:, e.g., comparing 4256: 4254: 4211: 4209: 4166: 4164: 4137: 4135: 4101: 4079: 4059: 4037: 4017: 3978: 3954: 3952: 3937: 3892: 3890: 3875: 3840: 3719: 3717: 3690: 3688: 3579: 3550: 3519: 3497: 3495: 3393: 3391: 3362: 3360: 3333: 3331: 3304: 3243: 3241: 3192: 3190: 3175: 3148: 3146: 3131: 3116: 3089: 3087: 3060: 3058: 3015: 2968: 2924: 2916: 2879: 2877: 2852: 2825: 2786: 2784: 2681: 2590: 2499: 2485: 2441: 2422: 2385: 2383: 2317: 2315: 2300: 2285: 2270: 2243: 2241: 2226: 2211: 2184: 2182: 2155: 2150: 2135: 2120: 2103:can be written as 2093: 2091: 2064: 2062: 2035: 2033: 2006: 2004: 1973: 1971: 1956: 1926:Historical notions 1854: 1849: 1828: 1797: 1792: 1777: 1762: 1747: 1704: 1692: 1649: 1647: 1591: 1589: 1556: 1554: 1527: 1507: 1472: 1470: 1414:, it is easier to 1412:mental calculation 1401:parts-per notation 1397:parts per thousand 1284:to the right of a 1082:top-heavy fraction 1062:The Ground of Arts 868: 866: 814:Forms of fractions 470: 469: 438: 437: 377: 126: 124: 97: 95: 52: 12447: 12446: 12226:. 5 January 2024. 12138:978-0-691-11485-9 12099:978-0-691-02397-7 12031:978-1-4020-2320-0 11891:978-0-03-029558-4 11718:978-0-470-59874-0 11509:10.11647/OBP.0071 11484:978-0-8218-5260-6 11425:978-0-7660-4247-6 11358:978-0-85950-159-0 11317:"Proper Fraction" 11254:978-0-85729-862-1 11204:978-0-486-20430-7 11109:978-0-88179-206-5 11056:978-2-940411-76-4 10998:978-0-88385-511-9 10859: 10858: 10855: 10854: 10851: 10850: 10847: 10846: 10836: 10835: 10832: 10831: 10828: 10827: 10824: 10823: 10793: 10792: 10789: 10788: 10785: 10784: 10781: 10780: 10774:Repeating decimal 10741: 10740: 10737: 10736: 10732:Negative integers 10726: 10725: 10722: 10721: 10717:Composite numbers 10450:{\displaystyle b} 10430:{\displaystyle a} 10406:{\displaystyle b} 10382:{\displaystyle a} 10329:Pedagogical tools 10294:Key to arithmetic 10246:decimal fractions 10223: 10064:⟨+⟩ 10060:⟨०⟩ 9910: 9909: 9902: 9884: 9789: 9762:complex fractions 9750:solidus fractions 9710:Special fractions 9656: 9650: 9625: 9608: 9580: 9567: 9534: 9506: 9503: 9485: 9463: 9460: 9435: 9432: 9392: 9386: 9361: 9344: 9316: 9303: 9270: 9242: 9239: 9221: 9199: 9196: 9171: 9168: 9115: 9109: 9093: 9092: 9087: 9076: 9075: 9061: 9060: 8981: 8960: 8923: 8866:{\displaystyle 2} 8832: 8762: 8735:. Similarly, the 8706: 8702: 8682: 8637: 8618: 8587:rational fraction 8571: 8500: 8481: 8448: 7930:Furthermore, the 7908: 7888: 7864: 7805: 7785: 7773: 7682: 7311:{\displaystyle a} 6986:repeating decimal 6980:Repeating decimal 6949: 6793: 6778: 6747: 6732: 6717: 6702: 6669: 6640: 6625: 6588: 6553: 6514: 6498: 6479: 6448: 6402: 6386: 6373: 6346: 6333: 6282: 6267: 6252: 6237: 6185: 6172: 6159: 6146: 6140: 6123: 6105: 6099: 6082: 6064: 6051: 6014: 6001: 5988: 5941: 5916: 5864: 5820: 5805: 5790: 5775: 5760: 5709: 5694: 5681: 5668: 5639: 5603: 5588: 5575: 5562: 5533: 5500: 5487: 5451: 5422: 5390: 5358: 5345: 5312: 5296: 5283: 5270: 5237: 5224: 5188: 5173: 5158: 5074: 5059: 5044: 5013: 4982: 4969: 4933: 4920: 4908: 4904: 4891: 4863: 4859: 4830: 4818: 4814: 4789:number apply the 4766: 4737: 4700: 4682: 4667: 4652: 4605: 4590: 4533: 4488: 4440: 4411: 4379: 4364: 4335: 4306: 4253: 4208: 4163: 4134: 4084: 4078: 4058: 4042: 4036: 4016: 3981:{\displaystyle 4} 3951: 3936: 3889: 3874: 3839: 3716: 3687: 3669:in simplest terms 3636:If one takes for 3522:{\displaystyle c} 3507:are divisible by 3494: 3390: 3359: 3330: 3307:{\displaystyle n} 3240: 3189: 3174: 3145: 3130: 3115: 3086: 3057: 3029:compound fraction 2915: 2876: 2851: 2777: 2764: 2751: 2718: 2687: 2680: 2656: 2643: 2630: 2611: 2598: 2589: 2569: 2556: 2543: 2530: 2517: 2500: 2498: 2484: 2421: 2382: 2314: 2299: 2284: 2269: 2240: 2225: 2210: 2181: 2149: 2134: 2119: 2090: 2061: 2032: 2003: 1985:ancient Egyptians 1970: 1955: 1937:Egyptian fraction 1931:Egyptian fraction 1848: 1827: 1791: 1776: 1761: 1746: 1718:improper fraction 1691: 1646: 1588: 1553: 1530:{\displaystyle 2} 1494: 1469: 1405:parts per million 1295:Decimal separator 1286:decimal separator 1078:improper fraction 865: 827:(also known as a 467: 450:rational fraction 435: 431: 397:and the notation 355:are integers and 323:In mathematics a 123: 94: 12467: 12424:Musical interval 12337: 12336: 12334: 12333: 12330: 12327: 12309: 12308: 12278: 12255: 12248: 12241: 12232: 12231: 12227: 12215: 12194: 12193: 12191: 12190: 12184: 12177: 12169: 12163: 12149: 12143: 12142: 12124: 12118: 12117: 12110: 12104: 12103: 12086: 12080: 12074: 12068: 12067: 12065: 12064: 12042: 12036: 12035: 12001: 11995: 11994: 11992: 11990: 11974: 11961: 11960: 11930: 11924: 11923: 11902: 11896: 11895: 11877: 11871: 11870: 11868: 11867: 11856:"piece fraction" 11852: 11846: 11845: 11843: 11842: 11831:"built fraction" 11827: 11821: 11820: 11818: 11817: 11811: 11778: 11769: 11763: 11762: 11760: 11759: 11744: 11738: 11737: 11735: 11733: 11696: 11690: 11689: 11687: 11686: 11671: 11665: 11664: 11662: 11661: 11642: 11636: 11635: 11633: 11631: 11622:. Archived from 11612: 11606: 11605: 11595: 11586: 11585: 11575: 11566: 11565: 11530: 11524: 11523: 11511: 11495: 11489: 11488: 11476: 11466: 11460: 11459: 11452: 11450: 11449: 11446: 11443: 11439: 11417: 11407: 11401: 11400: 11380: 11374: 11373: 11371: 11370: 11342: 11336: 11335: 11333: 11332: 11323:. Archived from 11313: 11307: 11306: 11304: 11303: 11283: 11277: 11276: 11265: 11259: 11258: 11238: 11232: 11231: 11215: 11209: 11208: 11188: 11179: 11178: 11158: 11152: 11151: 11139: 11133: 11132: 11120: 11114: 11113: 11098:(3rd ed.). 11091: 11085: 11079: 11073: 11071: 11069: 11068: 11036: 11027: 11026: 11024: 11023: 11009: 11003: 11002: 10990: 10980: 10974: 10955: 10946: 10916: 10913: 10907: 10899: 10885: 10872: 10808: 10807: 10799: 10798: 10756: 10755: 10747: 10746: 10690: 10689: 10686: 10684: 10683: 10678: 10676: 10656: 10655: 10652: 10651: 10648: 10646: 10645: 10640: 10638: 10618: 10617: 10614: 10613: 10610: 10608: 10607: 10602: 10600: 10580: 10579: 10576: 10575: 10572: 10570: 10569: 10564: 10562: 10542: 10541: 10538: 10537: 10534: 10532: 10531: 10526: 10524: 10504: 10498: 10495: 10494: 10491: 10490: 10456: 10454: 10453: 10448: 10436: 10434: 10433: 10428: 10416: 10414: 10413: 10412: 10410: 10409: 10404: 10392: 10389: 10388: 10386: 10385: 10380: 10310:Jamshīd al-Kāshī 10236: 10234: 10232: 10231: 10226: 10224: 10222: 10213: 10204: 10178: 10175: 10164: 10162: 10161: 10158: 10155: 10148: 10146: 10145: 10142: 10139: 10132: 10130: 10129: 10126: 10123: 10116: 10114: 10113: 10110: 10107: 10065: 10061: 10053: 10047: 10039: 10036: 10028: 10025: 10017: 10014: 9988: 9969: 9966: 9936:dyadic fractions 9929: 9926: 9905: 9898: 9894: 9891: 9885: 9883: 9842: 9818: 9810: 9800: 9798: 9797: 9792: 9790: 9782: 9737: 9735: 9734: 9731: 9728: 9682:Slash § Encoding 9667: 9665: 9664: 9659: 9657: 9652: 9651: 9646: 9634: 9626: 9624: 9613: 9609: 9604: 9586: 9581: 9579: 9578: 9577: 9568: 9563: 9552: 9551: 9546: 9539: 9535: 9530: 9512: 9507: 9505: 9504: 9499: 9487: 9486: 9481: 9469: 9464: 9462: 9461: 9456: 9441: 9436: 9434: 9433: 9428: 9413: 9403: 9401: 9400: 9395: 9393: 9388: 9387: 9382: 9370: 9362: 9360: 9349: 9345: 9340: 9322: 9317: 9315: 9314: 9313: 9304: 9299: 9288: 9287: 9282: 9275: 9271: 9266: 9248: 9243: 9241: 9240: 9235: 9223: 9222: 9217: 9205: 9200: 9198: 9197: 9192: 9177: 9172: 9170: 9169: 9164: 9149: 9126: 9124: 9123: 9118: 9116: 9111: 9110: 9105: 9099: 9094: 9088: 9083: 9082: 9077: 9071: 9067: 9062: 9056: 9052: 8994: 8992: 8990: 8989: 8984: 8982: 8980: 8966: 8961: 8959: 8945: 8934: 8932: 8931: 8926: 8924: 8922: 8915: 8914: 8904: 8896: 8886:partial fraction 8872: 8870: 8869: 8864: 8852: 8850: 8849: 8844: 8833: 8828: 8819: 8817: 8816: 8811: 8803: 8784: 8782: 8780: 8779: 8774: 8768: 8763: 8758: 8722:complex fraction 8719: 8717: 8715: 8714: 8709: 8707: 8705: 8704: 8695: 8685: 8684: 8675: 8665: 8650: 8648: 8646: 8645: 8640: 8638: 8636: 8629: 8628: 8608: 8607: 8584: 8582: 8580: 8579: 8574: 8572: 8570: 8554: 8553: 8543: 8535: 8513: 8511: 8509: 8508: 8503: 8501: 8499: 8492: 8491: 8471: 8470: 8459: 8457: 8456: 8451: 8449: 8447: 8431: 8430: 8420: 8412: 8374:is the field of 8370:, the generated 8323:More generally, 8311: 8301: 8297: 8293: 8291: 8290: 8287: 8284: 8269: 8267: 8266: 8261: 8253: 8242: 8225: 8214: 8149: 8147: 8146: 8141: 8135: 8124: 8087: 8085: 8084: 8079: 8073: 8062: 8011: 8009: 8008: 8003: 7926: 7924: 7923: 7918: 7916: 7909: 7906: 7889: 7886: 7883: 7882: 7881: 7865: 7862: 7859: 7835: 7834: 7806: 7803: 7786: 7783: 7780: 7779: 7778: 7774: 7771: 7768: 7705: 7703: 7702: 7697: 7683: 7680: 7609: 7607: 7606: 7601: 7534: 7532: 7531: 7526: 7449: 7447: 7446: 7441: 7346: 7344: 7343: 7338: 7317: 7315: 7314: 7309: 7294: 7292: 7291: 7286: 7248: 7246: 7245: 7242: 7239: 7217: 7197: 7193: 7178: 7160: 7150: 7136: 7121: 7106: 7077: 7069: 7061: 7054: 7047: 7027: 7020: 7013: 7006: 6998: 6964: 6962: 6961: 6956: 6951: 6942: 6926: 6922: 6920: 6919: 6916: 6913: 6906: 6904: 6903: 6900: 6897: 6890: 6886: 6882: 6878: 6874: 6870: 6866: 6864: 6863: 6860: 6857: 6850: 6846: 6842: 6838: 6834: 6830: 6828: 6827: 6824: 6821: 6805: 6803: 6802: 6797: 6795: 6786: 6780: 6777: 6766: 6755: 6749: 6740: 6734: 6725: 6719: 6710: 6704: 6695: 6681: 6679: 6678: 6673: 6671: 6662: 6652: 6650: 6649: 6644: 6642: 6633: 6627: 6624: 6610: 6601:and also equals 6600: 6598: 6597: 6592: 6590: 6581: 6571: 6569: 6568: 6563: 6555: 6546: 6528: 6526: 6525: 6520: 6515: 6507: 6499: 6491: 6480: 6472: 6449: 6441: 6413: 6411: 6410: 6405: 6403: 6395: 6387: 6379: 6374: 6369: 6352: 6347: 6339: 6334: 6326: 6294: 6292: 6291: 6286: 6284: 6275: 6269: 6260: 6254: 6245: 6239: 6230: 6196: 6194: 6193: 6188: 6186: 6178: 6173: 6165: 6160: 6152: 6147: 6145: 6144: 6138: 6136: 6128: 6127: 6121: 6119: 6111: 6106: 6104: 6103: 6097: 6095: 6087: 6086: 6080: 6078: 6070: 6065: 6057: 6052: 6044: 6025: 6023: 6022: 6017: 6015: 6007: 6002: 5994: 5989: 5981: 5956: 5954: 5953: 5948: 5943: 5934: 5925: 5924: 5918: 5909: 5900: 5899: 5866: 5857: 5832: 5830: 5829: 5824: 5822: 5813: 5807: 5798: 5792: 5783: 5777: 5768: 5762: 5753: 5724: 5722: 5721: 5716: 5714: 5710: 5702: 5695: 5687: 5682: 5674: 5669: 5667: 5656: 5645: 5640: 5638: 5627: 5616: 5608: 5604: 5596: 5589: 5581: 5576: 5568: 5563: 5561: 5550: 5539: 5534: 5532: 5521: 5510: 5501: 5493: 5488: 5480: 5463: 5461: 5460: 5455: 5453: 5444: 5434: 5432: 5431: 5426: 5424: 5415: 5401: 5399: 5398: 5393: 5391: 5389: 5381: 5364: 5359: 5351: 5346: 5338: 5323: 5321: 5320: 5315: 5313: 5305: 5297: 5289: 5284: 5276: 5271: 5263: 5248: 5246: 5245: 5240: 5238: 5230: 5225: 5217: 5202: 5200: 5198: 5197: 5192: 5190: 5181: 5175: 5166: 5160: 5151: 5139: 5137: 5136: 5133: 5130: 5124:Second, convert 5120: 5118: 5117: 5114: 5111: 5104: 5102: 5101: 5098: 5095: 5088: 5086: 5084: 5083: 5078: 5076: 5067: 5061: 5052: 5046: 5037: 5025: 5023: 5022: 5017: 5015: 5006: 4993: 4991: 4990: 4985: 4983: 4975: 4970: 4962: 4947: 4945: 4944: 4939: 4934: 4926: 4921: 4913: 4906: 4905: 4897: 4892: 4890: 4879: 4868: 4861: 4860: 4858: 4847: 4836: 4831: 4823: 4816: 4815: 4807: 4778: 4776: 4775: 4770: 4768: 4759: 4749: 4747: 4746: 4741: 4739: 4730: 4712: 4710: 4709: 4704: 4702: 4693: 4684: 4675: 4669: 4660: 4654: 4645: 4619: 4617: 4615: 4614: 4609: 4607: 4598: 4592: 4583: 4571: 4569: 4568: 4563: 4545: 4543: 4542: 4537: 4535: 4532: 4521: 4510: 4500: 4498: 4497: 4492: 4490: 4487: 4476: 4465: 4455: 4453: 4452: 4447: 4442: 4433: 4423: 4421: 4420: 4415: 4413: 4404: 4391: 4389: 4388: 4383: 4381: 4372: 4366: 4357: 4347: 4345: 4344: 4339: 4337: 4328: 4318: 4316: 4315: 4310: 4308: 4299: 4265: 4263: 4262: 4257: 4255: 4252: 4241: 4230: 4220: 4218: 4217: 4212: 4210: 4207: 4196: 4185: 4175: 4173: 4172: 4167: 4165: 4156: 4146: 4144: 4143: 4138: 4136: 4127: 4110: 4108: 4107: 4102: 4085: 4082: 4080: 4074: 4066: 4060: 4057: 4046: 4043: 4040: 4038: 4035: 4024: 4018: 4015: 4004: 3987: 3985: 3984: 3979: 3967: 3963: 3961: 3960: 3955: 3953: 3944: 3938: 3929: 3901: 3899: 3898: 3893: 3891: 3882: 3876: 3873: 3860: 3847: 3841: 3832: 3819: 3817: 3816: 3813: 3810: 3800: 3799: 3797: 3796: 3793: 3790: 3783: 3781: 3780: 3777: 3774: 3767: 3765: 3764: 3761: 3758: 3751: 3749: 3748: 3745: 3742: 3728: 3726: 3725: 3720: 3718: 3709: 3699: 3697: 3696: 3691: 3689: 3680: 3639: 3632: 3630: 3629: 3624: 3621: 3608: 3606: 3605: 3600: 3597: 3588: 3586: 3585: 3580: 3559: 3557: 3556: 3551: 3530: 3528: 3526: 3525: 3520: 3506: 3504: 3503: 3498: 3496: 3487: 3468: 3466: 3465: 3462: 3459: 3452: 3450: 3449: 3446: 3443: 3436: 3434: 3433: 3430: 3427: 3420: 3418: 3417: 3414: 3411: 3404: 3402: 3400: 3399: 3394: 3392: 3383: 3371: 3369: 3368: 3363: 3361: 3352: 3342: 3340: 3339: 3334: 3332: 3323: 3313: 3311: 3310: 3305: 3281:division by zero 3254: 3252: 3250: 3249: 3244: 3242: 3239: 3235: 3226: 3222: 3213: 3201: 3199: 3198: 3193: 3191: 3182: 3176: 3167: 3157: 3155: 3154: 3149: 3147: 3138: 3132: 3123: 3117: 3108: 3098: 3096: 3095: 3090: 3088: 3079: 3069: 3067: 3066: 3061: 3059: 3050: 3041:). For example, 3024: 3022: 3021: 3016: 3005: 2994: 2993: 2977: 2975: 2974: 2969: 2964: 2963: 2951: 2933: 2931: 2930: 2925: 2917: 2908: 2905: 2904: 2888: 2886: 2885: 2880: 2878: 2869: 2860: 2859: 2853: 2844: 2834: 2832: 2831: 2826: 2821: 2813: 2795: 2793: 2792: 2787: 2785: 2778: 2770: 2765: 2757: 2752: 2747: 2730: 2719: 2714: 2697: 2688: 2683: 2682: 2673: 2666: 2657: 2649: 2644: 2636: 2631: 2623: 2612: 2604: 2599: 2594: 2591: 2582: 2576: 2570: 2562: 2557: 2549: 2544: 2536: 2531: 2523: 2518: 2510: 2501: 2491: 2489: 2486: 2477: 2471: 2450: 2448: 2447: 2442: 2437: 2436: 2430: 2429: 2423: 2414: 2408: 2407: 2394: 2392: 2391: 2386: 2384: 2381: 2377: 2368: 2364: 2355: 2344:complex fraction 2326: 2324: 2323: 2318: 2316: 2307: 2301: 2292: 2286: 2277: 2271: 2262: 2252: 2250: 2249: 2244: 2242: 2233: 2227: 2218: 2212: 2203: 2193: 2191: 2190: 2185: 2183: 2174: 2164: 2162: 2161: 2156: 2151: 2142: 2136: 2127: 2121: 2112: 2102: 2100: 2099: 2094: 2092: 2083: 2073: 2071: 2070: 2065: 2063: 2054: 2044: 2042: 2041: 2036: 2034: 2025: 2015: 2013: 2012: 2007: 2005: 1996: 1982: 1980: 1979: 1974: 1972: 1963: 1957: 1948: 1916: 1914: 1913: 1908: 1905: 1884: 1882: 1881: 1878: 1875: 1871: 1863: 1861: 1860: 1855: 1850: 1841: 1829: 1820: 1806: 1804: 1803: 1798: 1793: 1784: 1778: 1769: 1763: 1754: 1748: 1739: 1713: 1711: 1710: 1705: 1700: 1699: 1693: 1684: 1675: 1674: 1658: 1656: 1655: 1650: 1648: 1639: 1619: 1617: 1616: 1613: 1610: 1606: 1600: 1598: 1597: 1592: 1590: 1581: 1565: 1563: 1562: 1557: 1555: 1546: 1536: 1534: 1533: 1528: 1516: 1514: 1513: 1508: 1503: 1492: 1481: 1479: 1478: 1473: 1471: 1462: 1370: 1368: 1367: 1364: 1361: 1347: 1343: 1339: 1335: 1333: 1320: 1318: 1317: 1314: 1311: 1307: 1276:decimal fraction 1260: 1258: 1257: 1254: 1251: 1244: 1242: 1241: 1238: 1235: 1191: 1189: 1188: 1185: 1182: 1171: 1169: 1168: 1165: 1162: 1152: 1150: 1149: 1146: 1143: 1129: 1127: 1126: 1123: 1120: 1113: 1111: 1110: 1107: 1104: 1041: 1039: 1038: 1035: 1032: 1025: 1023: 1022: 1019: 1016: 996: 994: 993: 990: 987: 969:common fractions 959: 957: 956: 953: 950: 943: 941: 940: 937: 934: 927: 925: 924: 921: 918: 911: 909: 908: 905: 902: 879: 877: 875: 874: 869: 867: 858: 809: 807: 806: 803: 800: 793: 791: 790: 787: 784: 769: 767: 766: 763: 760: 754:. (For example, 749: 747: 746: 743: 740: 733: 731: 730: 727: 724: 713: 711: 710: 707: 704: 689: 687: 686: 683: 680: 673: 671: 670: 667: 664: 650: 648: 647: 644: 641: 586: 585: 581: 576: 574: 573: 570: 567: 541: 539: 538: 535: 532: 479: 477: 476: 471: 468: 460: 447: 445: 444: 439: 436: 427: 426: 416: 414: 413: 408: 405: 388: 386: 384: 383: 378: 376: 346: 344: 343: 338: 335: 319: 317: 316: 313: 310: 303: 301: 300: 297: 294: 287: 285: 284: 281: 278: 271: 269: 268: 265: 262: 255: 253: 252: 249: 246: 239: 237: 236: 233: 230: 220: 216: 214: 213: 210: 207: 189: 187: 186: 183: 180: 173: 171: 170: 167: 164: 149: 148: 144: 135: 133: 132: 127: 125: 116: 106: 104: 103: 98: 96: 87: 50: 49: 47: 46: 43: 40: 12475: 12474: 12470: 12469: 12468: 12466: 12465: 12464: 12450: 12449: 12448: 12443: 12414:Just intonation 12341: 12331: 12328: 12325: 12324: 12322: 12321: 12310: 12306: 12301: 12279: 12268: 12259: 12218: 12206: 12203: 12198: 12197: 12188: 12186: 12182: 12175: 12171: 12170: 12166: 12160:Wayback Machine 12150: 12146: 12139: 12125: 12121: 12112: 12111: 12107: 12100: 12088: 12087: 12083: 12075: 12071: 12062: 12060: 12043: 12039: 12032: 12016:. p. 152. 12002: 11998: 11988: 11986: 11975: 11964: 11957:10.1119/1.11512 11931: 11927: 11920: 11903: 11899: 11892: 11878: 11874: 11865: 11863: 11854: 11853: 11849: 11840: 11838: 11829: 11828: 11824: 11815: 11813: 11809: 11795:10.2307/4145131 11776: 11770: 11766: 11757: 11755: 11746: 11745: 11741: 11731: 11729: 11719: 11697: 11693: 11684: 11682: 11673: 11672: 11668: 11659: 11657: 11644: 11643: 11639: 11629: 11627: 11614: 11613: 11609: 11596: 11589: 11576: 11569: 11531: 11527: 11520: 11496: 11492: 11485: 11467: 11463: 11447: 11444: 11441: 11440: 11437: 11435: 11426: 11408: 11404: 11397: 11381: 11377: 11368: 11366: 11359: 11343: 11339: 11330: 11328: 11315: 11314: 11310: 11301: 11299: 11284: 11280: 11266: 11262: 11255: 11239: 11235: 11216: 11212: 11205: 11189: 11182: 11159: 11155: 11141: 11140: 11136: 11122: 11121: 11117: 11110: 11092: 11088: 11080: 11076: 11066: 11064: 11057: 11037: 11030: 11021: 11019: 11011: 11010: 11006: 10999: 10981: 10977: 10971:Wayback Machine 10956: 10952: 10943: 10929:Weisstein, Eric 10925: 10920: 10919: 10914: 10910: 10893: 10879: 10873: 10869: 10864: 10672: 10666: 10663: 10662: 10634: 10628: 10625: 10624: 10596: 10590: 10587: 10586: 10558: 10552: 10549: 10548: 10520: 10514: 10511: 10510: 10467: 10442: 10439: 10438: 10422: 10419: 10418: 10398: 10395: 10394: 10393: 10390: 10374: 10371: 10370: 10369: 10368: 10366: 10359: 10339:Cuisenaire rods 10335:primary schools 10331: 10326: 10264:, published at 10250:Dirk Jan Struik 10214: 10205: 10203: 10201: 10198: 10197: 10196: 10176: 10159: 10156: 10153: 10152: 10150: 10143: 10140: 10137: 10136: 10134: 10127: 10124: 10121: 10120: 10118: 10111: 10108: 10105: 10104: 10102: 10080: 10063: 10059: 10037: 10026: 10015: 9998:Sthananga Sutra 9986: 9967: 9927: 9906: 9895: 9889: 9886: 9843: 9841: 9831: 9819: 9808: 9781: 9779: 9776: 9775: 9766:piece fractions 9732: 9729: 9726: 9725: 9723: 9684: 9678: 9645: 9635: 9633: 9614: 9603: 9587: 9585: 9573: 9569: 9562: 9547: 9542: 9541: 9540: 9529: 9513: 9511: 9498: 9488: 9480: 9470: 9468: 9455: 9445: 9440: 9427: 9417: 9412: 9410: 9407: 9406: 9381: 9371: 9369: 9350: 9339: 9323: 9321: 9309: 9305: 9298: 9283: 9278: 9277: 9276: 9265: 9249: 9247: 9234: 9224: 9216: 9206: 9204: 9191: 9181: 9176: 9163: 9153: 9148: 9146: 9143: 9142: 9104: 9100: 9098: 9081: 9066: 9051: 9049: 9046: 9045: 9023: 9015:Main articles: 9013: 8997:antiderivatives 8970: 8965: 8949: 8944: 8942: 8939: 8938: 8936: 8910: 8906: 8905: 8897: 8895: 8893: 8890: 8889: 8858: 8855: 8854: 8827: 8825: 8822: 8821: 8799: 8794: 8791: 8790: 8764: 8757: 8754: 8751: 8750: 8748: 8733:integral domain 8693: 8686: 8673: 8666: 8664: 8662: 8659: 8658: 8656: 8624: 8620: 8619: 8606: 8604: 8601: 8600: 8598: 8549: 8545: 8544: 8536: 8534: 8532: 8529: 8528: 8526: 8487: 8483: 8482: 8469: 8467: 8464: 8463: 8461: 8426: 8422: 8421: 8413: 8411: 8409: 8406: 8405: 8394: 8388: 8346:. For example, 8333:integral domain 8306: 8299: 8295: 8288: 8285: 8282: 8281: 8279: 8246: 8235: 8218: 8207: 8157: 8154: 8153: 8128: 8117: 8093: 8090: 8089: 8066: 8055: 8032: 8029: 8028: 7942: 7939: 7938: 7934:, specified as 7914: 7913: 7905: 7885: 7879: 7878: 7861: 7858: 7836: 7827: 7823: 7808: 7807: 7802: 7782: 7776: 7775: 7770: 7767: 7742: 7720: 7718: 7715: 7714: 7679: 7616: 7613: 7612: 7541: 7538: 7537: 7456: 7453: 7452: 7371: 7368: 7367: 7323: 7320: 7319: 7303: 7300: 7299: 7268: 7265: 7264: 7257: 7243: 7240: 7237: 7236: 7234: 7212: 7195: 7191: 7162: 7152: 7142: 7134: 7119: 7104: 7075: 7074:0.1523 + 0.0000 7067: 7059: 7052: 7045: 7025: 7018: 7011: 7004: 6996: 6982: 6976: 6970: 6940: 6932: 6929: 6928: 6924: 6917: 6914: 6911: 6910: 6908: 6901: 6898: 6895: 6894: 6892: 6891:. The fraction 6888: 6884: 6880: 6876: 6872: 6868: 6861: 6858: 6855: 6854: 6852: 6848: 6844: 6840: 6836: 6832: 6825: 6822: 6819: 6818: 6816: 6812: 6784: 6767: 6756: 6753: 6738: 6723: 6708: 6693: 6691: 6688: 6687: 6660: 6658: 6655: 6654: 6631: 6614: 6608: 6606: 6603: 6602: 6579: 6577: 6574: 6573: 6544: 6542: 6539: 6538: 6535: 6506: 6490: 6471: 6440: 6429: 6426: 6425: 6394: 6378: 6353: 6351: 6338: 6325: 6314: 6311: 6310: 6304: 6273: 6258: 6243: 6228: 6220: 6217: 6216: 6210: 6177: 6164: 6151: 6137: 6130: 6129: 6120: 6113: 6112: 6110: 6096: 6089: 6088: 6079: 6072: 6071: 6069: 6056: 6043: 6041: 6038: 6037: 6006: 5993: 5980: 5978: 5975: 5974: 5968: 5963: 5932: 5920: 5919: 5907: 5895: 5894: 5855: 5844: 5841: 5840: 5811: 5796: 5781: 5766: 5751: 5749: 5746: 5745: 5738: 5712: 5711: 5701: 5696: 5686: 5673: 5657: 5646: 5644: 5628: 5617: 5615: 5606: 5605: 5595: 5590: 5580: 5567: 5551: 5540: 5538: 5522: 5511: 5509: 5502: 5492: 5479: 5475: 5473: 5470: 5469: 5442: 5440: 5437: 5436: 5413: 5411: 5408: 5407: 5382: 5365: 5363: 5350: 5337: 5335: 5332: 5331: 5304: 5288: 5275: 5262: 5260: 5257: 5256: 5229: 5216: 5214: 5211: 5210: 5179: 5164: 5149: 5147: 5144: 5143: 5141: 5134: 5131: 5128: 5127: 5125: 5115: 5112: 5109: 5108: 5106: 5099: 5096: 5093: 5092: 5090: 5065: 5050: 5035: 5033: 5030: 5029: 5027: 5004: 5002: 4999: 4998: 4997:First, convert 4974: 4961: 4959: 4956: 4955: 4925: 4912: 4896: 4880: 4869: 4867: 4848: 4837: 4835: 4822: 4806: 4804: 4801: 4800: 4786: 4757: 4755: 4752: 4751: 4728: 4726: 4723: 4722: 4691: 4673: 4658: 4643: 4641: 4638: 4637: 4630: 4596: 4581: 4579: 4576: 4575: 4573: 4551: 4548: 4547: 4522: 4511: 4508: 4506: 4503: 4502: 4477: 4466: 4463: 4461: 4458: 4457: 4431: 4429: 4426: 4425: 4402: 4400: 4397: 4396: 4370: 4355: 4353: 4350: 4349: 4326: 4324: 4321: 4320: 4297: 4295: 4292: 4291: 4242: 4231: 4228: 4226: 4223: 4222: 4197: 4186: 4183: 4181: 4178: 4177: 4154: 4152: 4149: 4148: 4125: 4123: 4120: 4119: 4083: and 4081: 4067: 4064: 4050: 4044: 4039: 4028: 4022: 4008: 4002: 4000: 3997: 3996: 3973: 3970: 3969: 3965: 3942: 3927: 3925: 3922: 3921: 3915: 3880: 3861: 3848: 3845: 3830: 3828: 3825: 3824: 3814: 3811: 3808: 3807: 3805: 3801:, for example. 3794: 3791: 3788: 3787: 3785: 3778: 3775: 3772: 3771: 3769: 3762: 3759: 3756: 3755: 3753: 3746: 3743: 3740: 3739: 3737: 3736: 3707: 3705: 3702: 3701: 3678: 3676: 3673: 3672: 3671:. For example, 3646:absolute values 3637: 3625: 3622: 3617: 3616: 3614: 3601: 3598: 3593: 3592: 3590: 3565: 3562: 3561: 3536: 3533: 3532: 3514: 3511: 3510: 3508: 3485: 3483: 3480: 3479: 3475: 3463: 3460: 3457: 3456: 3454: 3447: 3444: 3441: 3440: 3438: 3431: 3428: 3425: 3424: 3422: 3415: 3412: 3409: 3408: 3406: 3381: 3379: 3376: 3375: 3373: 3350: 3348: 3345: 3344: 3321: 3319: 3316: 3315: 3314:, the fraction 3299: 3296: 3295: 3292: 3265: 3231: 3227: 3218: 3214: 3211: 3209: 3206: 3205: 3203: 3180: 3165: 3163: 3160: 3159: 3136: 3121: 3106: 3104: 3101: 3100: 3077: 3075: 3072: 3071: 3048: 3046: 3043: 3042: 3001: 2989: 2988: 2983: 2980: 2979: 2959: 2958: 2947: 2939: 2936: 2935: 2906: 2900: 2899: 2894: 2891: 2890: 2867: 2855: 2854: 2842: 2840: 2837: 2836: 2817: 2809: 2804: 2801: 2800: 2783: 2782: 2769: 2756: 2731: 2729: 2698: 2696: 2689: 2671: 2667: 2665: 2662: 2661: 2648: 2635: 2622: 2603: 2580: 2577: 2575: 2561: 2548: 2535: 2522: 2509: 2502: 2475: 2472: 2470: 2466: 2464: 2461: 2460: 2456:. For example: 2454:§ Division 2432: 2431: 2425: 2424: 2412: 2403: 2402: 2400: 2397: 2396: 2373: 2369: 2360: 2356: 2353: 2351: 2348: 2347: 2340: 2337:Complex numbers 2333: 2305: 2290: 2275: 2260: 2258: 2255: 2254: 2231: 2216: 2201: 2199: 2196: 2195: 2172: 2170: 2167: 2166: 2140: 2125: 2110: 2108: 2105: 2104: 2081: 2079: 2076: 2075: 2052: 2050: 2047: 2046: 2023: 2021: 2018: 2017: 1994: 1992: 1989: 1988: 1961: 1946: 1944: 1941: 1940: 1933: 1928: 1909: 1906: 1901: 1900: 1898: 1897:, the quotient 1895:rational number 1879: 1876: 1873: 1872: 1869: 1867: 1839: 1818: 1816: 1813: 1812: 1782: 1767: 1752: 1737: 1729: 1726: 1725: 1724:. For example, 1695: 1694: 1682: 1670: 1669: 1664: 1661: 1660: 1637: 1629: 1626: 1625: 1614: 1611: 1608: 1607: 1604: 1602: 1579: 1571: 1568: 1567: 1544: 1542: 1539: 1538: 1522: 1519: 1518: 1499: 1487: 1484: 1483: 1460: 1455: 1452: 1451: 1435:(also called a 1429: 1365: 1362: 1359: 1358: 1356: 1355:. For example, 1353:infinite series 1345: 1341: 1337: 1331: 1329: 1315: 1312: 1309: 1308: 1305: 1303: 1300:fractional part 1271: 1255: 1252: 1249: 1248: 1246: 1245:of the cars or 1239: 1236: 1233: 1232: 1230: 1198: 1186: 1183: 1180: 1179: 1177: 1166: 1163: 1160: 1159: 1157: 1147: 1144: 1141: 1140: 1138: 1124: 1121: 1118: 1117: 1115: 1108: 1105: 1102: 1101: 1099: 1090: 1080:, or sometimes 1070:proper fraction 1056: 1036: 1033: 1030: 1029: 1027: 1020: 1017: 1014: 1013: 1011: 1003:dyadic fraction 991: 988: 985: 984: 982: 954: 951: 948: 947: 945: 938: 935: 932: 931: 929: 922: 919: 916: 915: 913: 906: 903: 900: 899: 897: 856: 854: 851: 850: 848: 837:rational number 833:vulgar fraction 829:common fraction 825:simple fraction 821: 816: 804: 801: 798: 797: 795: 788: 785: 782: 781: 779: 764: 761: 758: 757: 755: 752:cardinal number 744: 741: 738: 737: 735: 728: 725: 722: 721: 719: 708: 705: 702: 701: 699: 684: 681: 678: 677: 675: 668: 665: 662: 661: 659: 656:ordinal numbers 645: 642: 639: 638: 636: 583: 579: 578: 571: 568: 565: 564: 562: 536: 533: 530: 529: 527: 500: 486: 459: 456: 453: 452: 425: 422: 419: 418: 409: 406: 401: 400: 398: 372: 370: 367: 366: 364: 339: 336: 331: 330: 328: 325:rational number 314: 311: 308: 307: 305: 298: 295: 292: 291: 289: 282: 279: 276: 275: 273: 266: 263: 260: 259: 257: 250: 247: 244: 243: 241: 234: 231: 228: 227: 225: 218: 211: 208: 205: 204: 202: 184: 181: 178: 177: 175: 168: 165: 162: 161: 159: 146: 142: 141: 114: 112: 109: 108: 85: 83: 80: 79: 44: 41: 38: 37: 35: 33: 24: 17: 12: 11: 5: 12473: 12463: 12462: 12445: 12444: 12442: 12441: 12436: 12431: 12426: 12421: 12416: 12411: 12410: 12409: 12399: 12394: 12393: 12392: 12382: 12377: 12372: 12367: 12362: 12357: 12352: 12346: 12343: 12342: 12340: 12339: 12318: 12316: 12312: 12311: 12304: 12302: 12300: 12299: 12285: 12283: 12280: 12273: 12270: 12269: 12258: 12257: 12250: 12243: 12235: 12229: 12228: 12216: 12202: 12201:External links 12199: 12196: 12195: 12164: 12144: 12137: 12119: 12105: 12098: 12081: 12069: 12037: 12030: 12006:Chemla, Karine 11996: 11962: 11943:(9): 896–906. 11925: 11918: 11906:Winkler, Peter 11897: 11890: 11872: 11847: 11822: 11789:(3): 238–242. 11764: 11739: 11717: 11691: 11666: 11652:. 2018-03-09. 11637: 11607: 11587: 11567: 11541:(4): 173–180. 11525: 11518: 11490: 11483: 11461: 11424: 11402: 11395: 11375: 11357: 11337: 11308: 11278: 11260: 11253: 11233: 11210: 11203: 11180: 11153: 11134: 11115: 11108: 11086: 11074: 11055: 11028: 11004: 10997: 10975: 10949: 10948: 10941: 10924: 10921: 10918: 10917: 10908: 10866: 10865: 10863: 10860: 10857: 10856: 10853: 10852: 10849: 10848: 10845: 10844: 10838: 10837: 10834: 10833: 10830: 10829: 10826: 10825: 10822: 10821: 10819:Transcendental 10815: 10814: 10805: 10795: 10794: 10791: 10790: 10787: 10786: 10783: 10782: 10779: 10778: 10776: 10770: 10769: 10763: 10762: 10760:Finite decimal 10753: 10743: 10742: 10739: 10738: 10735: 10734: 10728: 10727: 10724: 10723: 10720: 10719: 10713: 10712: 10706: 10705: 10698: 10697: 10687: 10675: 10670: 10649: 10637: 10632: 10611: 10599: 10594: 10573: 10561: 10556: 10535: 10523: 10518: 10501:Number systems 10489: 10488: 10483: 10478: 10473: 10466: 10463: 10446: 10426: 10402: 10378: 10358: 10355: 10343:pattern blocks 10330: 10327: 10325: 10322: 10316:mathematician 10308:mathematician 10221: 10217: 10212: 10208: 10099: 10098: 10095: 10092: 10085: 10084: 10081: 10078: 10075: 9932:unit fractions 9908: 9907: 9822: 9820: 9813: 9807: 9804: 9803: 9802: 9788: 9785: 9769: 9743: 9740:case fractions 9716:Case fractions 9713: 9677: 9674: 9669: 9668: 9655: 9649: 9644: 9641: 9638: 9632: 9629: 9623: 9620: 9617: 9612: 9607: 9602: 9599: 9596: 9593: 9590: 9584: 9576: 9572: 9566: 9561: 9558: 9555: 9550: 9545: 9538: 9533: 9528: 9525: 9522: 9519: 9516: 9510: 9502: 9497: 9494: 9491: 9484: 9479: 9476: 9473: 9467: 9459: 9454: 9451: 9448: 9444: 9439: 9431: 9426: 9423: 9420: 9416: 9404: 9391: 9385: 9380: 9377: 9374: 9368: 9365: 9359: 9356: 9353: 9348: 9343: 9338: 9335: 9332: 9329: 9326: 9320: 9312: 9308: 9302: 9297: 9294: 9291: 9286: 9281: 9274: 9269: 9264: 9261: 9258: 9255: 9252: 9246: 9238: 9233: 9230: 9227: 9220: 9215: 9212: 9209: 9203: 9195: 9190: 9187: 9184: 9180: 9175: 9167: 9162: 9159: 9156: 9152: 9128: 9127: 9114: 9108: 9103: 9097: 9091: 9086: 9080: 9074: 9070: 9065: 9059: 9055: 9012: 9009: 8979: 8976: 8973: 8969: 8964: 8958: 8955: 8952: 8948: 8921: 8918: 8913: 8909: 8903: 8900: 8881:coefficients. 8862: 8842: 8839: 8836: 8831: 8809: 8806: 8802: 8798: 8771: 8767: 8761: 8720:, is called a 8701: 8698: 8692: 8689: 8681: 8678: 8672: 8669: 8635: 8632: 8627: 8623: 8617: 8614: 8611: 8569: 8566: 8563: 8560: 8557: 8552: 8548: 8542: 8539: 8498: 8495: 8490: 8486: 8480: 8477: 8474: 8446: 8443: 8440: 8437: 8434: 8429: 8425: 8419: 8416: 8390:Main article: 8387: 8384: 8273: 8272: 8271: 8270: 8259: 8256: 8252: 8249: 8245: 8241: 8238: 8234: 8231: 8228: 8224: 8221: 8217: 8213: 8210: 8206: 8203: 8200: 8197: 8194: 8191: 8188: 8185: 8182: 8179: 8176: 8173: 8170: 8167: 8164: 8161: 8138: 8134: 8131: 8127: 8123: 8120: 8116: 8113: 8110: 8107: 8104: 8101: 8098: 8076: 8072: 8069: 8065: 8061: 8058: 8054: 8051: 8048: 8045: 8042: 8039: 8036: 8021:representative 8013: 8012: 8001: 7998: 7995: 7992: 7989: 7986: 7981: 7976: 7973: 7970: 7967: 7964: 7961: 7958: 7955: 7952: 7949: 7946: 7928: 7927: 7912: 7904: 7901: 7898: 7895: 7892: 7884: 7880: 7877: 7874: 7871: 7868: 7860: 7857: 7854: 7851: 7848: 7845: 7842: 7839: 7837: 7833: 7830: 7826: 7822: 7819: 7816: 7813: 7810: 7809: 7801: 7798: 7795: 7792: 7789: 7781: 7777: 7769: 7766: 7763: 7760: 7757: 7754: 7751: 7748: 7745: 7743: 7741: 7738: 7735: 7732: 7729: 7726: 7723: 7722: 7707: 7706: 7695: 7692: 7689: 7686: 7678: 7674: 7671: 7668: 7665: 7662: 7659: 7656: 7653: 7650: 7647: 7644: 7641: 7638: 7635: 7632: 7629: 7626: 7623: 7620: 7610: 7599: 7596: 7593: 7590: 7587: 7584: 7581: 7578: 7575: 7572: 7569: 7566: 7563: 7560: 7557: 7554: 7551: 7548: 7545: 7535: 7523: 7520: 7517: 7514: 7511: 7508: 7505: 7502: 7499: 7496: 7493: 7490: 7487: 7484: 7481: 7478: 7475: 7472: 7469: 7466: 7463: 7460: 7450: 7438: 7435: 7432: 7429: 7426: 7423: 7420: 7417: 7414: 7411: 7408: 7405: 7402: 7399: 7396: 7393: 7390: 7387: 7384: 7381: 7378: 7375: 7357:multiplication 7336: 7333: 7330: 7327: 7307: 7284: 7281: 7278: 7275: 7272: 7256: 7253: 7252: 7251: 7250: 7249: 7227:as a fraction 7221: 7220: 7219: 7210: 7200: 7199: 7198: 7139: 7138: 7137: 7124: 7123: 7122: 7109: 7108: 7107: 7087: 7086: 7079: 7078: 7064: 7063: 7056: 7049: 7038:trailing zeros 7030: 7029: 7022: 7015: 7008: 6969: 6966: 6954: 6948: 6945: 6939: 6936: 6887:decimals with 6847:"), to obtain 6811: 6808: 6792: 6789: 6783: 6776: 6773: 6770: 6765: 6762: 6759: 6752: 6746: 6743: 6737: 6731: 6728: 6722: 6716: 6713: 6707: 6701: 6698: 6668: 6665: 6639: 6636: 6630: 6623: 6620: 6617: 6613: 6587: 6584: 6561: 6558: 6552: 6549: 6534: 6531: 6530: 6529: 6518: 6513: 6510: 6505: 6502: 6497: 6494: 6489: 6486: 6483: 6478: 6475: 6470: 6467: 6464: 6461: 6458: 6455: 6452: 6447: 6444: 6439: 6436: 6433: 6415: 6414: 6401: 6398: 6393: 6390: 6385: 6382: 6377: 6372: 6368: 6365: 6362: 6359: 6356: 6350: 6345: 6342: 6337: 6332: 6329: 6324: 6321: 6318: 6303: 6300: 6296: 6295: 6281: 6278: 6272: 6266: 6263: 6257: 6251: 6248: 6242: 6236: 6233: 6227: 6224: 6209: 6206: 6198: 6197: 6184: 6181: 6176: 6171: 6168: 6163: 6158: 6155: 6150: 6143: 6134: 6126: 6117: 6109: 6102: 6093: 6085: 6076: 6068: 6063: 6060: 6055: 6050: 6047: 6027: 6026: 6013: 6010: 6005: 6000: 5997: 5992: 5987: 5984: 5967: 5964: 5962: 5961:Multiplication 5959: 5958: 5957: 5946: 5940: 5937: 5931: 5928: 5923: 5915: 5912: 5906: 5903: 5898: 5893: 5890: 5887: 5884: 5881: 5878: 5875: 5872: 5869: 5863: 5860: 5854: 5851: 5848: 5834: 5833: 5819: 5816: 5810: 5804: 5801: 5795: 5789: 5786: 5780: 5774: 5771: 5765: 5759: 5756: 5737: 5734: 5726: 5725: 5708: 5705: 5700: 5697: 5693: 5690: 5685: 5680: 5677: 5672: 5666: 5663: 5660: 5655: 5652: 5649: 5643: 5637: 5634: 5631: 5626: 5623: 5620: 5614: 5611: 5609: 5607: 5602: 5599: 5594: 5591: 5587: 5584: 5579: 5574: 5571: 5566: 5560: 5557: 5554: 5549: 5546: 5543: 5537: 5531: 5528: 5525: 5520: 5517: 5514: 5508: 5505: 5503: 5499: 5496: 5491: 5486: 5483: 5478: 5477: 5450: 5447: 5421: 5418: 5403: 5402: 5388: 5385: 5380: 5377: 5374: 5371: 5368: 5362: 5357: 5354: 5349: 5344: 5341: 5325: 5324: 5311: 5308: 5303: 5300: 5295: 5292: 5287: 5282: 5279: 5274: 5269: 5266: 5250: 5249: 5236: 5233: 5228: 5223: 5220: 5187: 5184: 5178: 5172: 5169: 5163: 5157: 5154: 5073: 5070: 5064: 5058: 5055: 5049: 5043: 5040: 5012: 5009: 4995: 4994: 4981: 4978: 4973: 4968: 4965: 4949: 4948: 4937: 4932: 4929: 4924: 4919: 4916: 4911: 4903: 4900: 4895: 4889: 4886: 4883: 4878: 4875: 4872: 4866: 4857: 4854: 4851: 4846: 4843: 4840: 4834: 4829: 4826: 4821: 4813: 4810: 4785: 4782: 4765: 4762: 4736: 4733: 4715: 4714: 4699: 4696: 4690: 4687: 4681: 4678: 4672: 4666: 4663: 4657: 4651: 4648: 4629: 4626: 4604: 4601: 4595: 4589: 4586: 4561: 4558: 4555: 4531: 4528: 4525: 4520: 4517: 4514: 4486: 4483: 4480: 4475: 4472: 4469: 4445: 4439: 4436: 4410: 4407: 4378: 4375: 4369: 4363: 4360: 4334: 4331: 4305: 4302: 4251: 4248: 4245: 4240: 4237: 4234: 4206: 4203: 4200: 4195: 4192: 4189: 4162: 4159: 4133: 4130: 4112: 4111: 4100: 4097: 4094: 4091: 4088: 4077: 4073: 4070: 4063: 4056: 4053: 4049: 4034: 4031: 4027: 4021: 4014: 4011: 4007: 3990: 3989: 3977: 3950: 3947: 3941: 3935: 3932: 3914: 3911: 3903: 3902: 3888: 3885: 3879: 3872: 3868: 3864: 3859: 3855: 3851: 3844: 3838: 3835: 3715: 3712: 3686: 3683: 3578: 3575: 3572: 3569: 3549: 3546: 3543: 3540: 3518: 3493: 3490: 3474: 3471: 3389: 3386: 3358: 3355: 3329: 3326: 3303: 3291: 3288: 3264: 3261: 3238: 3234: 3230: 3225: 3221: 3217: 3188: 3185: 3179: 3173: 3170: 3144: 3141: 3135: 3129: 3126: 3120: 3114: 3111: 3085: 3082: 3056: 3053: 3014: 3011: 3008: 3004: 3000: 2997: 2992: 2987: 2967: 2962: 2957: 2954: 2950: 2946: 2943: 2923: 2920: 2914: 2911: 2903: 2898: 2875: 2872: 2866: 2863: 2858: 2850: 2847: 2824: 2820: 2816: 2812: 2808: 2797: 2796: 2781: 2776: 2773: 2768: 2763: 2760: 2755: 2750: 2746: 2743: 2740: 2737: 2734: 2728: 2725: 2722: 2717: 2713: 2710: 2707: 2704: 2701: 2695: 2692: 2690: 2686: 2679: 2676: 2670: 2664: 2663: 2660: 2655: 2652: 2647: 2642: 2639: 2634: 2629: 2626: 2621: 2618: 2615: 2610: 2607: 2602: 2597: 2588: 2585: 2573: 2568: 2565: 2560: 2555: 2552: 2547: 2542: 2539: 2534: 2529: 2526: 2521: 2516: 2513: 2508: 2505: 2503: 2497: 2494: 2483: 2480: 2469: 2468: 2440: 2435: 2428: 2420: 2417: 2411: 2406: 2380: 2376: 2372: 2367: 2363: 2359: 2332: 2329: 2313: 2310: 2304: 2298: 2295: 2289: 2283: 2280: 2274: 2268: 2265: 2239: 2236: 2230: 2224: 2221: 2215: 2209: 2206: 2180: 2177: 2154: 2148: 2145: 2139: 2133: 2130: 2124: 2118: 2115: 2089: 2086: 2060: 2057: 2031: 2028: 2002: 1999: 1969: 1966: 1960: 1954: 1951: 1932: 1929: 1927: 1924: 1853: 1847: 1844: 1838: 1835: 1832: 1826: 1823: 1796: 1790: 1787: 1781: 1775: 1772: 1766: 1760: 1757: 1751: 1745: 1742: 1736: 1733: 1703: 1698: 1690: 1687: 1681: 1678: 1673: 1668: 1645: 1642: 1636: 1633: 1587: 1584: 1578: 1575: 1552: 1549: 1526: 1506: 1502: 1498: 1491: 1468: 1465: 1459: 1437:mixed fraction 1428: 1425: 1344:. Dividing by 1270: 1267: 1223: 1222: 1219: 1218:6 are red, and 1216: 1197: 1194: 1089: 1086: 1074:absolute value 1055: 1052: 1044: 1043: 998: 864: 861: 820: 817: 815: 812: 605:fraction slash 485: 482: 466: 463: 434: 430: 375: 122: 119: 93: 90: 15: 9: 6: 4: 3: 2: 12472: 12461: 12458: 12457: 12455: 12440: 12437: 12435: 12432: 12430: 12427: 12425: 12422: 12420: 12417: 12415: 12412: 12408: 12405: 12404: 12403: 12400: 12398: 12395: 12391: 12388: 12387: 12386: 12383: 12381: 12378: 12376: 12373: 12371: 12368: 12366: 12363: 12361: 12358: 12356: 12353: 12351: 12348: 12347: 12344: 12320: 12319: 12317: 12313: 12298: 12294: 12290: 12287: 12286: 12284: 12277: 12271: 12267: 12263: 12256: 12251: 12249: 12244: 12242: 12237: 12236: 12233: 12225: 12221: 12217: 12213: 12209: 12205: 12204: 12181: 12174: 12168: 12161: 12157: 12153: 12148: 12140: 12134: 12130: 12123: 12115: 12109: 12101: 12095: 12091: 12085: 12078: 12077:Cajori (1928) 12073: 12058: 12054: 12050: 12049: 12041: 12033: 12027: 12023: 12019: 12015: 12011: 12007: 12000: 11984: 11980: 11973: 11971: 11969: 11967: 11958: 11954: 11950: 11946: 11942: 11938: 11937: 11929: 11921: 11919:1-56881-201-9 11915: 11911: 11907: 11901: 11893: 11887: 11883: 11876: 11861: 11857: 11851: 11836: 11832: 11826: 11808: 11804: 11800: 11796: 11792: 11788: 11784: 11783: 11775: 11768: 11753: 11749: 11743: 11728: 11724: 11720: 11714: 11711:p. 120. 11710: 11706: 11702: 11695: 11680: 11676: 11670: 11655: 11651: 11647: 11641: 11626:on 2017-12-01 11625: 11621: 11617: 11611: 11603: 11602: 11594: 11592: 11584:. p. 65. 11583: 11582: 11574: 11572: 11564: 11562: 11556: 11552: 11548: 11544: 11540: 11536: 11529: 11521: 11519:9781783741373 11515: 11510: 11505: 11501: 11494: 11486: 11480: 11475: 11474: 11465: 11458: 11456: 11433: 11427: 11421: 11416: 11415: 11406: 11398: 11396:9781592571611 11392: 11388: 11387: 11379: 11364: 11360: 11354: 11350: 11349: 11341: 11327:on 2012-04-15 11326: 11322: 11318: 11312: 11297: 11293: 11289: 11282: 11274: 11273: 11264: 11256: 11250: 11246: 11245: 11237: 11229: 11225: 11221: 11220:Mathematics I 11214: 11206: 11200: 11196: 11195: 11187: 11185: 11176: 11172: 11168: 11164: 11157: 11149: 11145: 11138: 11130: 11126: 11119: 11111: 11105: 11101: 11100:Point Roberts 11097: 11090: 11083: 11082:Cajori (1928) 11078: 11062: 11058: 11052: 11048: 11044: 11043: 11035: 11033: 11018: 11014: 11008: 11000: 10994: 10989: 10988: 10979: 10972: 10968: 10964: 10960: 10954: 10950: 10947: 10944: 10942:1-58488-347-2 10938: 10934: 10930: 10912: 10905: 10904: 10897: 10891: 10890: 10883: 10877: 10871: 10867: 10843: 10840: 10839: 10820: 10817: 10816: 10813: 10810: 10809: 10806: 10804: 10801: 10800: 10797: 10796: 10777: 10775: 10772: 10771: 10768: 10765: 10764: 10761: 10758: 10757: 10754: 10752: 10749: 10748: 10745: 10744: 10733: 10730: 10729: 10718: 10715: 10714: 10711: 10710:Prime numbers 10708: 10707: 10703: 10700: 10699: 10695: 10692: 10691: 10688: 10668: 10661: 10658: 10657: 10654: 10653: 10650: 10630: 10623: 10620: 10619: 10616: 10615: 10612: 10592: 10585: 10582: 10581: 10578: 10577: 10574: 10554: 10547: 10544: 10543: 10540: 10539: 10536: 10516: 10509: 10506: 10505: 10502: 10497: 10496: 10493: 10492: 10487: 10484: 10482: 10479: 10477: 10474: 10472: 10469: 10468: 10462: 10460: 10444: 10424: 10400: 10376: 10364: 10354: 10352: 10348: 10344: 10340: 10336: 10321: 10319: 10315: 10311: 10307: 10301: 10299: 10295: 10291: 10287: 10283: 10279: 10275: 10271: 10267: 10263: 10259: 10253: 10251: 10247: 10242: 10240: 10219: 10215: 10210: 10206: 10194: 10190: 10186: 10182: 10171: 10166: 10096: 10093: 10090: 10089: 10088: 10082: 10076: 10073: 10072: 10071: 10069: 10057: 10052: 10043: 10032: 10021: 10016: AD 500 10010: 10006: 10001: 9999: 9995: 9991: 9984: 9980: 9976: 9973: 9962: 9959: 9956: 9952: 9948: 9944: 9939: 9937: 9933: 9923: 9919: 9915: 9904: 9901: 9893: 9882: 9879: 9875: 9872: 9868: 9865: 9861: 9858: 9854: 9851: – 9850: 9846: 9845:Find sources: 9839: 9835: 9829: 9828: 9821: 9817: 9812: 9811: 9786: 9783: 9773: 9770: 9767: 9763: 9759: 9755: 9751: 9747: 9744: 9741: 9721: 9717: 9714: 9711: 9708: 9707: 9706: 9703: 9701: 9697: 9693: 9689: 9683: 9673: 9653: 9647: 9642: 9639: 9636: 9630: 9627: 9621: 9618: 9615: 9605: 9600: 9597: 9594: 9588: 9582: 9574: 9564: 9559: 9553: 9548: 9543: 9531: 9526: 9523: 9520: 9514: 9508: 9500: 9495: 9492: 9489: 9482: 9477: 9474: 9471: 9465: 9457: 9452: 9449: 9446: 9442: 9437: 9429: 9424: 9421: 9418: 9414: 9405: 9389: 9383: 9378: 9375: 9372: 9366: 9363: 9357: 9354: 9351: 9341: 9336: 9333: 9330: 9324: 9318: 9310: 9300: 9295: 9289: 9284: 9279: 9267: 9262: 9259: 9256: 9250: 9244: 9236: 9231: 9228: 9225: 9218: 9213: 9210: 9207: 9201: 9193: 9188: 9185: 9182: 9178: 9173: 9165: 9160: 9157: 9154: 9150: 9141: 9140: 9139: 9137: 9133: 9112: 9106: 9101: 9095: 9089: 9084: 9078: 9072: 9068: 9063: 9057: 9053: 9044: 9043: 9042: 9040: 9036: 9032: 9028: 9022: 9018: 9008: 9006: 9002: 8998: 8977: 8974: 8971: 8967: 8962: 8956: 8953: 8950: 8946: 8919: 8916: 8911: 8907: 8901: 8898: 8887: 8882: 8880: 8876: 8860: 8840: 8837: 8834: 8829: 8820:since all of 8807: 8804: 8800: 8796: 8788: 8769: 8765: 8759: 8746: 8742: 8738: 8734: 8730: 8725: 8723: 8699: 8696: 8690: 8687: 8679: 8676: 8670: 8667: 8652: 8633: 8630: 8625: 8621: 8615: 8612: 8609: 8596: 8592: 8588: 8567: 8564: 8561: 8558: 8555: 8550: 8546: 8540: 8537: 8524: 8519: 8517: 8496: 8493: 8488: 8484: 8478: 8475: 8472: 8444: 8441: 8438: 8435: 8432: 8427: 8423: 8417: 8414: 8403: 8399: 8393: 8383: 8381: 8377: 8373: 8369: 8365: 8361: 8357: 8353: 8349: 8345: 8341: 8337: 8334: 8330: 8326: 8321: 8319: 8315: 8309: 8304: 8276: 8250: 8247: 8243: 8239: 8236: 8229: 8222: 8219: 8215: 8211: 8208: 8198: 8189: 8186: 8183: 8177: 8171: 8168: 8165: 8152: 8151: 8132: 8129: 8125: 8121: 8118: 8111: 8105: 8102: 8099: 8070: 8067: 8063: 8059: 8056: 8049: 8043: 8040: 8037: 8027: 8026: 8025: 8022: 8018: 7999: 7996: 7993: 7990: 7987: 7984: 7971: 7968: 7965: 7959: 7953: 7950: 7947: 7937: 7936: 7935: 7933: 7910: 7899: 7896: 7893: 7875: 7872: 7869: 7866: 7852: 7849: 7846: 7840: 7838: 7831: 7828: 7820: 7817: 7814: 7796: 7793: 7790: 7761: 7758: 7755: 7752: 7746: 7744: 7736: 7733: 7730: 7724: 7713: 7712: 7711: 7690: 7687: 7684: 7669: 7666: 7663: 7660: 7657: 7651: 7645: 7642: 7639: 7633: 7627: 7624: 7621: 7611: 7594: 7591: 7588: 7585: 7582: 7576: 7570: 7567: 7564: 7558: 7552: 7549: 7546: 7536: 7518: 7515: 7512: 7509: 7506: 7503: 7500: 7497: 7491: 7485: 7482: 7479: 7473: 7467: 7464: 7461: 7451: 7433: 7430: 7427: 7424: 7421: 7418: 7415: 7412: 7406: 7400: 7397: 7394: 7388: 7382: 7379: 7376: 7366: 7365: 7364: 7362: 7358: 7354: 7350: 7334: 7331: 7328: 7325: 7305: 7298: 7279: 7276: 7273: 7262: 7232: 7229: 7228: 7226: 7222: 7216: 7211: 7208: 7204: 7203: 7201: 7189: 7185: 7181: 7180: 7177: 7173: 7169: 7165: 7159: 7155: 7149: 7145: 7140: 7132: 7128: 7127: 7125: 7117: 7113: 7112: 7110: 7102: 7099: 7098: 7096: 7092: 7091: 7090: 7084: 7083: 7082: 7073: 7072: 7071: 7057: 7050: 7043: 7042: 7041: 7039: 7035: 7034:leading zeros 7023: 7016: 7009: 7002: 7001: 7000: 6994: 6989: 6987: 6981: 6975: 6965: 6952: 6946: 6943: 6937: 6934: 6807: 6790: 6787: 6781: 6774: 6771: 6768: 6763: 6760: 6757: 6750: 6744: 6741: 6735: 6729: 6726: 6720: 6714: 6711: 6705: 6699: 6696: 6685: 6666: 6663: 6637: 6634: 6628: 6621: 6618: 6615: 6611: 6585: 6582: 6559: 6556: 6550: 6547: 6516: 6511: 6508: 6503: 6500: 6495: 6492: 6487: 6484: 6481: 6476: 6473: 6468: 6465: 6462: 6459: 6456: 6453: 6450: 6445: 6442: 6437: 6434: 6431: 6424: 6423: 6422: 6420: 6399: 6396: 6391: 6388: 6383: 6380: 6375: 6370: 6366: 6363: 6360: 6357: 6354: 6348: 6343: 6340: 6335: 6330: 6327: 6322: 6319: 6316: 6309: 6308: 6307: 6299: 6279: 6276: 6270: 6264: 6261: 6255: 6249: 6246: 6240: 6234: 6231: 6225: 6222: 6215: 6214: 6213: 6205: 6203: 6182: 6179: 6174: 6169: 6166: 6161: 6156: 6153: 6148: 6141: 6132: 6124: 6115: 6107: 6100: 6091: 6083: 6074: 6066: 6061: 6058: 6053: 6048: 6045: 6036: 6035: 6034: 6031: 6011: 6008: 6003: 5998: 5995: 5990: 5985: 5982: 5973: 5972: 5971: 5944: 5938: 5935: 5929: 5926: 5913: 5910: 5904: 5901: 5891: 5885: 5882: 5879: 5876: 5873: 5867: 5861: 5858: 5852: 5849: 5846: 5839: 5838: 5837: 5817: 5814: 5808: 5802: 5799: 5793: 5787: 5784: 5778: 5772: 5769: 5763: 5757: 5754: 5744: 5743: 5742: 5733: 5731: 5706: 5703: 5698: 5691: 5688: 5683: 5678: 5675: 5670: 5664: 5661: 5658: 5653: 5650: 5647: 5641: 5635: 5632: 5629: 5624: 5621: 5618: 5612: 5610: 5600: 5597: 5592: 5585: 5582: 5577: 5572: 5569: 5564: 5558: 5555: 5552: 5547: 5544: 5541: 5535: 5529: 5526: 5523: 5518: 5515: 5512: 5506: 5504: 5497: 5494: 5489: 5484: 5481: 5468: 5467: 5466: 5448: 5445: 5419: 5416: 5386: 5383: 5378: 5375: 5372: 5369: 5366: 5360: 5355: 5352: 5347: 5342: 5339: 5330: 5329: 5328: 5309: 5306: 5301: 5298: 5293: 5290: 5285: 5280: 5277: 5272: 5267: 5264: 5255: 5254: 5253: 5234: 5231: 5226: 5221: 5218: 5209: 5208: 5207: 5204: 5185: 5182: 5176: 5170: 5167: 5161: 5155: 5152: 5122: 5071: 5068: 5062: 5056: 5053: 5047: 5041: 5038: 5010: 5007: 4979: 4976: 4971: 4966: 4963: 4954: 4953: 4952: 4935: 4930: 4927: 4922: 4917: 4914: 4909: 4901: 4898: 4893: 4887: 4884: 4881: 4876: 4873: 4870: 4864: 4855: 4852: 4849: 4844: 4841: 4838: 4832: 4827: 4824: 4819: 4811: 4808: 4799: 4798: 4797: 4794: 4792: 4763: 4760: 4734: 4731: 4719: 4697: 4694: 4688: 4685: 4679: 4676: 4670: 4664: 4661: 4655: 4649: 4646: 4636: 4635: 4634: 4625: 4621: 4602: 4599: 4593: 4587: 4584: 4559: 4556: 4553: 4529: 4526: 4523: 4518: 4515: 4512: 4484: 4481: 4478: 4473: 4470: 4467: 4443: 4437: 4434: 4408: 4405: 4393: 4376: 4373: 4367: 4361: 4358: 4332: 4329: 4303: 4300: 4289: 4285: 4281: 4277: 4273: 4269: 4249: 4246: 4243: 4238: 4235: 4232: 4204: 4201: 4198: 4193: 4190: 4187: 4160: 4157: 4131: 4128: 4116: 4098: 4095: 4092: 4089: 4086: 4075: 4071: 4068: 4061: 4054: 4051: 4047: 4032: 4029: 4025: 4019: 4012: 4009: 4005: 3995: 3994: 3993: 3988:are positive. 3975: 3948: 3945: 3939: 3933: 3930: 3920: 3919: 3918: 3910: 3908: 3886: 3883: 3877: 3870: 3866: 3862: 3857: 3853: 3849: 3842: 3836: 3833: 3823: 3822: 3821: 3802: 3733: 3731: 3713: 3710: 3684: 3681: 3670: 3666: 3662: 3661: 3655: 3653: 3652: 3647: 3643: 3634: 3628: 3620: 3612: 3604: 3596: 3576: 3573: 3570: 3567: 3547: 3544: 3541: 3538: 3516: 3491: 3488: 3470: 3387: 3384: 3356: 3353: 3327: 3324: 3301: 3287: 3284: 3282: 3278: 3274: 3270: 3260: 3256: 3236: 3232: 3228: 3223: 3219: 3215: 3186: 3183: 3177: 3171: 3168: 3142: 3139: 3133: 3127: 3124: 3118: 3112: 3109: 3083: 3080: 3054: 3051: 3040: 3039: 3034: 3030: 3025: 3012: 3006: 3002: 2998: 2985: 2965: 2952: 2948: 2944: 2921: 2918: 2912: 2909: 2896: 2873: 2870: 2864: 2861: 2848: 2845: 2822: 2818: 2814: 2810: 2806: 2779: 2774: 2771: 2766: 2761: 2758: 2753: 2748: 2744: 2741: 2738: 2735: 2732: 2726: 2723: 2720: 2715: 2711: 2708: 2705: 2702: 2699: 2693: 2691: 2684: 2677: 2674: 2668: 2658: 2653: 2650: 2645: 2640: 2637: 2632: 2627: 2624: 2619: 2616: 2613: 2608: 2605: 2600: 2595: 2586: 2583: 2571: 2566: 2563: 2558: 2553: 2550: 2545: 2540: 2537: 2532: 2527: 2524: 2519: 2514: 2511: 2506: 2504: 2495: 2492: 2481: 2478: 2459: 2458: 2457: 2455: 2438: 2418: 2415: 2409: 2378: 2374: 2370: 2365: 2361: 2357: 2345: 2338: 2328: 2311: 2308: 2302: 2296: 2293: 2287: 2281: 2278: 2272: 2266: 2263: 2237: 2234: 2228: 2222: 2219: 2213: 2207: 2204: 2178: 2175: 2152: 2146: 2143: 2137: 2131: 2128: 2122: 2116: 2113: 2087: 2084: 2058: 2055: 2029: 2026: 2000: 1997: 1986: 1967: 1964: 1958: 1952: 1949: 1938: 1923: 1921: 1912: 1904: 1896: 1892: 1891:metric system 1888: 1864: 1851: 1845: 1842: 1836: 1833: 1830: 1824: 1821: 1810: 1794: 1788: 1785: 1779: 1773: 1770: 1764: 1758: 1755: 1749: 1743: 1740: 1734: 1731: 1723: 1719: 1714: 1701: 1688: 1685: 1679: 1676: 1666: 1643: 1640: 1634: 1631: 1623: 1585: 1582: 1576: 1573: 1550: 1547: 1524: 1504: 1500: 1496: 1489: 1466: 1463: 1457: 1448: 1446: 1445:concatenation 1442: 1441:mixed numeral 1438: 1434: 1427:Mixed numbers 1424: 1421: 1417: 1413: 1408: 1406: 1402: 1398: 1394: 1393: 1387: 1385: 1381: 1377: 1372: 1354: 1349: 1327: 1322: 1301: 1297: 1296: 1291: 1287: 1283: 1278: 1277: 1266: 1264: 1226: 1221:4 are yellow, 1220: 1217: 1214: 1213: 1212: 1210: 1205: 1204: 1193: 1175: 1154: 1135: 1133: 1097: 1096: 1085: 1083: 1079: 1075: 1071: 1066: 1064: 1063: 1051: 1049: 1009: 1005: 1004: 999: 980: 979: 978:unit fraction 974: 973: 972: 970: 965: 963: 895: 891: 887: 883: 862: 859: 846: 842: 838: 834: 830: 826: 811: 777: 773: 753: 715: 696: 694: 657: 652: 634: 630: 626: 622: 618: 614: 610: 606: 602: 598: 594: 590: 560: 555: 553: 549: 545: 525: 521: 517: 513: 509: 505: 499: 495: 491: 481: 464: 461: 451: 432: 428: 412: 404: 396: 392: 362: 358: 354: 350: 342: 334: 326: 321: 222: 200: 196: 191: 157: 153: 139: 120: 117: 91: 88: 77: 73: 69: 65: 61: 57: 30: 26: 22: 12261: 12223: 12211: 12187:. Retrieved 12167: 12147: 12128: 12122: 12113: 12108: 12089: 12084: 12072: 12061:. Retrieved 12047: 12040: 12009: 11999: 11987:. Retrieved 11940: 11934: 11928: 11909: 11900: 11881: 11875: 11864:. Retrieved 11850: 11839:. Retrieved 11825: 11814:. Retrieved 11786: 11780: 11767: 11756:. Retrieved 11742: 11732:28 September 11730:. Retrieved 11704: 11694: 11683:. Retrieved 11669: 11658:. Retrieved 11649: 11640: 11628:. Retrieved 11624:the original 11619: 11610: 11600: 11580: 11560: 11558: 11538: 11534: 11528: 11499: 11493: 11472: 11464: 11454: 11431: 11429: 11413: 11405: 11385: 11378: 11367:. Retrieved 11347: 11340: 11329:. Retrieved 11325:the original 11320: 11311: 11300:. Retrieved 11291: 11281: 11270: 11263: 11243: 11236: 11219: 11213: 11193: 11169:(2): 83–88. 11166: 11162: 11156: 11147: 11143: 11137: 11128: 11124: 11118: 11095: 11089: 11077: 11065:. Retrieved 11041: 11020:. Retrieved 11016: 11007: 10986: 10978: 10958: 10953: 10932: 10926: 10911: 10901: 10887: 10870: 10750: 10458: 10360: 10332: 10303: 10293: 10274:Simon Stevin 10269: 10261: 10255: 10243: 10167: 10100: 10086: 10004: 10002: 9940: 9911: 9896: 9887: 9877: 9870: 9863: 9856: 9844: 9832:Please help 9827:verification 9824: 9771: 9765: 9749: 9745: 9739: 9719: 9715: 9709: 9704: 9696:Number Forms 9685: 9670: 9129: 9033:it (compare 9024: 8883: 8878: 8875:real numbers 8726: 8721: 8653: 8594: 8590: 8586: 8520: 8395: 8371: 8367: 8366:elements of 8363: 8359: 8355: 8351: 8343: 8335: 8328: 8324: 8322: 8317: 8313: 8307: 8277: 8274: 8014: 7929: 7708: 7258: 7230: 7224: 7214: 7206: 7190:= 1,523,987. 7187: 7183: 7175: 7171: 7167: 7163: 7157: 7153: 7147: 7143: 7133:= 1,523,987. 7130: 7115: 7100: 7094: 7088: 7080: 7065: 7055:= 392/999000 7031: 6990: 6983: 6851:. To change 6813: 6536: 6416: 6305: 6297: 6211: 6199: 6032: 6028: 5969: 5835: 5739: 5727: 5404: 5326: 5251: 5205: 5123: 4996: 4950: 4795: 4787: 4631: 4622: 4394: 4287: 4283: 4279: 4275: 4271: 4267: 4117: 4113: 3991: 3916: 3904: 3803: 3734: 3729: 3668: 3664: 3658: 3656: 3651:lowest terms 3649: 3635: 3626: 3618: 3610: 3602: 3594: 3476: 3293: 3285: 3277:distributive 3266: 3257: 3036: 3032: 3028: 3026: 2798: 2343: 2341: 1934: 1910: 1902: 1865: 1715: 1621: 1449: 1444: 1440: 1436: 1433:mixed number 1432: 1430: 1409: 1404: 1396: 1390: 1388: 1373: 1350: 1323: 1293: 1274: 1272: 1227: 1224: 1215:2 are white, 1208: 1201: 1199: 1173: 1155: 1136: 1093: 1091: 1081: 1077: 1069: 1067: 1060: 1057: 1048:Number Forms 1045: 1008:power of two 1001: 976: 968: 966: 893: 885: 881: 844: 840: 832: 828: 824: 822: 775: 716: 697: 653: 633:typefounding 628: 624: 559:fraction bar 558: 556: 515: 503: 501: 410: 402: 394: 360: 356: 352: 348: 340: 332: 322: 223: 192: 155: 137: 75: 71: 67: 55: 53: 25: 12402:Irreducible 12332:Denominator 11989:15 February 11707:. Hoboken: 11453:is read as 11013:"Fractions" 10290:sexagesimal 10278:Netherlands 10177: 1200 10038: 1150 10020:Brahmagupta 9996:wrote the " 9958:philosopher 9928: 1000 9914:reciprocals 9031:rationalize 9007:for more). 8523:polynomials 8348:polynomials 7353:subtraction 7218:= 1,522,464 7028:= 6291/9999 5736:Subtraction 4624:fractions. 3660:irreducible 3273:associative 3269:commutative 1403:, as in 75 1263:probability 839:written as 524:dēnōminātor 516:denominator 393:. The term 190:of a cake. 156:denominator 12434:Percentage 12429:Paper size 12338:= Quotient 12220:"Fraction" 12189:2013-10-10 12063:2017-08-30 11866:2013-06-18 11841:2013-06-18 11816:2010-01-27 11758:2012-08-15 11748:"Fraction" 11685:2018-03-13 11660:2018-03-13 11369:2014-07-29 11331:2011-12-31 11302:2014-10-30 11292:Math Forum 11175:1291644250 11142:"solidus, 11123:"virgule, 11067:2016-02-20 11022:2020-08-27 10923:References 10876:Bringhurst 10803:Irrational 10304:While the 10262:De Thiende 10068:Bhaskara I 10027: 628 10009:Aryabhatta 10005:bhinnarasi 9983:Metapontum 9968: 530 9961:Pythagoras 9860:newspapers 9849:"Fraction" 9758:half crown 9688:typography 9680:See also: 8314:equivalent 8024:fractions 7887:with 7784:with 7182:10,000,000 7129:10,000,000 6978:See also: 6684:reciprocal 1384:per centum 1376:percentage 1290:interpunct 1095:reciprocal 772:slash mark 609:typography 488:See also: 484:Vocabulary 12407:Reduction 12365:Continued 12350:Algebraic 12326:Numerator 12262:Fractions 11727:719886424 11630:29 August 11555:195210281 10842:Imaginary 10351:geoboards 10347:dot paper 10298:Samarkand 10260:pamphlet 10170:Al-Hassār 10062:or cross 9951:Followers 9890:June 2023 9640:− 9631:− 9619:− 9598:− 9554:− 9524:− 9493:− 9475:− 9466:⋅ 9367:− 9355:− 9290:− 9202:⋅ 9186:− 9158:− 9136:conjugate 9079:⋅ 8975:− 8917:− 8884:The term 8838:π 8797:π 8691:− 8631:− 8565:− 8494:− 8442:− 8358:. So for 8199:∼ 8112:∼ 8050:∼ 7980:⟺ 7960:∼ 7870:≠ 7829:− 7753:− 7725:− 7688:≠ 7634:÷ 7559:⋅ 7504:− 7474:− 7329:≠ 7213:9,990,000 7205:9,990,000 7062:= 12/9900 7021:= 264/999 6772:⋅ 6761:⋅ 6736:× 6706:÷ 6619:⋅ 6557:÷ 6469:× 6457:× 6435:× 6358:× 6349:× 6320:× 6256:× 6226:× 6162:× 6108:× 6054:× 5991:× 5905:− 5883:− 5877:− 5850:− 5794:− 5764:− 5662:⋅ 5651:⋅ 5633:⋅ 5622:⋅ 5556:⋅ 5545:⋅ 5527:⋅ 5516:⋅ 5162:× 5048:× 4885:× 4874:× 4853:× 4842:× 4557:× 4527:× 4516:× 4482:× 4471:× 4247:⋅ 4236:⋅ 4202:⋅ 4191:⋅ 4096:− 4087:− 4069:− 4052:− 4030:− 4010:− 3867:÷ 3854:÷ 3178:× 3119:× 2754:× 2736:× 2721:÷ 2703:× 2633:× 2614:÷ 2546:× 2520:÷ 1667:− 1632:− 1482:cakes or 1072:, if the 888:are both 512:numerātor 504:numerator 150:), and a 138:numerator 34:fraction 12454:Category 12380:Egyptian 12315:Fraction 12297:Quotient 12289:Dividend 12180:Archived 12156:Archived 12057:Archived 11983:Archived 11860:Archived 11835:Archived 11807:Archived 11752:Archived 11679:Archived 11654:Archived 11551:ProQuest 11436:⁠2 11363:Archived 11296:Archived 11171:ProQuest 11061:Archived 10967:Archived 10898:⟩ 10894:⟨ 10884:⟩ 10880:⟨ 10751:Fraction 10584:Rational 10486:FRACTRAN 10481:Multiple 10476:0.999... 10465:See also 10459:fraction 10314:Baghdadi 10286:Al-Kāshī 10270:La Disme 10252:states: 10149:(i.e., 1 10070:writes: 10042:Sanskrit 10031:Bhaskara 9979:Hippasus 9918:integers 9746:Shilling 9692:one half 9132:binomial 9039:monomial 9027:radicals 9017:Nth root 8789:such as 8525:, as in 8398:quotient 8251:′ 8240:′ 8223:′ 8212:′ 8133:′ 8122:′ 8071:′ 8060:′ 7932:relation 7361:division 7349:addition 7297:integers 7194:− 1,523. 7186:− 10,000 7118:= 1,523. 7103:= 0.1523 6993:vinculum 6881:1.000... 6869:1.000... 6533:Division 5089:. Since 4628:Addition 4501: ? 4424: ? 4319: ? 3966:3 > 2 3964:because 1868:⁠2 1603:⁠2 1420:accurate 1416:multiply 1392:permille 1304:⁠3 890:integers 880:, where 548:dividend 544:division 395:fraction 391:quotient 347:, where 199:division 154:integer 152:non-zero 56:fraction 12397:Integer 12370:Decimal 12335:⁠ 12323:⁠ 12293:Divisor 12079:, p. 89 11945:Bibcode 11803:4145131 11451:⁠ 10903:solidus 10900:as the 10889:virgule 10886:as the 10660:Natural 10622:Integer 10508:Complex 10415:⁠ 10367:⁠ 10306:Persian 10282:Chinese 10258:Flemish 10189:Morocco 10163:⁠ 10151:⁠ 10147:⁠ 10135:⁠ 10131:⁠ 10119:⁠ 10115:⁠ 10103:⁠ 10029:), and 9953:of the 9874:scholar 9806:History 9736:⁠ 9724:⁠ 9720:upright 9700:Unicode 8993:⁠ 8937:⁠ 8879:integer 8783:⁠ 8749:⁠ 8718:⁠ 8657:⁠ 8649:⁠ 8599:⁠ 8583:⁠ 8527:⁠ 8512:⁠ 8462:⁠ 8400:of two 8303:coprime 8292:⁠ 8280:⁠ 7247:⁠ 7244:9990000 7238:1522464 7235:⁠ 7161:, then 7014:= 62/99 6935:12.3456 6921:⁠ 6909:⁠ 6905:⁠ 6893:⁠ 6865:⁠ 6853:⁠ 6829:⁠ 6817:⁠ 6572:equals 5201:⁠ 5142:⁠ 5138:⁠ 5126:⁠ 5119:⁠ 5107:⁠ 5103:⁠ 5091:⁠ 5087:⁠ 5028:⁠ 4618:⁠ 4574:⁠ 3818:⁠ 3806:⁠ 3798:⁠ 3786:⁠ 3782:⁠ 3770:⁠ 3766:⁠ 3754:⁠ 3750:⁠ 3738:⁠ 3665:reduced 3631:⁠ 3615:⁠ 3607:⁠ 3591:⁠ 3529:⁠ 3509:⁠ 3467:⁠ 3455:⁠ 3451:⁠ 3439:⁠ 3435:⁠ 3423:⁠ 3419:⁠ 3407:⁠ 3403:⁠ 3374:⁠ 3253:⁠ 3204:⁠ 1915:⁠ 1899:⁠ 1883:⁠ 1618:⁠ 1369:⁠ 1357:⁠ 1319:⁠ 1259:⁠ 1247:⁠ 1243:⁠ 1231:⁠ 1190:⁠ 1178:⁠ 1170:⁠ 1158:⁠ 1151:⁠ 1139:⁠ 1128:⁠ 1116:⁠ 1112:⁠ 1100:⁠ 1050:block. 1040:⁠ 1028:⁠ 1024:⁠ 1012:⁠ 1010:, e.g. 995:⁠ 983:⁠ 958:⁠ 946:⁠ 942:⁠ 930:⁠ 926:⁠ 914:⁠ 910:⁠ 898:⁠ 878:⁠ 849:⁠ 808:⁠ 805:1000000 796:⁠ 792:⁠ 780:⁠ 768:⁠ 756:⁠ 748:⁠ 736:⁠ 732:⁠ 720:⁠ 712:⁠ 700:⁠ 693:percent 688:⁠ 676:⁠ 672:⁠ 660:⁠ 649:⁠ 637:⁠ 582:⁄ 575:⁠ 563:⁠ 552:divisor 540:⁠ 528:⁠ 415:⁠ 399:⁠ 387:⁠ 365:⁠ 345:⁠ 329:⁠ 318:⁠ 306:⁠ 302:⁠ 290:⁠ 286:⁠ 274:⁠ 270:⁠ 258:⁠ 254:⁠ 242:⁠ 238:⁠ 226:⁠ 215:⁠ 203:⁠ 188:⁠ 176:⁠ 172:⁠ 160:⁠ 145:⁄ 64:fractus 48:⁠ 36:⁠ 12390:Silver 12385:Golden 12375:Dyadic 12360:Binary 12355:Aspect 12266:ratios 12135: 12096: 12028: 11916: 11888: 11801: 11725: 11715: 11553: 11516: 11481: 11422: 11393: 11355: 11251: 11201: 11173: 11106: 11053: 10995: 10963:p. 374 10939: 10417:where 10266:Leyden 9987:150 BC 9943:Greeks 9876: 9869: 9862: 9855: 9847: 8593:). An 8310:> 0 8150:imply 8015:is an 7359:, and 7114:10,000 7048:= 5/90 6944:123456 6889:0.3333 6202:factor 6139: 6122: 6098: 6081: 4907: 4862: 4817: 4348:gives 3275:, and 2889:or as 1887:inches 1659:means 1493: 1378:(from 1282:digits 1196:Ratios 944:, and 615:" or " 597:stroke 595:), or 518:(from 506:(from 496:, and 195:ratios 76:simple 72:vulgar 68:common 58:(from 12183:(PDF) 12176:(PDF) 11810:(PDF) 11799:JSTOR 11777:(PDF) 10862:Notes 10183:from 10179:), a 10051:cheda 9994:India 9955:Greek 9881:JSTOR 9867:books 9003:(see 8741:field 8516:field 8294:with 7051:0.000 7007:= 5/9 6947:10000 6879:into 6843:into 3667:, or 2342:In a 1380:Latin 1330:6.023 1203:ratio 607:. In 589:slash 520:Latin 508:Latin 219:3 ÷ 4 74:, or 60:Latin 12439:Unit 12264:and 12133:ISBN 12094:ISBN 12026:ISBN 11991:2016 11914:ISBN 11886:ISBN 11734:2020 11723:OCLC 11713:ISBN 11632:2022 11514:ISBN 11479:ISBN 11420:ISBN 11391:ISBN 11353:ISBN 11249:ISBN 11199:ISBN 11104:ISBN 11051:ISBN 10993:ISBN 10937:ISBN 10704:: 1 10696:: 0 10694:Zero 10546:Real 10046:amsa 9990:Jain 9941:The 9853:news 9019:and 8873:are 8853:and 8589:(or 8460:and 8362:and 8327:and 8318:same 8305:and 8298:and 8088:and 7318:and 7151:and 7093:Let 7058:0.00 7026:6291 6849:0.25 6845:1.00 6833:1.00 5435:and 4594:> 4368:> 4282:and 4274:and 4221:and 4147:and 4093:< 4020:< 3940:> 3905:The 3640:the 2395:and 2253:and 2194:are 2045:and 1092:The 884:and 674:and 351:and 288:and 197:and 107:and 12419:LCD 12053:269 12018:doi 11953:doi 11791:doi 11787:111 11543:doi 11539:100 11504:doi 11432:and 11224:doi 11146:". 11127:". 10702:One 10333:In 10185:Fez 10174:fl. 10165:). 10117:, 1 10018:), 9981:of 9916:of 9836:by 9754:£sd 9748:or 8999:of 8382:). 8342:of 7295:of 7196:987 7192:987 7179:): 7135:987 7120:987 7105:987 7076:987 7068:987 7053:392 7044:0.0 7032:If 7019:264 6997:789 6871:by 6835:by 4793:1. 4721:If 3863:462 3837:462 3815:462 3795:100 3469:). 3255:.) 3070:of 2978:or 2922:10. 2775:104 1935:An 1622:and 1439:or 1395:or 1316:100 912:, − 847:or 831:or 789:117 776:not 695:". 617:nut 480:). 363:or 12456:: 12295:= 12291:÷ 12222:. 12210:. 12154:. 12055:. 12024:. 11981:. 11965:^ 11951:. 11941:46 11939:. 11805:. 11797:. 11785:. 11779:. 11721:. 11703:. 11648:. 11618:. 11590:^ 11570:^ 11557:. 11549:. 11537:. 11512:. 11428:. 11361:. 11319:. 11294:. 11290:. 11183:^ 11167:53 11165:. 11144:n. 11125:n. 11059:. 11049:. 11047:74 11031:^ 11015:. 10965:. 10349:, 10248:, 10235:". 10187:, 10044:: 10035:c. 10024:c. 10013:c. 9965:c. 9949:. 9925:c. 9774:: 9702:. 9654:11 9622:20 9390:11 9358:20 8724:. 8651:. 7355:, 7351:, 7233:= 7174:− 7170:= 7166:− 7156:= 7146:= 7060:12 7040:: 7024:0. 7017:0. 7012:62 7010:0. 7003:0. 6875:(" 6839:(" 6806:. 6638:15 6635:10 6612:10 6548:10 6381:33 6277:18 6012:12 5707:12 5704:19 5692:12 5689:10 5679:12 5601:12 5598:19 5586:24 5583:20 5573:24 5570:18 5310:15 5294:15 5291:19 5281:15 5278:10 5268:15 5203:. 5186:15 5183:10 5072:15 4931:12 4918:12 4902:12 4620:. 4603:17 4588:18 4560:17 4554:18 4530:17 4524:18 4513:18 4485:17 4479:18 4474:17 4438:17 4409:18 4392:. 4288:bd 4284:bc 4280:ad 4276:bc 4272:ad 4268:bd 4099:2. 3887:22 3871:21 3858:21 3850:63 3834:63 3809:63 3789:50 3784:= 3779:20 3773:10 3768:= 3752:= 3747:10 3730:is 3663:, 3654:. 3633:. 3603:ce 3595:cd 3560:, 3283:. 3271:, 3143:28 3140:15 3033:of 3027:A 3007:20 2999:10 2966:20 2953:10 2913:20 2910:10 2874:40 2862:20 2849:10 2823:20 2815:10 2772:51 2762:26 2733:12 2724:26 2700:12 2685:26 2669:12 2654:10 2439:26 2410:12 2327:. 2312:68 2238:68 2179:17 2176:13 2147:21 2016:, 1822:11 1786:11 1431:A 1382:: 1346:10 1342:10 1338:10 1334:10 1321:. 1310:75 1273:A 1240:12 1200:A 1192:. 1187:17 1161:17 1065:. 1026:= 1000:A 975:A 955:−5 933:−8 928:, 823:A 629:em 625:en 621:em 613:en 601:UK 593:US 554:. 522:: 510:: 492:, 315:−2 309:−1 299:−2 277:−1 272:, 118:17 70:, 62:: 54:A 12329:/ 12254:e 12247:t 12240:v 12214:. 12192:. 12141:. 12102:. 12066:. 12034:. 12020:: 11993:. 11959:. 11955:: 11947:: 11922:. 11894:. 11869:. 11844:. 11819:. 11793:: 11761:. 11736:. 11688:. 11663:. 11634:. 11604:. 11561:x 11545:: 11522:. 11506:: 11487:. 11457:. 11448:4 11445:/ 11442:1 11438:+ 11399:. 11372:. 11334:. 11305:. 11257:. 11230:. 11226:: 11207:. 11177:. 11112:. 11072:. 11070:. 11025:. 11001:. 10973:. 10945:. 10896:⁄ 10882:/ 10674:N 10669:: 10636:Z 10631:: 10598:Q 10593:: 10560:R 10555:: 10522:C 10517:: 10445:b 10425:a 10401:b 10391:/ 10377:a 10296:( 10220:3 10216:5 10211:1 10207:3 10172:( 10160:9 10157:/ 10154:8 10144:9 10141:/ 10138:1 10128:5 10125:/ 10122:1 10112:4 10109:/ 10106:1 10079:० 10033:( 10022:( 10011:( 9963:( 9903:) 9897:( 9892:) 9888:( 9878:· 9871:· 9864:· 9857:· 9830:. 9787:2 9784:1 9733:2 9730:/ 9727:1 9648:5 9643:6 9637:9 9628:= 9616:9 9611:) 9606:5 9601:2 9595:3 9592:( 9589:3 9583:= 9575:2 9571:) 9565:5 9560:2 9557:( 9549:2 9544:3 9537:) 9532:5 9527:2 9521:3 9518:( 9515:3 9509:= 9501:5 9496:2 9490:3 9483:5 9478:2 9472:3 9458:5 9453:2 9450:+ 9447:3 9443:3 9438:= 9430:5 9425:2 9422:+ 9419:3 9415:3 9384:5 9379:6 9376:+ 9373:9 9364:= 9352:9 9347:) 9342:5 9337:2 9334:+ 9331:3 9328:( 9325:3 9319:= 9311:2 9307:) 9301:5 9296:2 9293:( 9285:2 9280:3 9273:) 9268:5 9263:2 9260:+ 9257:3 9254:( 9251:3 9245:= 9237:5 9232:2 9229:+ 9226:3 9219:5 9214:2 9211:+ 9208:3 9194:5 9189:2 9183:3 9179:3 9174:= 9166:5 9161:2 9155:3 9151:3 9113:7 9107:7 9102:3 9096:= 9090:7 9085:7 9073:7 9069:3 9064:= 9058:7 9054:3 8978:1 8972:x 8968:1 8963:+ 8957:1 8954:+ 8951:x 8947:1 8920:1 8912:2 8908:x 8902:x 8899:2 8861:2 8841:, 8835:, 8830:2 8808:, 8805:2 8801:/ 8770:2 8766:/ 8760:2 8700:x 8697:1 8688:1 8680:x 8677:1 8671:+ 8668:1 8634:3 8626:2 8622:x 8616:2 8613:+ 8610:x 8568:3 8562:x 8559:2 8556:+ 8551:2 8547:x 8541:x 8538:3 8497:3 8489:2 8485:x 8479:2 8476:+ 8473:x 8445:3 8439:x 8436:2 8433:+ 8428:2 8424:x 8418:x 8415:3 8368:P 8364:b 8360:a 8356:P 8352:D 8344:R 8336:R 8329:b 8325:a 8308:b 8300:b 8296:a 8289:b 8286:/ 8283:a 8258:) 8255:) 8248:d 8244:, 8237:c 8233:( 8230:+ 8227:) 8220:b 8216:, 8209:a 8205:( 8202:( 8196:) 8193:) 8190:d 8187:, 8184:c 8181:( 8178:+ 8175:) 8172:b 8169:, 8166:a 8163:( 8160:( 8137:) 8130:d 8126:, 8119:c 8115:( 8109:) 8106:d 8103:, 8100:c 8097:( 8075:) 8068:b 8064:, 8057:a 8053:( 8047:) 8044:b 8041:, 8038:a 8035:( 8000:, 7997:c 7994:b 7991:= 7988:d 7985:a 7975:) 7972:d 7969:, 7966:c 7963:( 7957:) 7954:b 7951:, 7948:a 7945:( 7911:. 7903:) 7900:b 7897:, 7894:b 7891:( 7876:, 7873:0 7867:a 7856:) 7853:a 7850:, 7847:b 7844:( 7841:= 7832:1 7825:) 7821:b 7818:, 7815:a 7812:( 7800:) 7797:b 7794:, 7791:0 7788:( 7765:) 7762:b 7759:, 7756:a 7750:( 7747:= 7740:) 7737:b 7734:, 7731:a 7728:( 7694:) 7691:0 7685:c 7677:( 7673:) 7670:c 7667:b 7664:, 7661:d 7658:a 7655:( 7652:= 7649:) 7646:d 7643:, 7640:c 7637:( 7631:) 7628:b 7625:, 7622:a 7619:( 7598:) 7595:d 7592:b 7589:, 7586:c 7583:a 7580:( 7577:= 7574:) 7571:d 7568:, 7565:c 7562:( 7556:) 7553:b 7550:, 7547:a 7544:( 7522:) 7519:d 7516:b 7513:, 7510:c 7507:b 7501:d 7498:a 7495:( 7492:= 7489:) 7486:d 7483:, 7480:c 7477:( 7471:) 7468:b 7465:, 7462:a 7459:( 7437:) 7434:d 7431:b 7428:, 7425:c 7422:b 7419:+ 7416:d 7413:a 7410:( 7407:= 7404:) 7401:d 7398:, 7395:c 7392:( 7389:+ 7386:) 7383:b 7380:, 7377:a 7374:( 7335:, 7332:0 7326:b 7306:a 7283:) 7280:b 7277:, 7274:a 7271:( 7241:/ 7231:x 7225:x 7215:x 7207:x 7188:x 7184:x 7176:d 7172:b 7168:c 7164:a 7158:d 7154:c 7148:b 7144:a 7131:x 7116:x 7101:x 7095:x 7046:5 7005:5 6953:. 6938:= 6925:1 6918:3 6915:/ 6912:1 6902:4 6899:/ 6896:1 6885:4 6877:3 6873:3 6862:3 6859:/ 6856:1 6841:4 6837:4 6826:4 6823:/ 6820:1 6791:3 6788:2 6782:= 6775:3 6769:2 6764:4 6758:1 6751:= 6745:3 6742:4 6730:2 6727:1 6721:= 6715:4 6712:3 6700:2 6697:1 6667:3 6664:2 6629:= 6622:5 6616:3 6586:3 6583:2 6560:5 6551:3 6517:. 6512:4 6509:1 6504:8 6501:= 6496:4 6493:9 6488:+ 6485:6 6482:= 6477:4 6474:3 6466:3 6463:+ 6460:2 6454:3 6451:= 6446:4 6443:3 6438:2 6432:3 6400:4 6397:1 6392:8 6389:= 6384:4 6376:= 6371:4 6367:3 6364:+ 6361:4 6355:2 6344:1 6341:3 6336:= 6331:4 6328:3 6323:2 6317:3 6280:4 6271:= 6265:4 6262:3 6250:1 6247:6 6241:= 6235:4 6232:3 6223:6 6183:2 6180:1 6175:= 6170:2 6167:1 6157:1 6154:1 6149:= 6142:2 6133:4 6125:1 6116:3 6101:1 6092:3 6084:1 6075:2 6067:= 6062:4 6059:3 6049:3 6046:2 6009:6 6004:= 5999:4 5996:3 5986:3 5983:2 5945:. 5939:4 5936:1 5930:1 5927:= 5922:) 5914:4 5911:3 5902:1 5897:( 5892:+ 5889:) 5886:1 5880:2 5874:4 5871:( 5868:= 5862:4 5859:3 5853:2 5847:4 5818:6 5815:1 5809:= 5803:6 5800:3 5788:6 5785:4 5779:= 5773:2 5770:1 5758:3 5755:2 5699:= 5684:+ 5676:9 5671:= 5665:6 5659:2 5654:5 5648:2 5642:+ 5636:3 5630:4 5625:3 5619:3 5613:= 5593:= 5578:+ 5565:= 5559:6 5553:4 5548:5 5542:4 5536:+ 5530:6 5524:4 5519:6 5513:3 5507:= 5498:6 5495:5 5490:+ 5485:4 5482:3 5449:6 5446:5 5420:4 5417:3 5387:d 5384:b 5379:b 5376:c 5373:+ 5370:d 5367:a 5361:= 5356:d 5353:c 5348:+ 5343:b 5340:a 5307:4 5302:1 5299:= 5286:= 5273:+ 5265:9 5235:3 5232:2 5227:+ 5222:5 5219:3 5177:= 5171:5 5168:5 5156:3 5153:2 5135:3 5132:/ 5129:2 5116:3 5113:/ 5110:3 5100:3 5097:/ 5094:3 5069:9 5063:= 5057:3 5054:3 5042:5 5039:3 5011:5 5008:3 4980:3 4977:2 4972:+ 4967:5 4964:3 4936:. 4928:7 4923:= 4915:4 4910:+ 4899:3 4894:= 4888:4 4882:3 4877:4 4871:1 4865:+ 4856:3 4850:4 4845:3 4839:1 4833:= 4828:3 4825:1 4820:+ 4812:4 4809:1 4764:4 4761:1 4735:2 4732:1 4713:. 4698:4 4695:1 4689:1 4686:= 4680:4 4677:5 4671:= 4665:4 4662:3 4656:+ 4650:4 4647:2 4600:4 4585:5 4519:4 4468:5 4444:, 4435:4 4406:5 4377:6 4374:3 4362:6 4359:4 4333:2 4330:1 4304:3 4301:2 4250:d 4244:b 4239:c 4233:b 4205:d 4199:b 4194:d 4188:a 4161:d 4158:c 4132:b 4129:a 4090:3 4076:b 4072:a 4062:= 4055:b 4048:a 4033:4 4026:2 4013:4 4006:3 3976:4 3949:4 3946:2 3934:4 3931:3 3884:3 3878:= 3843:= 3812:/ 3792:/ 3776:/ 3763:2 3760:/ 3757:1 3744:/ 3741:5 3714:8 3711:3 3685:9 3682:3 3638:c 3627:e 3623:/ 3619:d 3611:c 3599:/ 3577:e 3574:c 3571:= 3568:b 3548:d 3545:c 3542:= 3539:a 3517:c 3492:b 3489:a 3464:2 3461:/ 3458:1 3448:4 3445:/ 3442:2 3432:2 3429:/ 3426:1 3416:4 3413:/ 3410:2 3388:2 3385:1 3357:n 3354:n 3328:n 3325:n 3302:n 3237:5 3233:/ 3229:7 3224:4 3220:/ 3216:3 3187:7 3184:5 3172:4 3169:3 3134:= 3128:7 3125:5 3113:4 3110:3 3084:7 3081:5 3055:4 3052:3 3013:. 3010:) 3003:/ 2996:( 2991:/ 2986:5 2961:/ 2956:) 2949:/ 2945:5 2942:( 2919:= 2902:/ 2897:5 2871:1 2865:= 2857:/ 2846:5 2819:/ 2811:/ 2807:5 2780:. 2767:= 2759:1 2749:4 2745:3 2742:+ 2739:4 2727:= 2716:4 2712:3 2709:+ 2706:4 2694:= 2678:4 2675:3 2659:, 2651:3 2646:= 2641:5 2638:1 2628:2 2625:3 2620:= 2617:5 2609:2 2606:3 2601:= 2596:5 2587:2 2584:3 2572:, 2567:2 2564:3 2559:= 2554:1 2551:3 2541:2 2538:1 2533:= 2528:3 2525:1 2515:2 2512:1 2507:= 2496:3 2493:1 2482:2 2479:1 2434:/ 2427:) 2419:4 2416:3 2405:( 2379:3 2375:/ 2371:1 2366:2 2362:/ 2358:1 2339:. 2309:1 2303:+ 2297:6 2294:1 2288:+ 2282:4 2279:1 2273:+ 2267:3 2264:1 2235:1 2229:+ 2223:4 2220:1 2214:+ 2208:2 2205:1 2153:. 2144:1 2138:+ 2132:6 2129:1 2123:+ 2117:2 2114:1 2088:7 2085:5 2059:4 2056:3 2030:3 2027:2 2001:2 1998:1 1968:3 1965:1 1959:+ 1953:2 1950:1 1911:q 1907:/ 1903:p 1880:2 1877:/ 1874:1 1870:+ 1852:. 1846:4 1843:3 1837:+ 1834:2 1831:= 1825:4 1795:. 1789:4 1780:= 1774:4 1771:3 1765:+ 1759:4 1756:8 1750:= 1744:4 1741:3 1735:+ 1732:2 1702:. 1697:) 1689:4 1686:3 1680:+ 1677:2 1672:( 1644:4 1641:3 1635:2 1615:4 1612:/ 1609:3 1605:+ 1586:4 1583:3 1577:+ 1574:2 1551:4 1548:3 1525:2 1505:4 1501:/ 1497:3 1490:2 1467:4 1464:3 1458:2 1366:3 1363:/ 1360:1 1332:× 1313:/ 1306:+ 1256:3 1253:/ 1250:1 1237:/ 1234:4 1209:n 1184:/ 1181:1 1167:1 1164:/ 1148:7 1145:/ 1142:7 1125:3 1122:/ 1119:7 1109:7 1106:/ 1103:3 1042:. 1037:2 1034:/ 1031:1 1021:8 1018:/ 1015:1 992:7 989:/ 986:1 952:/ 949:8 939:5 936:/ 923:5 920:/ 917:8 907:2 904:/ 901:1 894:b 886:b 882:a 863:b 860:a 845:b 843:/ 841:a 802:/ 799:6 786:/ 783:1 765:1 762:/ 759:3 745:5 742:/ 739:1 729:5 726:/ 723:2 709:1 706:/ 703:3 685:5 682:/ 679:3 669:5 666:/ 663:2 646:2 643:/ 640:1 599:( 591:( 584:9 580:4 572:3 569:/ 566:1 537:5 534:/ 531:8 465:x 462:1 433:2 429:2 411:b 407:/ 403:a 374:Q 361:Q 357:b 353:b 349:a 341:b 337:/ 333:a 312:/ 296:/ 293:1 283:2 280:/ 267:2 264:/ 261:1 251:2 248:/ 245:1 235:2 232:/ 229:1 212:4 209:/ 206:3 185:4 182:/ 179:3 169:4 166:/ 163:3 147:2 143:1 121:3 92:2 89:1 45:4 42:/ 39:1 23:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index