Half Of 5 3 8

Ratio of 2 numbers

A cake with one quarter (one fourth) removed. The remaining three fourths are shown by dotted lines and labeled by the fraction one / 4

A fraction (from Latin: fractus, "broken") represents a part of a whole or, more more often than not, any number of equal parts. When spoken in everyday English language, a fraction describes how many parts of a certain size there are, for example, half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: ${\displaystyle {\tfrac {ane}{ii}}}$ and ${\displaystyle {\tfrac {17}{3}}}$ ) consists of a numerator, displayed above a line (or before a slash like 1⁄two ), and a non-zip denominator, displayed below (or after) that line. Numerators and denominators are as well used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, considering goose egg parts can never brand up a whole. For instance, in the fraction three / 4 , the numerator 3 indicates that the fraction represents three equal parts, and the denominator 4 indicates that 4 parts make upwards a whole. The picture to the right illustrates 3 / 4 of a block.

A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percentage, or with a negative exponent. For example, 0.01, 1%, and x⁻ⁱⁱ are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for case, 7 equals 7/i).

Other uses for fractions are to represent ratios and division.^[ane] Thus the fraction 3 / 4 can also be used to represent the ratio 3:four (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided past four). The non-nix denominator rule, which applies when representing a division every bit a fraction, is an example of the rule that sectionalisation by aught is undefined.

We tin can besides write negative fractions, which correspond the opposite of a positive fraction. For example, if ane / 2 represents a half-dollar profit, then − i / 2 represents a half-dollar loss. Because of the rules of sectionalization of signed numbers (which states in part that negative divided by positive is negative), − i / two , −1 / 2 and 1 / −2 all represent the aforementioned fraction – negative half. And because a negative divided by a negative produces a positive, −1 / −two represents positive one-half.

In mathematics the set of all numbers that can be expressed in the form a / b , where a and b are integers and b is not goose egg, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. A number is a rational number precisely when information technology tin can exist written in that course (i.e., as a common fraction). All the same, the give-and-take fraction can also be used to draw mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as ${\textstyle {\frac {\sqrt {2}}{2}}}$ (encounter square root of 2) and π / iv (encounter proof that π is irrational).

Vocabulary [edit]

In a fraction, the number of equal parts existence described is the numerator (from Latin: numerātor, "counter" or "numberer"), and the blazon or variety of the parts is the denominator (from Latin: dēnōminātor, "thing that names or designates").^{[2] [3]} Every bit an example, the fraction 8 / five amounts to eight parts, each of which is of the type named "fifth". In terms of segmentation, the numerator corresponds to the dividend, and the denominator corresponds to the divisor.

Informally, the numerator and denominator may be distinguished by placement solitary, but in formal contexts they are usually separated by a fraction bar. The fraction bar may exist horizontal (every bit in 1 / 3 ), oblique (as in 2/5), or diagonal (as in 4⁄9 ).^[four] These marks are respectively known as the horizontal bar; the virgule, slash (US), or stroke (UK); and the fraction bar, solidus,^[5] or fraction slash.^{[north 1]} In typography, fractions stacked vertically are also known equally "en" or "nut fractions", and diagonal ones as "em" or "mutton fractions", based on whether a fraction with a single-digit numerator and denominator occupies the proportion of a narrow en square, or a wider em square.^[iv] In traditional typefounding, a piece of type bearing a consummate fraction (eastward.thousand. 1 / two ) was known as a "instance fraction", while those representing just part of fraction were called "piece fractions".

The denominators of English language fractions are generally expressed equally ordinal numbers, in the plural if the numerator is non one. (For case, 2 / 5 and 3 / 5 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 exist alternatively expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be alternatively expressed every bit "hundredth"/"hundredths" or "percent".

When the denominator is 1, it may be expressed in terms of "wholes" but is more than commonly ignored, with the numerator read out as a whole number. For example, 3 / 1 may be described as "three wholes", or simply as "three". When the numerator is 1, it may be omitted (every bit in "a tenth" or "each quarter").

The entire fraction may be expressed as a single limerick, in which example information technology is hyphenated, or as a number of fractions with a numerator of ane, in which case they are non. (For case, "two-fifths" is the fraction 2 / 5 and "two fifths" is the same fraction understood as two instances of ane / 5 .) Fractions should ever be hyphenated when used as adjectives. Alternatively, a fraction may be described past reading it out as the numerator "over" the denominator, with the denominator expressed equally a fundamental number. (For example, 3 / 1 may also be expressed every bit "three over one".) The term "over" is used even in the example of solidus fractions, where the numbers are placed left and right of a slash mark. (For instance, ane/2 may be read "one-half", "i half", or "one over two".) Fractions with large denominators that are non powers of 10 are often rendered in this fashion (eastward.thousand., 1 / 117 equally "1 over ane hundred seventeen"), while those with denominators divisible by 10 are typically read in the normal ordinal fashion (e.thou., 6 / 1000000 as "six-millionths", "six millionths", or "half-dozen one-millionths").

Forms of fractions [edit]

Simple, common, or vulgar fractions [edit]

A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a/b or ${\displaystyle {\tfrac {a}{b}}}$ , where a and b are both integers.^[ix] As with other fractions, the denominator (b) cannot exist zero. Examples include ${\displaystyle {\tfrac {1}{2}}}$ , ${\displaystyle -{\tfrac {viii}{5}}}$ , ${\displaystyle {\tfrac {-8}{v}}}$ , and ${\displaystyle {\tfrac {8}{-5}}}$ . The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.^[10]

Common fractions can be positive or negative, and they tin can exist proper or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are non common fractions; though, unless irrational, they tin be evaluated to a common fraction.

In Unicode, precomposed fraction characters are in the Number Forms cake.

Proper and improper fractions [edit]

Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is chosen proper if the numerator is less than the denominator, and improper otherwise.^{[11] [12]} The concept of an "improper fraction" is a late development, with the terminology deriving from the fact that "fraction" means "a slice", so a proper fraction must exist less than i.^[x] This was explained in the 17th century textbook The Basis of Arts.^{[13] [14]}

In general, a common fraction is said to exist a proper fraction, if the accented value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.^{[15] [16]} It is said to exist an improper fraction, or sometimes top-heavy fraction,^[17] if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are ii/3, −3/four, and 4/9, whereas examples of improper fractions are nine/four, −four/3, and three/iii.

Reciprocals and the "invisible denominator" [edit]

The reciprocal of a fraction is some other fraction with the numerator and denominator exchanged. The reciprocal of ${\displaystyle {\tfrac {three}{7}}}$ , for instance, is ${\displaystyle {\tfrac {7}{3}}}$ . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are non equal) is a proper fraction.

When the numerator and denominator of a fraction are equal (for case, ${\displaystyle {\tfrac {7}{7}}}$ ), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to 1 and improper.

Any integer tin exist written every bit a fraction with the number i as denominator. For case, 17 can be written as ${\displaystyle {\tfrac {17}{i}}}$ , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer, except for zippo, has a reciprocal. For example. the reciprocal of 17 is ${\displaystyle {\tfrac {1}{17}}}$ .

Ratios [edit]

A ratio is a human relationship between two or more than numbers that tin 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 every bit "group 1 to group 2 ... to group due north". For example, if a car lot had 12 vehicles, of which

• 2 are white,
• 6 are cerise, and
• four are xanthous,

then the ratio of cherry to white to yellowish cars is 6 to 2 to iv. The ratio of yellowish cars to white cars is 4 to two and may exist expressed equally 4:2 or 2:ane.

A ratio is often converted to a fraction when information technology is expressed every bit a ratio to the whole. In the higher up example, the ratio of yellowish cars to all the cars on the lot is iv:12 or 1:iii. Nosotros can convert these ratios to a fraction, and say that 4 / 12 of the cars or one / 3 of the cars in the lot are yellow. Therefore, if a person randomly chose 1 car on the lot, and so at that place is a 1 in three chance or probability that information technology would exist yellowish.

Decimal fractions and percentages [edit]

A decimal fraction is a fraction whose denominator is not given explicitly, just is understood to exist an integer power of ten. Decimal fractions are usually expressed using decimal note in which the implied denominator is determined by the number of digits to the right of a decimal separator, the appearance of which (e.yard., a period, an interpunct (·), a comma) depends on the locale (for examples, see decimal separator). Thus, for 0.75 the numerator is 75 and the implied denominator is x 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 every bit 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this example). 3.75 tin can be written either as an improper fraction, 375/100, or as a mixed number, ${\displaystyle iii{\tfrac {75}{100}}}$ .

Decimal fractions can too exist expressed using scientific notation with negative exponents, such as 6.023×10⁻⁷ , which represents 0.0000006023. The 10⁻⁷ represents a denominator of 10⁷ . Dividing by x⁷ moves the decimal point 7 places to the left.

Decimal fractions with infinitely many digits to the correct of the decimal separator represent an infinite serial. For case, one / 3 = 0.333... represents the infinite series three/x + 3/100 + 3/1000 + ....

Some other kind of fraction is the percentage (from Latin: percentage, pregnant "per hundred", represented by the symbol %), in which the unsaid denominator is always 100. Thus, 51% ways 51/100. Percentages greater than 100 or less than null are treated in the same way, eastward.chiliad. 311% equals 311/100, and −27% equals −27/100.

The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation, equally in 75 parts per 1000000 (ppm), ways that the proportion is 75/one,000,000.

Whether mutual fractions or decimal fractions are used is often a thing of taste and context. Common fractions are used most often when the denominator is relatively modest. By mental adding, information technology is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more than accurate to multiply 15 by 1/3, for example, than it is to multiply fifteen by any decimal approximation of one 3rd. Budgetary values are unremarkably expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. Still, as noted above, in pre-decimal British currency, shillings and pence were oftentimes given the class (only not the meaning) of a fraction, every bit, for example, "iii/half-dozen" (read "three and six") pregnant three shillings and six pence, and having no relationship to the fraction 3/6.

Mixed numbers [edit]

A mixed numeral (as well called a mixed fraction or mixed number) is a traditional denotation of the sum of a non-zero integer and a proper fraction (having the same sign). It is used primarily in measurement: ${\displaystyle two{\tfrac {3}{xvi}}}$ inches, for example. Scientific measurements most invariably apply decimal notation rather than mixed numbers. The sum can be implied without the use of a visible operator such as the appropriate "+". For instance, in referring to ii entire cakes and three quarters of some other cake, the numerals denoting the integer part and the fractional part of the cakes tin can be written next to each other as ${\displaystyle 2{\tfrac {3}{4}}}$ instead of the unambiguous annotation ${\displaystyle 2+{\tfrac {3}{iv}}.}$ Negative mixed numerals, as in ${\displaystyle -2{\tfrac {3}{iv}}}$ , are treated like ${\displaystyle \scriptstyle -\left(ii+{\frac {3}{4}}\correct).}$ Whatsoever such sum of a whole plus a part tin be converted to an improper fraction by applying the rules of adding unlike quantities.

This tradition is, formally, in disharmonize with the notation in algebra where adjacent symbols, without an explicit infix operator, denote a product. In the expression ${\displaystyle 2x}$ , the "understood" operation is multiplication. If x is replaced by, for instance, the fraction ${\displaystyle {\tfrac {three}{4}}}$ , the "understood" multiplication needs to be replaced by explicit multiplication, to avoid the appearance of a mixed number.

When multiplication is intended, ${\displaystyle 2{\tfrac {b}{c}}}$ may be written as

${\displaystyle 2\cdot {\frac {b}{c}},\quad }$ or ${\displaystyle \quad two\times {\frac {b}{c}},\quad }$ or ${\displaystyle \quad 2\left({\frac {b}{c}}\right),\;\ldots }$

An improper fraction can be converted to a mixed number every bit follows:

1. Using Euclidean division (division with remainder), split the numerator past the denominator. In the example, ${\displaystyle {\tfrac {xi}{4}}}$ , divide 11 by four. xi ÷ 4 = ii remainder 3.
2. The quotient (without the balance) becomes the whole number function of the mixed number. The remainder becomes the numerator of the partial part. In the example, 2 is the whole number part and 3 is the numerator of the fractional office.
3. The new denominator is the aforementioned as the denominator of the improper fraction. In the example, it is iv. Thus, ${\displaystyle {\tfrac {11}{4}}=two{\tfrac {3}{four}}}$ .

Historical notions [edit]

Egyptian fraction [edit]

An Egyptian fraction is the sum of distinct positive unit fractions, for example ${\displaystyle {\tfrac {ane}{2}}+{\tfrac {1}{3}}}$ . This definition derives from the fact that the ancient Egyptians expressed all fractions except ${\displaystyle {\tfrac {one}{2}}}$ , ${\displaystyle {\tfrac {2}{3}}}$ and ${\displaystyle {\tfrac {iii}{4}}}$ in this fashion. Every positive rational number can exist expanded as an Egyptian fraction. For example, ${\displaystyle {\tfrac {5}{seven}}}$ can be written as ${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {i}{21}}.}$ Any positive rational number can be written every bit a sum of unit fractions in infinitely many ways. Two ways to write ${\displaystyle {\tfrac {xiii}{17}}}$ are ${\displaystyle {\tfrac {1}{ii}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}$ and ${\displaystyle {\tfrac {1}{iii}}+{\tfrac {1}{4}}+{\tfrac {one}{6}}+{\tfrac {1}{68}}}$ .

Complex and chemical compound fractions [edit]

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number,^{[18] [19]} corresponding to partitioning of fractions. For example, ${\displaystyle {\frac {\tfrac {1}{2}}{\tfrac {ane}{3}}}}$ and ${\displaystyle {\frac {12{\tfrac {3}{4}}}{26}}}$ are complex fractions. To reduce a complex fraction to a elementary fraction, treat the longest fraction line as representing division. For example:

${\displaystyle {\frac {\tfrac {1}{2}}{\tfrac {1}{three}}}={\tfrac {1}{2}}\times {\tfrac {three}{ane}}={\tfrac {3}{two}}}$

${\displaystyle {\frac {12{\tfrac {3}{4}}}{26}}=12{\tfrac {iii}{four}}\cdot {\tfrac {1}{26}}={\tfrac {12\cdot iv+three}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{4}}\cdot {\tfrac {i}{26}}={\tfrac {51}{104}}}$

${\displaystyle {\frac {\tfrac {3}{two}}{v}}={\tfrac {iii}{two}}\times {\tfrac {1}{v}}={\tfrac {iii}{10}}}$

${\displaystyle {\frac {8}{\tfrac {1}{3}}}=8\times {\tfrac {3}{one}}=24.}$

If, in a complex fraction, in that location is no unique way to tell which fraction lines takes precedence, so this expression is improperly formed, because of ambivalence. And then five/x/20/40 is non a valid mathematical expression, because of multiple possible interpretations, due east.g. every bit

${\displaystyle 5/(10/(20/40))={\frac {5}{ten/{\tfrac {20}{xl}}}}={\frac {1}{iv}}\quad }$ or as ${\displaystyle \quad (5/10)/(twenty/40)={\frac {\tfrac {5}{10}}{\tfrac {20}{40}}}=i}$

A compound fraction is a fraction of a fraction, or whatsoever number of fractions connected with the give-and-take of,^{[18] [19]} corresponding to multiplication of fractions. To reduce a chemical compound fraction to a simple fraction, just carry out the multiplication (see the department on multiplication). For instance, ${\displaystyle {\tfrac {iii}{iv}}}$ of ${\displaystyle {\tfrac {5}{7}}}$ is a compound fraction, respective to ${\displaystyle {\tfrac {iii}{iv}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}$ . The terms chemical compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the chemical compound fraction ${\displaystyle {\tfrac {iii}{four}}\times {\tfrac {5}{7}}}$ is equivalent to the complex fraction ${\displaystyle {\tfrac {3/iv}{seven/5}}}$ .)

Still, "circuitous fraction" and "compound fraction" may both be considered outdated^[20] and now used in no well-defined manner, partly even taken synonymously for each other^[21] or for mixed numerals.^[22] They have lost their meaning as technical terms and the attributes "complex" and "compound" tend to be used in their every day significant of "consisting of parts".

Arithmetic with fractions [edit]

Similar whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against partitioning past cypher.

Equivalent fractions [edit]

Multiplying the numerator and denominator of a fraction past the same (not-nothing) number results in a fraction that is equivalent to the original fraction. This is true because for any non-aught number ${\displaystyle due north}$ , the fraction ${\displaystyle {\tfrac {northward}{n}}}$ equals ${\displaystyle 1}$ . Therefore, multiplying by ${\displaystyle {\tfrac {due north}{northward}}}$ is the same every bit multiplying past one, and any number multiplied by one has the aforementioned value as the original number. Past style of an example, start with the fraction ${\displaystyle {\tfrac {1}{two}}}$ . When the numerator and denominator are both multiplied by 2, the result is ${\displaystyle {\tfrac {2}{4}}}$ , which has the aforementioned value (0.5) as ${\displaystyle {\tfrac {one}{2}}}$ . To picture this visually, imagine cutting a cake into four pieces; two of the pieces together ( ${\displaystyle {\tfrac {2}{4}}}$ ) brand upwardly half the cake ( ${\displaystyle {\tfrac {i}{two}}}$ ).

Simplifying (reducing) fractions [edit]

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 cistron) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction ${\displaystyle {\tfrac {a}{b}}}$ are divisible by ${\displaystyle c,}$ then they tin can exist written as ${\displaystyle a=cd}$ and ${\displaystyle b=ce,}$ and the fraction becomes ${\displaystyle {\tfrac {cd}{ce}}}$ , which can be reduced by dividing both the numerator and denominator by ${\displaystyle c}$ to give the reduced fraction ${\displaystyle {\tfrac {d}{e}}.}$

If ane takes for c the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its lowest terms.

If the numerator and the denominator practise non share any factor greater than 1, the fraction is already reduced to its everyman terms, and it is said to be irreducible, reduced, or in simplest terms. For case, ${\displaystyle {\tfrac {3}{9}}}$ is not in lowest terms because both 3 and ix tin be exactly divided by 3. In contrast, ${\displaystyle {\tfrac {3}{8}}}$ is in lowest terms—the merely positive integer that goes into both 3 and viii evenly is ane.

Using these rules, we tin can show that ${\displaystyle {\tfrac {v}{10}}={\tfrac {1}{two}}={\tfrac {10}{20}}={\tfrac {fifty}{100}}}$ , for example.

Equally another case, since the greatest common divisor of 63 and 462 is 21, the fraction ${\displaystyle {\tfrac {63}{462}}}$ can be reduced to lowest terms by dividing the numerator and denominator by 21:

${\displaystyle {\tfrac {63}{462}}={\tfrac {63\,\div \,21}{462\,\div \,21}}={\tfrac {3}{22}}}$

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

Comparing fractions [edit]

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

${\displaystyle {\tfrac {iii}{4}}>{\tfrac {2}{4}}}$ considering three > 2, and the equal denominators ${\displaystyle four}$ are positive.

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

${\displaystyle {\tfrac {3}{-4}}<{\tfrac {2}{-4}}{\text{ because }}{\tfrac {a}{-b}}={\tfrac {-a}{b}}{\text{ and }}-3<-two.}$

If 2 positive fractions have the aforementioned 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 slice must be larger. When two positive fractions have the same numerator, they stand for the aforementioned number of parts, but in the fraction with the smaller denominator, the parts are larger.

One style to compare fractions with different numerators and denominators is to observe a common denominator. To compare ${\displaystyle {\tfrac {a}{b}}}$ and ${\displaystyle {\tfrac {c}{d}}}$ , these are converted to ${\displaystyle {\tfrac {a\cdot d}{b\cdot d}}}$ and ${\displaystyle {\tfrac {b\cdot c}{b\cdot d}}}$ (where the dot signifies multiplication and is an alternative symbol to ×). And then bd is a common denominator and the numerators ad and bc can be compared. It is not necessary to make up one's mind the value of the common denominator to compare fractions – 1 tin can just compare ad and bc, without evaluating bd, e.chiliad., comparing ${\displaystyle {\tfrac {ii}{3}}}$  ? ${\displaystyle {\tfrac {1}{2}}}$ gives ${\displaystyle {\tfrac {4}{vi}}>{\tfrac {3}{half-dozen}}}$ .

For the more laborious question ${\displaystyle {\tfrac {5}{xviii}}}$  ? ${\displaystyle {\tfrac {4}{17}},}$ multiply summit and lesser of each fraction by the denominator of the other fraction, to become a common denominator, yielding ${\displaystyle {\tfrac {5\times 17}{eighteen\times 17}}}$  ? ${\displaystyle {\tfrac {eighteen\times 4}{18\times 17}}}$ . Information technology is non necessary to calculate ${\displaystyle 18\times 17}$ – only the numerators demand to be compared. Since five×17 (= 85) is greater than 4×xviii (= 72), the result of comparing is ${\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}$ .

Because every negative number, including negative fractions, is less than goose egg, and every positive number, including positive fractions, is greater than zero, it follows that whatsoever negative fraction is less than whatever positive fraction. This allows, together with the above rules, to compare all possible fractions.

Addition [edit]

The first rule of addition is that only like quantities tin be added; for example, various quantities of quarters. Unlike quantities, such as calculation thirds to quarters, must first be converted to similar quantities as described below: Imagine a pocket containing 2 quarters, and another pocket containing 3 quarters; in total, in that location are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

${\displaystyle {\tfrac {ii}{4}}+{\tfrac {3}{iv}}={\tfrac {v}{iv}}=1{\tfrac {1}{iv}}}$ .

If ${\displaystyle {\tfrac {1}{ii}}}$ of a block is to be added to ${\displaystyle {\tfrac {1}{4}}}$ of a block, the pieces demand to be converted into comparable quantities, such as cake-eighths or cake-quarters.

Adding dissimilar quantities [edit]

To add fractions containing dissimilar quantities (e.g. quarters and thirds), information technology is necessary to convert all amounts to similar quantities. It is piece of cake to piece of work out the chosen type of fraction to convert to; just multiply together the 2 denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator ${\displaystyle 1.}$

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

${\displaystyle {\frac {1}{4}}\ +{\frac {i}{3}}={\frac {1\times 3}{4\times 3}}\ +{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}\ +{\frac {four}{12}}={\frac {7}{12}}.}$

Consider adding the post-obit two quantities:

${\displaystyle {\frac {3}{5}}+{\frac {2}{iii}}}$

Get-go, convert ${\displaystyle {\tfrac {3}{v}}}$ into fifteenths by multiplying both the numerator and denominator by iii: ${\displaystyle {\tfrac {3}{v}}\times {\tfrac {iii}{3}}={\tfrac {nine}{15}}}$ . Since ${\displaystyle {\tfrac {3}{3}}}$ equals 1, multiplication by ${\displaystyle {\tfrac {three}{3}}}$ does not change the value of the fraction.

2d, catechumen ${\displaystyle {\tfrac {ii}{3}}}$ into fifteenths by multiplying both the numerator and denominator by five: ${\displaystyle {\tfrac {2}{3}}\times {\tfrac {5}{5}}={\tfrac {ten}{xv}}}$ .

Now it tin be seen that:

${\displaystyle {\frac {3}{v}}+{\frac {2}{three}}}$

is equivalent to:

${\displaystyle {\frac {9}{15}}+{\frac {ten}{xv}}={\frac {nineteen}{15}}=i{\frac {4}{xv}}}$

This method can be expressed algebraically:

${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {advertizement+cb}{bd}}}$

This algebraic method always works, thereby guaranteeing that the sum of simple fractions is e'er again a simple fraction. However, if the single denominators contain a common cistron, a smaller denominator than the product of these can exist used. For example, when adding ${\displaystyle {\tfrac {3}{iv}}}$ and ${\displaystyle {\tfrac {5}{6}}}$ the single denominators accept a common gene ${\displaystyle 2,}$ and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not merely reducing the denominator in the result, just also the factors in the numerator.

${\displaystyle {\begin{aligned}{\frac {3}{4}}+{\frac {5}{six}}&={\frac {3\cdot half dozen}{iv\cdot 6}}+{\frac {4\cdot 5}{4\cdot 6}}={\frac {18}{24}}+{\frac {xx}{24}}&={\frac {19}{12}}\\&={\frac {three\cdot 3}{4\cdot three}}+{\frac {2\cdot 5}{ii\cdot 6}}={\frac {9}{12}}+{\frac {10}{12}}&={\frac {19}{12}}\cease{aligned}}}$

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

Subtraction [edit]

The process for subtracting fractions is, in essence, the aforementioned every bit that of adding them: find a mutual denominator, and modify each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will exist the consequence of subtracting the numerators of the original fractions. For instance,

${\displaystyle {\tfrac {two}{three}}-{\tfrac {1}{2}}={\tfrac {4}{vi}}-{\tfrac {3}{6}}={\tfrac {1}{half dozen}}}$

Multiplication [edit]

Multiplying a fraction by another fraction [edit]

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

${\displaystyle {\frac {2}{3}}\times {\frac {iii}{iv}}={\frac {half dozen}{12}}}$

To explain the process, consider one 3rd of one quarter. Using the instance of a cake, if three small slices of equal size brand up a quarter, and 4 quarters brand up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as 1 third. Since one tertiary of a quarter is one 12th, two thirds of a quarter is ii twelfth. The 2nd fraction, three quarters, is three times as large equally one quarter, so 2 thirds of three quarters is three times as large as ii thirds of one quarter. Thus two thirds times three quarters is half dozen twelfths.

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

${\displaystyle {\frac {two}{3}}\times {\frac {three}{four}}={\frac {{\abolish {2}}^{~1}}{{\cancel {three}}^{~1}}}\times {\frac {{\cancel {3}}^{~1}}{{\cancel {4}}^{~2}}}={\frac {i}{1}}\times {\frac {ane}{two}}={\frac {1}{2}}}$

A ii is a mutual gene in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.

Multiplying a fraction past a whole number [edit]

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

${\displaystyle vi\times {\tfrac {3}{four}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}}$

This method works considering the fraction 6/ane ways six equal parts, each one of which is a whole.

Multiplying mixed numbers [edit]

When multiplying mixed numbers, it is considered preferable to convert the mixed number into an improper fraction.^[23] For instance:

${\displaystyle 3\times two{\frac {3}{4}}=3\times \left({\frac {8}{4}}+{\frac {3}{4}}\right)=3\times {\frac {11}{four}}={\frac {33}{4}}=viii{\frac {one}{4}}}$

In other words, ${\displaystyle ii{\tfrac {3}{four}}}$ is the same as ${\displaystyle {\tfrac {8}{4}}+{\tfrac {iii}{4}}}$ , making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is ${\displaystyle 8{\tfrac {1}{4}}}$ , since 8 cakes, each made of quarters, is 32 quarters in total.

Partitioning [edit]

To dissever a fraction past a whole number, you may either divide the numerator by the number, if information technology goes evenly into the numerator, or multiply the denominator by the number. For instance, ${\displaystyle {\tfrac {ten}{3}}\div 5}$ equals ${\displaystyle {\tfrac {2}{3}}}$ and also equals ${\displaystyle {\tfrac {ten}{three\cdot 5}}={\tfrac {ten}{fifteen}}}$ , which reduces to ${\displaystyle {\tfrac {2}{3}}}$ . To dissever a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, ${\displaystyle {\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{ii}}\times {\tfrac {4}{3}}={\tfrac {ane\cdot 4}{2\cdot 3}}={\tfrac {2}{three}}}$ .

Converting betwixt decimals and fractions [edit]

To change a mutual fraction to a decimal, exercise 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 reply to the desired accurateness. For example, to alter one / 4 to a decimal, divide 1.00 by 4 (" 4 into 1.00"), to obtain 0.25. To change 1 / three to a decimal, divide 1.000... by iii (" 3 into 1.000..."), and stop when the desired accuracy is obtained, east.k., at four decimals with 0.3333. The fraction 1 / iv tin can exist written exactly with two decimal digits, while the fraction one / 3 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 1 followed by every bit many zeroes as there are digits to the correct of the decimal point, and write in the numerator all the digits of the original decimal, just omitting the decimal point. Thus ${\displaystyle 12.3456={\tfrac {123456}{10000}}.}$

Converting repeating decimals to fractions [edit]

Decimal numbers, while arguably more useful to piece of work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to catechumen repeating decimals into fractions.

A conventional mode to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0.789 = 0.789789789... For repeating patterns that begin immediately subsequently the decimal betoken, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For case:

0.5 = 5/ix

0.62 = 62/99

0.264 = 264/999

0.6291 = 6291/9999

If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:

0.05 = 5/90

0.000392 = 392/999000

0.0012 = 12/9900

If a non-repeating prepare of decimals precede the design (such as 0.1523987), 1 may write the number as the sum of the non-repeating and repeating parts, respectively:

0.1523 + 0.0000987

And so, convert both parts to fractions, and add together them using the methods described above:

1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra can exist used, such as below:

1. Let x = the repeating decimal:

   x = 0.1523987

2. Multiply both sides past the ability of 10 simply great enough (in this case x^iv) to move the decimal signal merely before the repeating role of the decimal number:

   x,00010 = 1,523.987

3. Multiply both sides by the power of 10 (in this case x³) that is the same every bit the number of places that repeat:

   10,000,000x = i,523,987.987

4. Subtract the two equations from each other (if a = b and c = d, then a − c = b − d):

   10,000,000ten − 10,000x = ane,523,987.987 − 1,523.987

5. Go along the subtraction operation to clear the repeating decimal:

   9,990,000x = one,523,987 − one,523

   9,990,000x = 1,522,464

6. Divide both sides by 9,990,000 to stand for x every bit a fraction

   10 = 1522464 / 9990000

Fractions in abstruse mathematics [edit]

In improver to being of great applied importance, fractions are as well studied by mathematicians, who cheque that the rules for fractions given above are consistent and reliable. Mathematicians ascertain a fraction as an ordered pair ${\displaystyle (a,b)}$ of integers ${\displaystyle a}$ and ${\displaystyle b\neq 0,}$ for which the operations addition, subtraction, multiplication, and division are defined equally follows:^[24]

${\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,}$

${\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,}$

${\displaystyle (a,b)\cdot (c,d)=(air conditioning,bd)}$

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

These definitions agree in every case with the definitions given in a higher place; merely the notation is different. Alternatively, instead of defining subtraction and partition as operations, the "inverse" fractions with respect to add-on and multiplication might be defined as:

${\displaystyle {\begin{aligned}-(a,b)&=(-a,b)&&{\text{condiment inverse fractions,}}\\&&&{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-one}&=(b,a)&&{\text{multiplicative inverse fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}}$

Furthermore, the relation, specified every bit

${\displaystyle (a,b)\sim (c,d)\quad \iff \quad advertizing=bc,}$

is an equivalence relation of fractions. Each fraction from ane equivalence class may exist considered as a representative for the whole course, and each whole grade may exist considered as one abstract fraction. This equivalence is preserved past the higher up defined operations, i.eastward., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions

${\displaystyle (a,b)\sim (a',b')\quad }$ and ${\displaystyle \quad (c,d)\sim (c',d')\quad }$ imply

${\displaystyle ((a,b)+(c,d))\sim ((a',b')+(c',d'))}$

and similarly for the other operations.

In the instance of fractions of integers, the fractions a / b with a and b coprime and b > 0 are often taken as uniquely adamant representatives for their equivalent fractions, which are considered to be the aforementioned rational number. This way the fractions of integers brand up the field of the rational numbers.

More by and large, a and b may exist elements of any integral domain R, in which case a fraction is an chemical element of the field of fractions of R. For example, polynomials in i indeterminate, with coefficients from some integral domain D, are themselves an integral domain, telephone call it P. So for a and b elements of P, the generated field of fractions is the field of rational fractions (also known as the field of rational functions).

Algebraic fractions [edit]

An algebraic fraction is the indicated caliber of two algebraic expressions. Equally with fractions of integers, the denominator of an algebraic fraction cannot be goose egg. Ii examples of algebraic fractions are ${\displaystyle {\frac {3x}{x^{2}+2x-three}}}$ and ${\displaystyle {\frac {\sqrt {x+2}}{x^{ii}-3}}}$ . Algebraic fractions are subject to the same field properties as arithmetic fractions.

If the numerator and the denominator are polynomials, as in ${\displaystyle {\frac {3x}{10^{2}+2x-3}}}$ , the algebraic fraction is chosen a rational fraction (or rational expression). An irrational fraction is 1 that is not rational, as, for instance, one that contains the variable under a fractional exponent or root, as in ${\displaystyle {\frac {\sqrt {ten+2}}{x^{two}-iii}}}$ .

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 just factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, incorporate a fraction, such every bit ${\displaystyle {\frac {1+{\tfrac {1}{ten}}}{1-{\tfrac {i}{x}}}}}$ , is called a complex fraction.

The field of rational numbers is the field of fractions of the integers, while the integers themselves are non a field but rather an integral domain. Similarly, the rational fractions with coefficients in a field grade the field of fractions of polynomials with coefficient in that field. Because the rational fractions with existent coefficients, radical expressions representing numbers, such as ${\displaystyle \textstyle {\sqrt {2}}/ii,}$ are also rational fractions, as are a transcendental numbers such as ${\textstyle \pi /2,}$ since all of ${\displaystyle {\sqrt {2}},\pi ,}$ and ${\displaystyle 2}$ are existent numbers, and thus considered every bit coefficients. These same numbers, however, are not rational fractions with integer coefficients.

The term partial fraction is used when decomposing rational fractions into sums of simpler fractions. For case, the rational fraction ${\displaystyle {\frac {2x}{x^{2}-1}}}$ can exist decomposed as the sum of two fractions: ${\displaystyle {\frac {1}{x+1}}+{\frac {1}{x-one}}.}$ This is useful for the computation of antiderivatives of rational functions (encounter partial fraction decomposition for more than).

Radical expressions [edit]

A fraction may also comprise radicals in the numerator or the denominator. If the denominator contains radicals, information technology tin can be helpful to rationalize it (compare Simplified form of a radical expression), especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is too more user-friendly if division is to be done manually. When the denominator is a monomial square root, information technology can exist rationalized past multiplying both the top and the bottom of the fraction by the denominator:

${\displaystyle {\frac {three}{\sqrt {vii}}}={\frac {iii}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {vii}}}{7}}}$

The process of rationalization of binomial denominators involves multiplying the top and the lesser of a fraction past the conjugate of the denominator and so that the denominator becomes a rational number. For case:

${\displaystyle {\frac {3}{iii-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {v}}}{3+ii{\sqrt {5}}}}={\frac {3(three+2{\sqrt {5}})}{{3}^{2}-(two{\sqrt {five}})^{2}}}={\frac {three(three+ii{\sqrt {5}})}{ix-twenty}}=-{\frac {9+vi{\sqrt {5}}}{11}}}$

${\displaystyle {\frac {iii}{3+2{\sqrt {five}}}}={\frac {3}{iii+two{\sqrt {five}}}}\cdot {\frac {three-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(three-2{\sqrt {5}})}{{three}^{ii}-(ii{\sqrt {5}})^{2}}}={\frac {3(three-two{\sqrt {5}})}{9-20}}=-{\frac {9-half-dozen{\sqrt {5}}}{eleven}}}$

Even if this process results in the numerator being irrational, similar 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.

Typographical variations [edit]

In reckoner displays and typography, simple fractions are sometimes printed equally a unmarried character, eastward.g. ½ (ane half). Run into the commodity on Number Forms for information on doing this in Unicode.

Scientific publishing distinguishes four ways to prepare fractions, together with guidelines on utilize:^[25]

• Special fractions: fractions that are presented as a single graphic symbol 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 consequence, especially for small-sized fonts. These are not used in modernistic mathematical notation, but in other contexts.
• Case fractions: similar to special fractions, these are rendered every bit a single typographical character, but with a horizontal bar, thus making them upright. An example would be ${\displaystyle {\tfrac {1}{two}}}$ , but rendered with the same height as other characters. Some sources include all rendering of fractions as instance fractions if they take only one typographical space, regardless of the direction of the bar.^[26]
• Shilling or solidus fractions: 1/2, and so called because this notation was used for pre-decimal British currency (£sd), equally in "2/6" for a half crown, meaning two shillings and six pence. While the notation "two shillings and six pence" did not correspond a fraction, the frontward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is as well used for fractions within fractions (complex fractions) or within exponents to increase legibility. Fractions written this way, also known as slice fractions,^[27] are written all on one typographical line, but take iii or more than typographical spaces.
• Congenital-upwards fractions: ${\displaystyle {\frac {1}{2}}}$ . This notation uses 2 or more lines of ordinary text and results in a variation in spacing betwixt lines when included within other text. While large and legible, these tin can be confusing, peculiarly for simple fractions or inside complex fractions.

History [edit]

The primeval fractions were reciprocals of integers: ancient symbols representing one role of two, one office of three, one part of four, and so on.^[28] The Egyptians used Egyptian fractions c.  m BC. About 4000 years ago, Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the aforementioned respond as mod methods.^[29] The Egyptians as well had a different notation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus issues.

The Greeks used unit fractions and (later) continued fractions. Followers of the Greek philosopher Pythagoras (c.  530 BC) discovered that the square root of 2 cannot be expressed as a fraction of integers. (This is unremarkably though probably erroneously ascribed to Hippasus of Metapontum, who is said to have been executed for revealing this fact.) In 150 BC Jain mathematicians in Bharat wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions.

A modern expression of fractions known as bhinnarasi seems to accept originated in India in the piece of work of Aryabhatta (c.  AD 500),^{[ citation needed ]} Brahmagupta (c.  628), and Bhaskara (c.  1150).^[xxx] Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators ( cheda ), but without a bar betwixt them.^[30] In Sanskrit literature, fractions were always expressed as an addition to or subtraction from an integer.^{[ citation needed ]} 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 ⟨०⟩ or cantankerous ⟨+⟩, it is subtracted from the integer; if no such sign appears, it is understood to exist added. For instance, Bhaskara I writes:^[31]

६  १  २

१  १  १_०

४  ५  ९

which is the equivalent of

6  1  ii

1  1  −1

four  v  9

and would be written in mod notation every bit 6 i / 4 , one 1 / 5 , and 2 − one / 9 (i.eastward., 1 8 / 9 ).

The horizontal fraction bar is starting time attested in the work of Al-Hassār (fl.  1200),^[30] a Muslim mathematician from Fez, Kingdom of morocco, who specialized in Islamic inheritance jurisprudence. In his discussion he writes: "for example, if you are told to write 3-fifths and a third of a fifth, write thus, ${\displaystyle {\frac {3\quad 1}{five\quad iii}}}$ ". ^[32] The same partial notation—with the fraction given before the integer^[30]—appears before long afterward in the work of Leonardo Fibonacci in the 13th century.^[33]

In discussing the origins of decimal fractions, Dirk Jan Struik states:^[34]

The introduction of decimal fractions as a common computational do can exist dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), so settled in the Northern Netherlands. Information technology is true that decimal fractions were used past the Chinese many centuries earlier Stevin and that the Western farsi astronomer Al-Kāshī used both decimal and sexagesimal fractions with slap-up ease in his Key to arithmetics (Samarkand, early on fifteenth century).^[35]

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were commencement used v centuries before him by the Baghdadi mathematician Abu'50-Hasan al-Uqlidisi as early every bit the 10th century.^{[36] [due north ii]}

In formal education [edit]

Pedagogical tools [edit]

In master schools, fractions have been demonstrated through Cuisenaire rods, Fraction Confined, fraction strips, fraction circles, newspaper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboards, counters and estimator software.

Documents for teachers [edit]

Several states in the The states have adopted learning trajectories from the Common Cadre State Standards Initiative's guidelines for mathematics educational activity. 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 ${\displaystyle a}$ / ${\displaystyle b}$ where ${\displaystyle a}$ is a whole number and ${\displaystyle b}$ is a positive whole number. (The word fraction in these standards always refers to a non-negative number.)"^[38] The document itself also refers to negative fractions.

See also [edit]

• Cross multiplication
• 0.999...
• Multiple
• FRACTRAN

Number systems

Complex ${\displaystyle :\;\mathbb {C} }$

Real ${\displaystyle :\;\mathbb {R} }$ Rational ${\displaystyle :\;\mathbb {Q} }$ Integer ${\displaystyle :\;\mathbb {Z} }$ Natural ${\displaystyle :\;\mathbb {Due north} }$ Goose egg: 0 One: 1 Prime numbers Composite numbers Negative integers Fraction Finite decimal Dyadic (finite binary) Repeating decimal Irrational Algebraic irrational Transcendental Imaginary

Notes [edit]

1. ^ Some typographers such as Bringhurst mistakenly distinguish the slash ⟨/⟩ as the virgule and the fraction slash ⟨⁄⟩ as the solidus,^{[half dozen]} although in fact both are synonyms for the standard slash.^{[7] [eight]}
2. ^ While there is some disagreement among history of mathematics scholars as to the primacy of al-Uqlidisi'due south contribution, there is no question as to his major contribution to the concept of decimal fractions.^[37]

References [edit]

1. ^ H. Wu, "The Mis-Educational activity of Mathematics Teachers", Notices of the American Mathematical Gild, Volume 58, Issue 03 (March 2011), p. 374. Archived 2017-08-twenty at the Wayback Motorcar.
2. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English . Mathematical Association of America. ISBN978-0-88385-511-9.
3. ^ "Fractions". world wide web.mathsisfun.com . Retrieved 2020-08-27 .
4. ^ ^{a b} Ambrose, Gavin; et al. (2006). The Fundamentals of Typography (2nd ed.). Lausanne: AVA Publishing. p. 74. ISBN978-2-940411-76-iv. Archived from the original on 2016-03-04. Retrieved 2016-02-20 . .
5. ^ Weisstein, Eric W. "Fraction". mathworld.wolfram.com . Retrieved 2020-08-27 .
6. ^ Bringhurst, Robert (2002). "5.two.5: Use the Virgule with Words and Dates, the Solidus with Dissever-level Fractions". The Elements of Typographic Way (tertiary ed.). Point Roberts: Hartley & Marks. pp. 81–82. ISBN978-0-88179-206-5.
7. ^ "virgule, n.". Oxford English Dictionary (1st ed.). Oxford: Oxford University Press. 1917.
8. ^ "solidus, northward.¹ ". Oxford English Dictionary (1st ed.). Oxford: Oxford Academy Printing. 1913.
9. ^ Weisstein, Eric West. "Common Fraction". MathWorld.
10. ^ ^{a b} David E. Smith (1 June 1958). History of Mathematics. Courier Corporation. p. 219. ISBN978-0-486-20430-7.
11. ^ "World Broad Words: Vulgar fractions". World Broad Words. Archived from the original on 2014-10-30. Retrieved 2014-10-xxx .
12. ^ Weisstein, Eric West. "Improper Fraction". MathWorld.
13. ^ Jack Williams (19 Nov 2011). Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation. Springer Science & Business Media. pp. 87–. ISBN978-0-85729-862-1.
14. ^ Tape, Robert (1654). Record'southward Arithmetick: Or, the Ground of Arts: Teaching the Perfect Work and Practise of Arithmetick ... Made by Mr. Robert Record ... Afterward Augmented past Mr. John Dee. And Since Enlarged with a Third Part of Rules of Practise ... By John Mellis. And Now Diligently Perused, Corrected ... and Enlarged ; with an Appendix of Figurative Numbers ... with Tables of Board and Timber Measure ... the Commencement Calculated by R. C. But Corrected, and the Latter ... Calculated by Ro. Hartwell ... James Flesher, and are to be sold by Edward Dod. pp. 266–.
15. ^ Laurel (31 March 2004). "Math Forum – Ask Dr. Math: Tin can Negative Fractions Also Be Proper or Improper?". Archived from the original on 9 Nov 2014. Retrieved 2014-x-xxx .
16. ^ "New England Compact Math Resources". Archived from the original on 2012-04-15. Retrieved 2011-12-31 .
17. ^ Greer, A. (1986). New comprehensive mathematics for 'O' level (2nd ed., reprinted ed.). Cheltenham: Thornes. p. 5. ISBN978-0-85950-159-0. Archived from the original on 2019-01-xix. Retrieved 2014-07-29 .
18. ^ ^{a b} Trotter, James (1853). A complete system of arithmetics. p. 65.
19. ^ ^{a b} Barlow, Peter (1814). A new mathematical and philosophical dictionary.
20. ^ "complex fraction". Collins English Dictionary. Archived from the original on 2017-12-01. Retrieved 29 August 2022.
21. ^ "Complex fraction definition and pregnant". Collins English Lexicon. 2018-03-09. Archived from the original on 2017-12-01. Retrieved 2018-03-13 .
22. ^ "Compound Fractions". Sosmath.com. 1996-02-05. Archived from the original on 2018-03-fourteen. Retrieved 2018-03-13 .
23. ^ Schoenborn, Barry; Simkins, Bradley (2010). "8. Fun with Fractions". Technical Math For Dummies. Hoboken: Wiley Publishing Inc. p. 120. ISBN978-0-470-59874-0. OCLC 719886424. Retrieved 28 September 2020.
24. ^ "Fraction". Encyclopedia of Mathematics. 2012-04-06. Archived from the original on 2014-ten-21. Retrieved 2012-08-15 .
25. ^ Galen, Leslie Blackwell (March 2004). "Putting Fractions in Their Identify" (PDF). American Mathematical Monthly. 111 (3): 238–242. doi:10.2307/4145131. JSTOR 4145131. Archived (PDF) from the original on 2011-07-xiii. Retrieved 2010-01-27 .
26. ^ "built fraction". allbusiness.com glossary. Archived from the original on 2013-05-26. Retrieved 2013-06-eighteen .
27. ^ "piece fraction". allbusiness.com glossary. Archived from the original on 2013-05-21. Retrieved 2013-06-18 .
28. ^ Eves, Howard (1990). An introduction to the history of mathematics (sixth ed.). Philadelphia: Saunders College Pub. ISBN978-0-03-029558-4.
29. ^ Milo Gardner (Dec 19, 2005). "Math History". Archived from the original on December nineteen, 2005. Retrieved 2006-01-18 . Come across for examples and an explanation.
30. ^ ^{a b c d} Miller, Jeff (22 December 2014). "Earliest Uses of Diverse Mathematical Symbols". Archived from the original on xx February 2016. Retrieved 15 Feb 2016.
31. ^ Filliozat, Pierre-Sylvain (2004). "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature". In Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.). History of Scientific discipline, History of Text. Boston Series in the Philosophy of Science. Vol. 238. Dordrecht: Springer Netherlands. p. 152. doi:10.1007/1-4020-2321-9_7. ISBN978-1-4020-2320-0.
32. ^ Cajori, Florian (1928). A History of Mathematical Notations. Vol. i. La Salle, Illinois: Open Courtroom Publishing Company. p. 269. Archived from the original on 2014-04-14. Retrieved 2017-08-30 .
33. ^ Cajori (1928), p. 89
34. ^ A Source Book in Mathematics 1200–1800. New Jersey: Princeton University Press. 1986. ISBN978-0-691-02397-7.
35. ^ Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī. Wiesbaden: Steiner. 1951.
36. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN978-0-691-11485-ix.
37. ^ "MacTutor'due south al-Uqlidisi biography". Archived 2011-eleven-15 at the Wayback Machine. Retrieved 2011-11-22.
38. ^ "Mutual Core Country Standards for Mathematics" (PDF). Common Core State Standards Initiative. 2010. p. 85. Archived (PDF) from the original on 2013-10-19. Retrieved 2013-10-10 .

External links [edit]

Wikimedia Eatables has media related to Fractions.

Await up denominator in Wiktionary, the costless dictionary.

Wait up numerator in Wiktionary, the free dictionary.

• "Fraction, arithmetical". The Online Encyclopaedia of Mathematics.
• "Fraction". Encyclopædia Britannica.
• "Fraction (mathematics)". Citizendium.
• "Fraction". PlanetMath. Archived from the original on 25 Oct 2019. Retrieved 29 September 2019.

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