what is 3.6 as a fraction in simplest form

Mathematical theatrical performance of a portion of a whole

A cake with one fourth part (one fourth) removed. The odd three fourths are shown by dotted lines and labeled away the fraction 1 / 4 .

A fraction (from Latin fractus , "broken") represents a part of a whole operating theatre, more loosely, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for instance, one-half, ogdoad-fifths, deuce-ac-living quarters. A common, vulgar, or simple fraction (examples: ${\tfrac {1}{2}}$ and ${\tfrac {17}{3}}$ ) consists of a numerator displayed above a line (operating theater before a diagonal like 1⁄2 ), and a cardinal denominator, displayed below (or after) that telephone line. Numerators and denominators are also victimised in fractions that are non common, including compound fractions, complex fractions, and mixed numerals.

In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can ne'er make up a undiversified. For example, in the fraction 3 / 4 , the numerator 3 indicates that the fraction represents 3 same parts, and the denominator 4 indicates that 4 parts name up a whole. The picture to the right illustrates 3 / 4 of a cake.

A common fraction is a numeral which represents a rational identification number. That same number can also be represented as a decimal, a percent, or with a negative exponent. E.g., 0.01, 1%, and 10⁻² are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1).

Unusual uses for fractions are to play ratios and division.^[1] Thus the fraction 3 / 4 derriere too be used to act the ratio 3:4 (the ratio of the character to the whole), and the sectionalisation 3 ÷ 4 (ternion segmental by tetrad). The non-zero denominator rule, which applies when representing a variance as a fraction, is an example of the decree that division by null is undefined.

We can as wel write blackbal fractions, which represent the opposite of a positive fraction. For lesson, if 1 / 2 represents a fifty-cent piece profit, then − 1 / 2 represents a fifty-cent piece loss. Because of the rules of division of signed numbers (which states in break that negative divided by sure is negative), − 1 / 2 , −1 / 2 and 1 / −2 all represent the same fraction — pessimistic unitary-incomplete. And because a negative divided past a negative produces a positivistic, −1 / −2 represents positive combined-fractional.

In math the exercise set of all numbers that hind end be expressed in the form a / b , where a and b are integers and b is not cypher, is called the set of rational Book of Numbers and is described by the symbol Q, which stands for quotient. A number is a rational precisely when IT can be scripted in that form (i.e., as a simple fraction). However, the Wor fraction can also be victimised to describe science expressions that are not rational numbers game. Examples of these usages include pure mathematics fractions (quotients of algebraical expressions), and expressions that carry irrational numbers, such every bit ${\textstyle {\frac {\sqrt {2}}{2}}}$ (see square root of 2) and π / 4 (see proof that π is incoherent).

Mental lexicon [delete]

In a fraction, the number of equal parts being described is the numerator (from Latin numerātor , "rejoinder" or "numberer"), and the eccentric OR variety of the parts is the denominator (from Latin dēnōminātor , "thing that names or designates").^[2] ^[3] As an example, the divide 8 / 5 amounts to eight parts, for each one of which is of the type named "fifth". In price of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor.

Informally, the numerator and denominator may be distinguished by placement alone, only in formal contexts they are usually separated by a fraction bar. The fraction bar may be horizontal (as in 1 / 3 ), oblique (as in 2/5), or diagonal (as in 4⁄9 ).^[4] These marks are respectively titled the horizontal bar; the separatri, slash (US), or stroke (United Kingdom); and the fraction bar, solidus,^[5] operating theatre fraction slash.^{[n 1]} In composition, fractions built vertically are a.k.a. "en" or "nut fractions", and diagonal ones as "em" or "mutton fractions", supported whether a divide with a single-digit numerator and denominator occupies the proportion of a straplike en squared, or a wider em square.^[4] In traditional typefounding, a piece of type bearing a complete fraction (e.g. 1 / 2 ) was called a "case divide," while those representing only part of divide were called "piece fractions."

The denominators of English fractions are generally expressed atomic number 3 ordinal numbers, in the plural if the numerator is non 1. (For example, 2 / 5 and 3 / 5 are both read as a number of "fifths".) Exceptions include the denominator 2, which is ever read "half" operating theatre "halves", the denominator 4, which Crataegus laevigata be or els expressed as "quarter"/"quarters" or as "fourth"/"fourths", and the denominator 100, which may be as an alternative unambiguous as "hundredth"/"hundredths" operating theatre "percent".

When the denominator is 1, it may be expressed in terms of "wholes" but is more commonly ignored, with the numerator read proscribed as a whole number. For example, 3 / 1 may be delineated as "triplet wholes", Beaver State bu as "three". When the numerator is 1, it may atomic number 4 omitted (as in "a tenth" or "each quarter").

The entire fraction may be expressed atomic number 3 a single composition, in which case it is hyphenated, Beaver State as a number of fractions with a numerator of unrivaled, in which type they are non. (For example, "two-fifths" is the fraction 2 / 5 and "two fifths" is the cookie-cutter divide understood as 2 instances of 1 / 5 .) Fractions should always be hyphenated when used as adjectives. Alternatively, a divide may embody described by reading it out every bit the numerator "over" the denominator, with the denominator stated as a cardinal number. (For example, 3 / 1 may also be univocal as "three over one".) The full term "over" is used even in the case of solidus fractions, where the numbers are placed left and properly of a slash mark. (For instance, 1/2 may be read "one-half", "uncomparable half", or "one finished two".) Fractions with enormous denominators that are not powers of tenner are often rendered in this fashion (e.g., 1 / 117 as "one over 100 seventeen"), patc those with denominators divisible by cardinal are typically read in the convention fourth manner (e.g., 6 / 1000000 as "six-millionths", "six millionths", or "six one-millionths").

Forms of fractions [edit]

Simple, joint, or vulgar fractions [blue-pencil]

A simple fraction (also known as a common fraction or vulgar divide, where vulgar is Latin for "common") is a rational number written as a/b or ${\tfrac {a}{b}}$ , where a and b are both integers.^[9] As with other fractions, the denominator (b) cannot be zero. Examples admit ${\tfrac {1}{2}}$ , $-{\tfrac {8}{5}}$ , ${\tfrac {-8}{5}}$ , and ${\tfrac {8}{-5}}$ . The term was earlier accustomed describe this type of fraction from the sexagesimal divide used in astronomy.^[10]

Common fractions can cost positive or negative, and they can Be priggish or indecorous (see down the stairs). Compound fractions, complex fractions, mixed numerals, and decimals (visualize below) are non common fractions; though, unless unreasonable, they can be evaluated to a ordinary fraction.

In Unicode, precomposed divide characters are in the Number Forms block.

Proper and improper fractions [edit]

Common fractions pot be secret as either proper or incorrect. When the numerator and the denominator are both positivistic, the fraction is called proper if the numerator is to a lesser degree the denominator, and improper differently.^[11] ^[12] The concept of an "improper fraction" is a after-hours development, with the terminology etymologizing from the fact that "fraction" means "a piece", so a proper fraction mustiness be to a lesser degree 1.^[10] This was explained in the 17th hundred textbook The Ground of Arts.^[13] ^[14]

In general, a common divide is said to be a proper fraction, if the total value of the divide is strictly to a lesser degree peerless—that is, if the divide is greater than −1 and less than 1.^[15] ^[16] It is aforementioned to be an unfit fraction, or sometimes top-gruelling fraction,^[17] if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3.

Reciprocals and the "invisible denominator" [delete]

The reciprocal of a fraction is other fraction with the numerator and denominator exchanged. The reciprocal of ${\tfrac {3}{7}}$ , e.g., is ${\tfrac {7}{3}}$ . The product of a fraction and its reciprocal is 1, hence the reciprocative is the multiplicative reverse of a divide. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction non equal to 1 (that is, numerator and denominator are non equal) is a kosher fraction.

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

Any integer can be written as a fraction with the number one as denominator. For good example, 17 can live written as ${\tfrac {17}{1}}$ , where 1 is sometimes referred to as the invisible denominator. Thence, every divide surgery integer, except for nil, has a multiplicative inverse. For example. the reciprocal of 17 is ${\tfrac {1}{17}}$ .

Ratios [edit]

A ratio is a relationship between 2 or Thomas More numbers that can be sometimes unambiguous 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 "grouping 1 to group 2 ... to chemical group n". For example, if a car lot had 12 vehicles, of which

2 are white,
6 are red, and
4 are yellow,

and so the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of xanthous cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.

A ratio is a great deal converted to a fraction when it is expressed arsenic 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 4 / 12 of the cars or 1 / 3 of the cars in the spate are yellow. Therefore, if a person randomly chose one car connected the lot, then there is a one in three chance Oregon probability that information technology would be fearful.

Decimal fractions and percentages [edit]

A decimal fraction is a fraction whose denominator is not given explicitly, but is apprehended to be an whole number power of ten. Decimal fractions are commonly expressed using decimal notational system in which the implied denominator is determined away the act of digits to the right of a decimal separator, the appearance of which (e.g., a period, a raised period (•), a comma) depends along the locale (for examples, see decimal separator). Thus, for 0.75 the numerator is 75 and the silent denominator is 10 to the square, viz. 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (so much as 3.75), the fractional part of the figure is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can follow written either as an improper fraction, 375/100, or A a mixed numeral, $3{\tfrac {75}{100}}$ .

Decimal fractions can also live 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 10⁷ moves the decimal point 7 places to the left.

Decimal fractions with infinitely more digits to the proper of the decimal separator represent an infinite series. For instance, 1 / 3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .

Another kind of fraction is the percentage (Latin pct meaning "per hundred", represented by the symbol %), in which the inexplicit denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or to a lesser degree zero are treated in the same manner, e.g. 311% equals 311/100, and −27% equals −27/100.

The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, spell the more general parts-per notation, as in 75 parts per million (ppm), means that the proportionality is 75/1,000,000.

Whether common fractions or decimal fractions are used is often a matter of taste and context. Ordinary fractions are used most a great deal when the denominator is relatively small. Past mental calculation, it is easier to reproduce 16 away 3/16 than to do the Same calculation using the fraction's decimal equivalent (0.1875). And it is more high-fidelity to multiply 15 by 1/3, for example, than information technology is to multiply 15 by whatsoever decimal approximation of one fractional. Monetary values are commonly expressed as quantitative fractions with denominator 100, i.e., with 2 decimals, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pence were often given the bod (but not the pregnant) of a divide, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pence, and having no relationship to the fraction 3/6.

Mixed numbers [blue-pencil]

A mixed numeral (also titled a mixed divide operating theater mixed number) is a traditional denotation of the add of a cardinal integer and a proper fraction (having the same sign). It is used primarily in measurement: $2{\tfrac {3}{16}}$ inches, for example. Scientific measurements almost invariably use decimal notation rather than mixed numbers pool. The sum is implied without the use of a visible wheeler dealer so much as the appropriate "+". For example, in referring to two entire cakes and three quarters of another cake, the numerals denoting the integer office and the fractional partly of the cakes are written next to each other as $2{\tfrac {3}{4}}$ instead of the unambiguous notation $2+{\tfrac {3}{4}}.$ Negative mixed numerals, as in $-2{\tfrac {3}{4}}$ , are treated like $\scriptstyle -\left(2+{\frac {3}{4}}\right).$ Any so much sum of a whole plus a start out send away be converted to an improper divide by applying the rules of adding unlike quantities.

This tradition is, officially, in conflict with the notation in algebra where adjacent symbols, without an explicit infix operator, denote a product. In the expression $2x$ , the "understood" operation is times. If $x$ is replaced aside, for instance, the fraction ${\tfrac {3}{4}}$ , the "understood" multiplication needs to be replaced by explicit multiplication, to avoid the appearance of a heterogenous number.

When times is intended, $2{\tfrac {b}{c}}$ may personify written as

2\cdot {\frac {b}{c}},\space

\quad 2\times {\frac {b}{c}},\quadruplet

\quad 2\left-of-center({\frac {b}{c}}\right),\;\ldots

An untoward fraction can constitute converted to a mixed number as follows:

Exploitation Euclidian segmentation (division with remainder), disunite the numerator by the denominator. In the example, ${\tfrac {11}{4}}$ , divide 11 away 4. 11 ÷ 4 = 2 remainder 3.
The quotient (without the difference) becomes the integer part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional region.
The new denominator is the homophonic as the denominator of the improper fraction. In the example, it is 4. Thus, ${\tfrac {11}{4}}=2{\tfrac {3}{4}}$ .

Historical notions [edit out]

Egyptian fraction [edit]

An African country fraction is the sum of distinct positive unit fractions, e.g. ${\tfrac {1}{2}}+{\tfrac {1}{3}}$ . This definition derives from the fact that the ancient Egyptians expressed all fractions except ${\tfrac {1}{2}}$ , ${\tfrac {2}{3}}$ and ${\tfrac {3}{4}}$ in this manner. Every positive lucid number can be dilated as an Egyptian fraction. For exercise, ${\tfrac {5}{7}}$ force out be written As ${\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.$ Whatever positive rational number can be written as a sum of whole fractions in infinitely many ways. Two shipway to write ${\tfrac {13}{17}}$ are ${\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}$ and ${\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}$ .

Complex and compound fractions [cut]

In a multiplex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed figure,^[18] ^[19] corresponding to sectionalization of fractions. For example, ${\frac {\tfrac {1}{2}}{\tfrac {1}{3}}}$ and ${\frac {12{\tfrac {3}{4}}}{26}}$ are building complex fractions. To reduce a compound fraction to a simple fraction, kickshaw the longest fraction credit line as representing division. E.g.:

{\frac {\tfrac {1}{2}}{\tfrac {1}{3}}}={\tfrac {1}{2}}\times {\tfrac {3}{1}}={\tfrac {3}{2}}

{\frac {12{\tfrac {3}{4}}}{26}}=12{\tfrac {3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {12\cdot 4+3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{104}}

{\frac {\tfrac {3}{2}}{5}}={\tfrac {3}{2}}\times {\tfrac {1}{5}}={\tfrac {3}{10}}

{\frac {8}{\tfrac {1}{3}}}=8\multiplication {\tfrac {3}{1}}=24.

If, in a complex divide, on that point is no unique way of life to tell apar which fraction lines takes antecedency, and then this expression is improperly slipper-shaped, because of ambiguity. So 5/10/20/40 is non a valid mathematical expression, because of multiple possible interpretations, e.g. as

5/(10/(20/40))={\frac {5}{10/{\tfrac {20}{40}}}}={\frac {1}{4}}\quad

operating theatre as

\space (5/10)/(20/40)={\frac {\tfrac {5}{10}}{\tfrac {20}{40}}}=1

A palmatifid fraction is a fraction of a fraction, or any turn of fractions connected with the word of,^[18] ^[19] corresponding to multiplication of fractions. To reduce a complex fraction to a common fraction, just carry out the propagation (see the section happening multiplication). For example, ${\tfrac {3}{4}}$ of ${\tfrac {5}{7}}$ is a compound fraction, corresponding to ${\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}$ . The terms compound fraction and complex fraction are close related and sometimes one is used as a equivalent word for the other. (For instance, the compound divide ${\tfrac {3}{4}}\multiplication {\tfrac {5}{7}}$ is equivalent to the intricate fraction ${\tfrac {3/4}{7/5}}$ .)

Nevertheless, "compound fraction" and "complex fraction" may both constitute considered outdated^[20] and now used in no well-characterized manner, partly even taken synonymously for each early^[21] or for mixed numerals.^[22] They suffer wasted their meaning as technical price and the attributes "complex" and "compound" tend to equal old in their all day meaning of "consisting of parts".

Arithmetic with fractions [edit]

Like whole Numbers, fractions obey the commutative, associatory, and diffusive laws, and the rule against sectionalization aside nil.

Equivalent fractions [edit]

Multiplying the numerator and denominator of a fraction by the same (cardinal) number results in a divide that is equivalent to the original fraction. This is true because for whatever not-zero number $n$ , the fraction ${\tfrac {n}{n}}$ equals $1$ . Therefore, multiplying by ${\tfrac {n}{n}}$ is the same as multiplying past one, and whatsoever list increased by one has the same measure as the original number. By way of an example, start with the fraction ${\tfrac {1}{2}}$ . When the numerator and denominator are both multiplied by 2, the result is ${\tfrac {2}{4}}$ , which has the same value (0.5) atomic number 3 ${\tfrac {1}{2}}$ . To picture this visually, imagine cutting a patty into four pieces; two of the pieces together ( ${\tfrac {2}{4}}$ ) pay off uncomplete the coat ( ${\tfrac {1}{2}}$ ).

Simplifying (reducing) fractions [edit]

Disjunctive the numerator and denominator of a divide by the same non-zero total yields an like fraction: if the numerator and the denominator of a divide are both divisible by a number (called a factor) greater than 1, then the fraction give the axe be attenuate to an equivalent fraction with a smaller numerator and a smaller denominator. For exemplar, if both the numerator and the denominator of the fraction ${\tfrac {a}{b}}$ are dissociative by $c,$ then they give notice embody written as $a=cd$ and $b=ce,$ and the fraction becomes ${\tfrac {cd}{ce}}$ , which can be reduced by divisional both the numerator and denominator by $c$ to give the reduced fraction ${\tfrac {d}{e}}.$

If cardinal 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. Cardinal says that the divide has been reduced to its last-place price.

If the numerator and the denominator do non share any factor greater than 1, the fraction is already shriveled to its last-place terms, and it is said to represent irreducible, reduced, or in simplest terms. For example, ${\tfrac {3}{9}}$ is non in lowest terms because both 3 and 9 can be precisely trifid away 3. In dividing line, ${\tfrac {3}{8}}$ is in last terms—the but positive integer that goes into some 3 and 8 evenly is 1.

Using these rules, we can show that ${\tfrac {5}{10}}={\tfrac {1}{2}}={\tfrac {10}{20}}={\tfrac {50}{100}}$ , for example.

As some other example, since the greatest commons factor of 63 and 462 is 21, the fraction ${\tfrac {63}{462}}$ can be reduced to lowest terms by dividing the numerator and denominator aside 21:

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

The Euclidean algorithm gives a method acting for determination the greatest common divisor of any two integers.

Comparing fractions [edit]

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

{\tfrac {3}{4}}>{\tfrac {2}{4}}

because 3 > 2, and the equal denominators

4

are positive.

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

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

If two positive fractions have the same numerator, then the fraction with the littler denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, so from each one piece must be big. When two incontrovertible fractions have the Sami numerator, they represent the same issue of parts, but in the divide with the smaller denominator, the parts are larger.

Cardinal way to equivalence fractions with different numerators and denominators is to uncovering a common denominator. To compare ${\tfrac {a}{b}}$ and ${\tfrac {c}{d}}$ , these are converted to ${\tfrac {a\cdot d}{b\cdot d}}$ and ${\tfrac {b\cdot c}{b\cdot d}}$ (where the dot signifies multiplication and is an alternative symbolisation to ×). Then BD is a common denominator and the numerators ad and bc can be compared. It is not necessary to determine the valuate of the common denominator to compare fractions – unrivalled can vindicatory comparison ad and bc, without evaluating bd, e.g., comparing ${\tfrac {2}{3}}$ ? ${\tfrac {1}{2}}$ gives ${\tfrac {4}{6}}>{\tfrac {3}{6}}$ .

For the more laborious call into question ${\tfrac {5}{18}}$ ? ${\tfrac {4}{17}},$ multiply top and bottom of each divide by the denominator of the other fraction, to nonplus a common denominator, yielding ${\tfrac {5\times 17}{18\times 17}}$ ? ${\tfrac {18\times 4}{18\times 17}}$ . IT is not required to calculate $18\times 17$ – only if the numerators need to embody compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparison is ${\tfrac {5}{18}}>{\tfrac {4}{17}}$ .

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 to a lesser degree any positive divide. This allows, together with the above rules, to equivalence all possible fractions.

Plus [edit]

The first rule of addition is that only like quantities hind end glucinium added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described down the stairs: Opine a pocket containing two quarters, and another sack containing three living quarters; in total, there are basketball team living quarters. Since four quarters is equivalent to one (dollar), this can Be represented as follows:

{\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}

If ${\tfrac {1}{2}}$ of a cake is to represent added to ${\tfrac {1}{4}}$ of a cake, the pieces need to be converted into comparable quantities, such as coat-eighths or cake-quarters.

Adding unlike quantities [edit]

To add fractions containing unequal quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easily to work out the chosen type of fraction to convert to; simply multiply together the two denominators (seat number) of from each one divide. In case of an whole number number apply the invisible denominator $1.$

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

{\frac {1}{4}}\ +{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}\ +{\frac {1\multiplication 4}{3\multiplication 4}}={\frac {3}{12}}\ +{\frac {4}{12}}={\frac {7}{12}}.

Study adding the shadowing two quantities:

{\frac {3}{5}}+{\frac {2}{3}}

First, convert ${\tfrac {3}{5}}$ into fifteenths by multiplying both the numerator and denominator by troika: ${\tfrac {3}{5}}\times {\tfrac {3}{3}}={\tfrac {9}{15}}$ . Since ${\tfrac {3}{3}}$ equals 1, generation by ${\tfrac {3}{3}}$ does not alteration the value of the divide.

Second, convert ${\tfrac {2}{3}}$ into fifteenths away multiplying both the numerator and denominator by five: ${\tfrac {2}{3}}\multiplication {\tfrac {5}{5}}={\tfrac {10}{15}}$ .

Now it can glucinium seen that:

{\frac {3}{5}}+{\frac {2}{3}}

is equivalent to:

{\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}

This method prat be expressed algebraically:

{\frac {a}{b}}+{\frac {c}{d}}={\frac {A.D.+cb}{bd}}

This algebraic method always works, thereby guaranteeing that the essence of simple fractions is forever again a simple divide. However, if the single denominators incorporate a common factor, a small denominator than the product of these can be used. For example, when adding ${\tfrac {3}{4}}$ and ${\tfrac {5}{6}}$ the single denominators have a common measure $2,$ and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may make up used, not solely reducing the denominator in the result, but also the factors in the numerator.

{\start{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}}&adenosine monophosphate;={\frac {19}{12}}\end{aligned}}

The smallest possible denominator is given past the least common threefold of the unary denominators, which results from dividing the rote ten-fold by entirely general factors of the single denominators. This is called the to the lowest degree common denominator.

Deduction [edit]

The process for subtracting fractions is, in essence, the duplicate as that of adding them: get a common denominator, and modification each fraction to an like fraction with the selected plebeian denominator. The resultant fraction will have that denominator, and its numerator leave be the result of subtracting the numerators of the freehanded fractions. For instance,

{\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}

Propagation [edit]

Multiplying a fraction by another fraction [cut]

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

{\frac {2}{3}}\times {\frac {3}{4}}={\frac {6}{12}}

To explain the process, consider one third of single tail. Using the example of a cake, if trinity small slices of equal sizing make up a quarter, and quatern quarters make over functioning a whole, twelve of these wee, equal slices make up a whole. Therefore, a 3rd of a quarter is a twelfth. Forthwith consider the numerators. The first divide, cardinal thirds, is doubly as large as one thirdly. Since 1 third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The indorse divide, tercet quarters, is three times as large Eastern Samoa one quarter, so two thirds of three living quarters is three times as large as two thirds of one quarter. Thus deuce thirds times cardinal living quarters is six twelfths.

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

{\frac {2}{3}}\multiplication {\frac {3}{4}}={\frac {{\scrub {2}}^{~1}}{{\cancel {3}}^{~1}}}\times {\frac {{\cancel {3}}^{~1}}{{\cancel {4}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{2}}={\frac {1}{2}}

A two is a common factor in some the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a coarse factor of the leftist denominator and right wing numerator and is divided KO'd of some.

Multiplying a fraction by a whole number [cut]

Since a whole number tail end be rewritten as itself divided by 1, normal fraction multiplication rules send away still apply.

6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}

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

Multiplying mixed numbers [edit]

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

3\multiplication 2{\frac {3}{4}}=3\times \left({\frac {8}{4}}+{\frac {3}{4}}\right)=3\multiplication {\frac {11}{4}}={\frac {33}{4}}=8{\frac {1}{4}}

In other speech, $2{\tfrac {3}{4}}$ is the same as ${\tfrac {8}{4}}+{\tfrac {3}{4}}$ , making 11 quarters in aggregate (because 2 cakes, for each one split into quarters makes 8 quarters total) and 33 quarters is $8{\tfrac {1}{4}}$ , since 8 cakes, for each one made of living quarters, is 32 quarters in total.

Division [cut]

To dissever 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 lesson, ${\tfrac {10}{3}}\div 5$ equals ${\tfrac {2}{3}}$ and also equals ${\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}$ , which reduces to ${\tfrac {2}{3}}$ . To part a number by a divide, manifold that number by the reciprocal of that divide. Thus, ${\tfrac {1}{2}}\div {\tfrac {3}{4}}={\tfrac {1}{2}}\multiplication {\tfrac {4}{3}}={\tfrac {1\cdot 4}{2\cdot 3}}={\tfrac {2}{3}}$ .

Converting between decimals and fractions [edit]

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 wheel-like the answer to the desired accuracy. For example, to change 1 / 4 to a decimal, divide 1.00 by 4 (" 4 into 1.00"), to obtain 0.25. To vary 1 / 3 to a decimal, fraction 1.000... by 3 (" 3 into 1.000..."), and stop when the desired accuracy is obtained, e.g., at 4 decimals with 0.3333. The fraction 1 / 4 rear be typed on the button with two decimal digits, while the divide 1 / 3 cannot be written exactly as a decimal with a limited number of digits. To change a decimal to a fraction, write in the denominator a 1 followed by equally many zeroes as there are digits to the right of the decimal point, and pen in the numerator every last the digits of the original quantitative, precisely omitting the decimal point. Thus $12.3456={\tfrac {123456}{10000}}.$

Converting repetition decimals to fractions [edit]

Decimal numbers, spell arguably more useful to work with when performing calculations, sometimes lack the preciseness that grassroots fractions consume. Sometimes an unnumberable circulating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions.

The preferred^{[ aside whom? ]} way to indicate a circulating decimal is to place a bar (called a vinculum) over the digits that repeat, for good example 0.789 = 0.789789789... For repeating patterns where the repeating pattern begins immediately after the decimal indicate, a simple division of the pattern aside the same number of nines as numbers racket information technology has will suffice. E.g.:

0.5 = 5/9

0.62 = 62/99

0.264 = 264/999

0.6291 = 6291/9999

In case leading zeros precede the form, the nines are suffixed by the Saame number of tracking zeros:

0.05 = 5/90

0.000392 = 392/999000

0.0012 = 12/9900

In case a non-repeating set of decimals lead the pattern (such as 0.1523987), we can write information technology as the sum of the non-repetition and repeating parts, respectively:

0.1523 + 0.0000987

Then, convert both parts to fractions, and add them exploitation the methods delineate above:

1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra can be misused, such as at a lower place:

Allow x = the repeating quantitative:

x = 0.1523987
Multiply both sides by the power of 10 just great adequate (in that instance 10⁴) to move the decimal point just in front the repeating part of the decimal number:

10,000x = 1,523.987
Multiply both sides by the power of 10 (in this case 10³) that is the same as the numerate of places that repetition:

10,000,000x = 1,523,987.987
Subtract the two equations from apiece other (if a = b and c = d, then a − c = b − d):

10,000,000x − 10,000x = 1,523,987.987 − 1,523.987
Continue the deduction military operation to clear the repetition decimal fraction:

9,990,000x = 1,523,987 − 1,523

9,990,000x = 1,522,464
Divide both sides by 9,990,000 to represent x as a divide

x = 1522464 / 9990000

Fractions in sneak mathematics [edit]

In increase to beingness of great concrete importance, fractions are also studied aside mathematicians, WHO check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an sequential pair $(a,b)$ of integers $a$ and $b\neq 0,$ for which the operations addition, subtraction, propagation, and partitioning are delimited as follows:^[24]

(a,b)+(c,d)=(ad+bc,Bachelor of Divinity)\,

(a,b)-(c,d)=(ad-bc,bd)\,

(a,b)\cdot (c,d)=(ac,Bachelor of Divinity)

(a,b)\div (c,d)=(adver,bc)\quadruplet ({\text{with, in addition, }}c\neq 0)

These definitions agree in every causa with the definitions given in a higher place; only the annotation is different. Alternatively, instead of defining subtraction and division as operations, the "inverse" fractions with regard to add-on and propagation might follow defined as:

{\set out{aligned}-(a,b)&=(-a,b)&&{\text{additive inverse fractions,}}\\&&&ere;{\text{with }}(0,b){\text{ as additive unities, and}}\\(a,b)^{-1}&adenosine monophosphate;=(b,a)&ere;&{\text{reciprocal fractions, for }}a\neq 0,\\&ere;&&adenosine monophosphate;{\text{with }}(b,b){\text{ as increasing unities}}.\end{aligned}}

Furthermore, the relation, specified as

(a,b)\sim (c,d)\quad \iff \quad ad=bc,

is an equivalence relation of fractions. To each one divide from one equivalence class may be considered as a example for the whole class, and each whole class May atomic number 4 considered as uncomparable abstract fraction. This compare is preserved by the in a higher place outlined operations, i.e., the results of operating on fractions are independent of the selection of representatives from their equivalence class. Formally, for addition of fractions

(a,b)\sim (a',b')\space

and

\quad (c,d)\sim (c',d')\quad

imply

((a,b)+(c,d))\sim ((a',b')+(c',d'))

and likewise for the other trading operations.

In the case of fractions of integers, the fractions a / b with $a$ and $b$ coprime and $b > 0$ are often taken as unambiguously determined representatives for their equivalent fractions, which are thoughtful to be the same rational number. This mode the fractions of integers make up the field of the rational numbers.

More generally, a and b may be elements of any integral domain R, in which example a divide is an element of the field of fractions of R. For example, polynomials in one indeterminate, with coefficients from extraordinary integral domain D, are themselves an integral area, call information technology P. And so for a and b elements of P, the generated field of fractions is the field of rational fractions (alias the subject of rational functions).

Algebraic fractions [cut]

An algebraic fraction is the indicated quotient of two pure mathematics expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zip. Two examples of algebraic fractions are ${\frac {3x}{x^{2}+2x-3}}$ and ${\frac {\sqrt {x+2}}{x^{2}-3}}$ . Pure mathematics fractions are subject to the Sami battleground properties atomic number 3 arithmetical fractions.

If the numerator and the denominator are polynomials, every bit in ${\frac {3x}{x^{2}+2x-3}}$ , the algebraic fraction is titled a rational fraction (operating theater rational expression). An irrational fraction is one that is not rational, as, for example, one that contains the adaptable under a halfway exponent or stem, as in ${\frac {\sqrt {x+2}}{x^{2}-3}}$ .

The language used to distinguish algebraic fractions is similar to that used for quotidian fractions. For instance, an algebraic fraction is in lowest footing if the only factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, much as ${\frac {1+{\tfrac {1}{x}}}{1-{\tfrac {1}{x}}}}$ , is called a complex fraction.

The field of rational numbers is the branch of knowledg of fractions of the integers, while the integers themselves are not a field but rather an integral domain. Similarly, the sane fractions with coefficients in a field form the field of fractions of polynomials with coefficient in this field. Considering the sensible fractions with real coefficients, radical expressions representing numbers, such as $\textstyle {\sqrt {2}}/2,$ are too rational fractions, as are a otherworldly Numbers such as ${\textstyle \pi /2,}$ since all of ${\sqrt {2}},\private detective ,$ and $2$ are real Book of Numbers, and thus well-advised as coefficients. These same numbers, however, are not coherent fractions with integer coefficients.

The term inclined fraction is used when decomposing rational fractions into sums of simpler fractions. For lesson, the rational divide ${\frac {2x}{x^{2}-1}}$ can be decomposed as the sum of two fractions: ${\frac {1}{x+1}}+{\frac {1}{x-1}}.$ This is useable for the figuring of antiderivatives of rational functions (get word partial fraction decomposition for more).

Ultra expressions [edit]

A fraction may also moderate radicals in the numerator or the denominator. If the denominator contains radicals, it can be helpful to rationalize it (equate Simplified form of a radical expression), especially if further operations, such as adding or comparing that divide to another, are to be carried out. It is also more convenient if sectionalization is to be done manually. When the denominator is a monomial square theme, it can be rationalized by multiplying both the top and the bottom of the divide by the denominator:

{\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}

The process of rationalization of binomial denominators involves multiplying the top and the bottom of a fraction by the conjugate of the denominator thusly that the denominator becomes a rational. E.g.:

{\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}}

{\frac {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}

Even if this process results in the numerator being blind, like in the examples above, the process Crataegus oxycantha unmoving alleviate subsequent manipulations by reducing the number of irrationals one has to form with in the denominator.

Typographic variations [blue-pencil]

In computer displays and typography, simple fractions are sometimes printed as a single character, e.g. ½ (one half). Envision the clause on Number Forms for entropy happening doing this in Unicode.

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

special fractions: fractions that are bestowed as a single character with a unfair bar, with roughly the same height and breadth as other characters in the text. Generally used for simple fractions, much as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are small, legibility stern be an subject, especially for small-sized fonts. These are non utilized in modern mathematical notation, but in other contexts.
case fractions: similar to special fractions, these are rendered as a unvarying typographical character, just with a naiant bar, hence making them upright. An exercise would personify ${\tfrac {1}{2}}$ , but rendered with the same height As other characters. Any sources include all rendering of fractions as slip fractions if they take merely indefinite typographical space, regardless of the direction of the bar.^[26]
shilling or solidus fractions: 1/2, so called because this annotation was used for pre-decimal British up-to-dateness (£sd), as in 2/6 for a half crown, meaning ii shillings and six pence. While the notation "two shillings and sextuplet pence" did not represent a fraction, the forward slash is at once used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also misused for fractions inside fractions (complex fractions) or inside exponents to increase legibility. Fractions shorthand this way, also known as piece fractions,^[27] are written all on indefinite typographical line, but lead 3 or more typographical spaces.
assembled-up fractions: ${\frac {1}{2}}$ . This annotation uses cardinal or more lines of ordinary text, and results in a variation in spatial arrangement between lines when included within other text. While large and legible, these buttocks be turbulent, particularly for simple fractions or within involved fractions.

History [edit]

The earliest fractions were reciprocals of integers: ancient symbols representing one share of two, i break u of three, one divide of four, and so on.^[28] The Egyptians victimized African country fractions c. 1000 B.C.. About 4000 years ago, Egyptians segmental with fractions using slightly incompatible methods. They exploited least common multiples with unit fractions. Their methods gave the comparable resolve as modern methods.^[29] The Egyptians also had a different notation for dyadic fractions in the Akhmim Awkward Tablet and respective Rhind Possible Papyrus problems.

The Greeks used whole fractions and (later) continued fractions. Followers of the Greek philosopher Pythagoras (c. 530 BC) discovered that the square origin of two cannot be expressed as a divide of integers. (This is usually though probably erroneously ascribed to Hippasus of Metapontum, who is aforesaid to have been executed for revealing this fact.) In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains mold on the possibility of numbers, arithmetical trading operations, and trading operations with fractions.

A modern expression of fractions renowned as bhinnarasi seems to have originated in India in the act upon of Aryabhatta (c. AD 500),^{[ citation needed ]} Brahmagupta (c. 628), and Bhaskara (c. 1150).^[30] Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators (cheda), but without a BAR between them.^[30] In Sanskrit literature, fractions were always stated as an addition to or subtraction from an whole number.^{[ acknowledgment needed ]} The integer was holographic on one line and the divide in its two parts on the next line. If the fraction was marked by a small rotary ⟨०⟩ operating theatre cross ⟨+⟩, IT is subtracted from the integer; if none such sign appears, it is understood to be added. E.g., Bhaskara I writes:^[31]

६ १ २

१ १ १_०

४ ५ ९

which is the equivalent of

6 1 2

1 1 −1

4 5 9

and would be statute in modern notation A 6 1 / 4 , 1 1 / 5 , and 2 − 1 / 9 (i.e., 1 8 / 9 ).

The horizontal fraction bar is first documented in the work of Al-Hassār (Florida. 1200),^[30] a Islamic mathematician from Tarboosh, Morocco, WHO specialized in Islamic inheritance jurisprudence. In his discussion helium writes, "... for example, if you are told to write three-fifths and a third of a fifth part, write frankincense, ${\frac {3\musculus quadriceps femoris 1}{5\quad 3}}$ ."^[32] The same fractional notation—with the fraction given ahead the whole number^[30]—appears soon aft in the work of Leonardo da Vinci Fibonacci in the 13th hundred.^[33]

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

"The introduction of denary fractions as a common procedure practice can follow dated back to the Flemish pamphlet De Thiende, publicized at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), then settled in the Northern Netherlands. It is typical that decimal fractions were used by the Chinese many centuries ahead Stevin and that the Farsi astronomer Alabama-Kāshī put-upon both decimal fraction and simple fraction fractions with great simpleness in his Key to pure mathematics (Samarkand, early 15th century)."^[35]

While the Asian nation mathematician Jamshīd al-Kāshī claimed to feature discovered decimal fractions himself in the 15th one C, J. Lennart Berggren notes that he was wrong, as decimal fractions were commencement victimised five centuries before him by the Baghdadi mathematician Abu'l-Hasan AL-Uqlidisi Eastern Samoa early as the 10th century.^[36] ^{[n 2]}

In full-dress education [edit]

Pedagogical tools [edit]

In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for fold OR cutting), pattern blocks, pie-attribute pieces, formative rectangles, grid paper, dot paper, geoboards, counters and figurer software.

Documents for teachers [edit]

Several states in the United States of America take up adopted learning trajectories from the Common Core State Standards Initiative'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 phone number utterable in the form $a$ / $b$ where $a$ is a whole number and $b$ is a positive whole number. (The word of honor fraction in these standards always refers to a non-negative number.)"^[38] The papers itself also refers to negative fractions.

Project also [edit]

Cross multiplication
0.999...
Multiple
FRACTRAN

Number systems

Complex

:\;\mathbb {C}

Literal

:\;\mathbb {R}

Rational

:\;\mathbb {Q}

Integer

:\;\mathbb {Z}

Innate

:\;\mathbb {N}

Zero: 0

One: 1

Prime Numbers

Composite Numbers

Negative integers

Fraction

Finite decimal
Dyadic (finite binary)
Repeating quantitative

Irrational

Algebraic

Nonnatural

Complex quantity

Notes [redact]

^ Some typographers such Eastern Samoa Bringhurst mistakenly distinguish the slash ⟨/⟩ as the virgule and the fraction slash ⟨⁄⟩ every bit the stroke,^[6] although in point of fact both are synonyms for the standard whip.^[7] ^[8]
^ While in that location is few disagreement among history of mathematics scholars American Samoa to the primacy of Camellia State-Uqlidisi's share, there is no call into question as to his prima contribution to the concept of denary fractions.^[37]

References [edit]

^ H. Wu, "The Mis-Education of Mathematics Teachers", Notices of the American Mathematical Society, Volume 58, Issue 03 (Borderland 2011), p. 374 Archived 2017-08-20 at the Wayback Machine
^ Schwartzman, Steven (1994). The Wrangle of Mathematics: An Etymological Dictionary of Mathematical Terms In use in English . Mathematical Association of America. ISBN978-0-88385-511-9.
^ "Fractions". www.mathsisfun.com . Retrieved 2020-08-27 .
^ ^a ^b Saint Ambrose, Gavin; et AL. (2006). The Fundamentals of Typography (2nd erectile dysfunction.). Lausanne: AVA Publishing. p. 74. ISBN978-2-940411-76-4. Archived from the original on 2016-03-04. Retrieved 2016-02-20 . .
^ Weisstein, Eric W. "Fraction". mathworld.wolfram.com . Retrieved 2020-08-27 .
^ Bringhurst, Robert (2002). "5.2.5: Use the Virgule with Words and Dates, the Solidus with Split-tier Fractions". The Elements of Typographic Style (3rd ed.). Item Roberts: Hartley & Marks. pp. 81–82. ISBN978-0-88179-206-5.
^ "virgule, n.". Oxford English Lexicon (1st ed.). Oxford: Oxford Pressur. 1917.
^ "solidus, n.¹ ". Oxford English Dictionary (1st erectile dysfunction.). Oxford: Oxford Agitat. 1913.
^ Weisstein, Eric W. "Common Fraction". MathWorld.
^ ^a ^b David E. Kate Smith (1 June 1958). History of Mathematics. Courier Corporation. p. 219. ISBN978-0-486-20430-7.
^ "World Wide Words: Vulgar fractions". World Thick Words. Archived from the original on 2014-10-30. Retrieved 2014-10-30 .
^ Weisstein, Eric W. "Improper Divide". MathWorld.
^ Manual laborer Williams (19 November 2011). Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation. Springer spaniel Scientific discipline & Business Media. pp. 87–. ISBN978-0-85729-862-1.
^ Record, Robert (1654). Record's Arithmetick: Oregon, the Ground of Arts: Teaching the Perfect Work and Practise of Arithmetick ... Made by Mr. Robert Record ... Afterward Augmented away Mr. John Dee. And Since Enlarged with a Thirdly Part of Rules of Practise ... By Can Mellis. And Now Diligently Perused, Corrected ... and Enlarged ; with an Appendix of Figurative Numbers ... with Tables of Board and Timber Assess ... the First Calculated by R. C. But Rectified, and the Last mentioned ... Measured by Ro. Hartwell ... James Flesher, and are to be sold aside Edward Dod. pp. 266–.
^ Arthur Stanley Jefferson Laurel (31 March 2004). "Math Forum – Require Dr. Math: Tin Negative Fractions Besides Be Proper or Improper?". Archived from the original on 9 November 2014. Retrieved 2014-10-30 .
^ "Revolutionary England Compact Math Resources". Archived from the original on 2012-04-15. Retrieved 2011-12-31 .
^ Greer, A. (1986). New comprehensive mathematics for 'O' level (2nd erectile dysfunction., reprinted ed.). Cheltenham: Thornes. p. 5. ISBN978-0-85950-159-0. Archived from the underived on 2019-01-19. Retrieved 2014-07-29 .
^ ^a ^b Trotting horse, James (1853). A complete organisation of arithmetic. p. 65.
^ ^a ^b Barlow, Saint Peter (1814). A new mathematical and philosophical dictionary.
^ https://www.collinsdictionary.com/lexicon/english/complex-fraction Archived 2017-12-01 at the Wayback Machine et al.
^ "Complex fraction definition and pregnant". Collins English Lexicon. 2018-03-09. Archived from the original along 2017-12-01. Retrieved 2018-03-13 .
^ "Compound Fractions". Sosmath.com. 1996-02-05. Archived from the underivative on 2018-03-14. Retrieved 2018-03-13 .
^ Schoenborn, Barry; Simkins, Omar 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.
^ "Fraction". Encyclopedia of Maths. 2012-04-06. Archived from the seminal on 2014-10-21. Retrieved 2012-08-15 .
^ Galen, Leslie Blackwell (March 2004). "Putting Fractions in Their Commit" (PDF). American Mathematical Each month. 111 (3): 238–242. Interior:10.2307/4145131. JSTOR 4145131. Archived (PDF) from the underivative on 2011-07-13. Retrieved 2010-01-27 .
^ "built divide". allbusiness.com glossary. Archived from the original on 2013-05-26. Retrieved 2013-06-18 .
^ "piece divide". allbusiness.com glossary. Archived from the original along 2013-05-21. Retrieved 2013-06-18 .
^ Eves, Leslie Howard Stainer (1990). An introduction to the chronicle of mathematics (6th male erecticle dysfunction.). Philadelphia: Saunders College Pub. ISBN978-0-03-029558-4.
^ Milo Gardner (December 19, 2005). "Maths Account". Archived from the original on December 19, 2005. Retrieved 2006-01-18 . Run into for examples and an explanation.
^ ^a ^b ^c ^d Miller, Jeff (22 December 2014). "Earliest Uses of Single Possible Symbols". Archived from the original on 20 February 2016. Retrieved 15 February 2016.
^ Filliozat, Capital of South Dakota-Sylvain (2004). "Antediluvian Sanskrit Math: An Oral Tradition and a Printed Literature". In Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et atomic number 13. (EDS.). Account of Science, History of Schoolbook. Boston Series in the Philosophy of Science. 238. Dordrecht: Springer Netherlands. p. 152. doi:10.1007/1-4020-2321-9_7. ISBN978-1-4020-2320-0.
^ Cajori, Florian (1928). A History of Mathematical Notations. 1. La Salle, Illinois: Open Court Publication Company. p. 269. Archived from the pilot on 2014-04-14. Retrieved 2017-08-30 .
^ Cajori (1928), p. 89
^ A Source Book in Mathematics 1200–1800. Jersey: Princeton University Press. 1986. ISBN978-0-691-02397-7.
^ Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī. Wiesbaden: Steiner. 1951.
^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Maths of Egypt, Mesopotamia, China, Republic of India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN978-0-691-11485-9.
^ "MacTutor's al-Uqlidisi biography" Archived 2011-11-15 at the Wayback Machine. Retrieved 2011-11-22.
^ "Common Core State Standards for Mathematics" (PDF). Common Core State Standards Maiden. 2010. p. 85. Archived (PDF) from the original on 2013-10-19. Retrieved 2013-10-10 .

External links [edit]

Wikimedia Commons has media related to Fractions.

Look up denominator in Wiktionary, the justify dictionary.

Consult numerator in Wiktionary, the free dictionary.

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

what is 3.6 as a fraction in simplest form

Source: https://en.wikipedia.org/wiki/Fraction