6 Percent In Decimal Form

Ratio of ii numbers

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

A fraction (from Latin: fractus, "broken") represents a office of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for instance, one-half, eight-fifths, three-quarters. A mutual, vulgar, or elementary fraction (examples: ${\displaystyle {\tfrac {1}{two}}}$ and ${\displaystyle {\tfrac {17}{iii}}}$ ) consists of a numerator, displayed above a line (or earlier a slash like 1⁄2 ), and a non-zero denominator, displayed beneath (or subsequently) that line. Numerators and denominators are also used in fractions that are not mutual, 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 upwards a unit or a whole. The denominator cannot be zero, because zero parts can never make upwards a whole. For instance, in the fraction 3 / 4 , the numerator iii indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that iv parts make upwardly a whole. The picture to the right illustrates 3 / four of a cake.

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

Other uses for fractions are to stand for ratios and sectionalisation.^[1] Thus the fraction three / 4 can also exist used to represent the ratio iii:four (the ratio of the part to the whole), and the sectionalization 3 ÷ 4 (three divided by four). The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that division by nil is undefined.

We can as well write negative fractions, which correspond the opposite of a positive fraction. For example, if one / 2 represents a half-dollar profit, then − 1 / two represents a one-half-dollar loss. Because of the rules of division of signed numbers (which states in function that negative divided past positive is negative), − 1 / 2 , −1 / ii and one / −2 all represent the same fraction – negative half. And because a negative divided past a negative produces a positive, −i / −ii represents positive one-half.

In mathematics the prepare of all numbers that can exist expressed in the grade a / b , where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for caliber. A number is a rational number precisely when information technology can be written in that grade (i.due east., as a common fraction). Nevertheless, the give-and-take fraction tin also be used to describe 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 ${\textstyle {\frac {\sqrt {ii}}{2}}}$ (see foursquare root of ii) and π / four (see 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 type or diversity of the parts is the denominator (from Latin: dēnōminātor, "matter that names or designates").^{[ii] [3]} As an instance, the fraction 8 / five amounts to eight parts, each of which is of the type named "fifth". In terms 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 ane / 3 ), oblique (as in 2/v), or diagonal (as in four⁄9 ).^[iv] These marks are respectively known every bit the horizontal bar; the virgule, slash (U.s.), or stroke (Britain); and the fraction bar, solidus,^[5] or fraction slash.^{[northward 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 foursquare, or a wider em square.^[4] In traditional typefounding, a piece of type begetting a complete fraction (e.grand. 1 / ii ) was known as a "example fraction", while those representing only part of fraction were called "slice fractions".

The denominators of English fractions are by and large expressed as ordinal numbers, in the plural if the numerator is non ane. (For example, 2 / 5 and 3 / 5 are both read equally a number of "fifths".) Exceptions include the denominator 2, which is ever read "one-half" or "halves", the denominator 4, which may be alternatively expressed as "quarter"/"quarters" or as "4th"/"fourths", and the denominator 100, which may be alternatively expressed as "hundredth"/"hundredths" or "pct".

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

The unabridged fraction may be expressed as a single limerick, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction 2 / 5 and "two fifths" is the same fraction understood as 2 instances of i / 5 .) Fractions should ever exist hyphenated when used equally adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed equally a key number. (For example, three / 1 may as well exist expressed as "three over 1".) The term "over" is used even in the case of solidus fractions, where the numbers are placed left and right of a slash mark. (For example, 1/2 may be read "half", "one half", or "one over two".) Fractions with large denominators that are not powers of ten are often rendered in this mode (due east.g., i / 117 as "ane over 1 hundred seventeen"), while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., six / 1000000 as "six-millionths", "half-dozen millionths", or "vi one-millionths").

Forms of fractions [edit]

Simple, common, or vulgar fractions [edit]

A elementary fraction (also known every bit 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 ${\displaystyle {\tfrac {i}{2}}}$ , ${\displaystyle -{\tfrac {8}{v}}}$ , ${\displaystyle {\tfrac {-8}{5}}}$ , and ${\displaystyle {\tfrac {8}{-5}}}$ . The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.^[10]

Mutual fractions can exist positive or negative, and they tin be proper or improper (see below). Compound fractions, circuitous fractions, mixed numerals, and decimals (see below) are not common fractions; though, unless irrational, they can be evaluated to a mutual fraction.

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

Proper and improper fractions [edit]

Common fractions can exist classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.^{[xi] [12]} The concept of an "improper fraction" is a late evolution, with the terminology deriving from the fact that "fraction" means "a slice", so a proper fraction must exist less than 1.^[10] This was explained in the 17th century textbook The Footing of Arts.^{[thirteen] [14]}

In general, a mutual fraction is said to exist a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.^{[fifteen] [16]} Information technology is said to be an improper fraction, or sometimes top-heavy fraction,^[17] if the absolute value of the fraction is greater than or equal to ane. Examples of proper fractions are two/iii, −3/4, and 4/nine, whereas examples of improper fractions are ix/four, −4/3, and 3/three.

Reciprocals and the "invisible denominator" [edit]

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of ${\displaystyle {\tfrac {3}{7}}}$ , 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 example, ${\displaystyle {\tfrac {7}{7}}}$ ), its value is 1, and the fraction therefore is improper. Its reciprocal is identical and hence also equal to ane and improper.

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

Ratios [edit]

A ratio is a relationship between two or more numbers that tin exist sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as "group 1 to grouping 2 ... to group n". For example, if a car lot had 12 vehicles, of which

• two are white,
• 6 are cherry-red, and
• 4 are yellow,

and so the ratio of ruby-red to white to yellow cars is 6 to two to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as four:2 or 2:one.

A ratio is oftentimes converted to a fraction when information technology is expressed equally a ratio to the whole. In the above example, the ratio of yellowish 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 four / 12 of the cars or 1 / 3 of the cars in the lot are xanthous. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.

Decimal fractions and percentages [edit]

A decimal fraction is a fraction whose denominator is not given explicitly, merely is understood to exist an integer power of ten. Decimal fractions are commonly expressed using decimal annotation in which the unsaid denominator is determined by the number of digits to the correct of a decimal separator, the appearance of which (e.one thousand., a period, an interpunct (·), a comma) depends on the locale (for examples, run into decimal separator). Thus, for 0.75 the numerator is 75 and the implied denominator is x to the 2nd ability, namely, 100, because in that location are two digits to the right of the decimal separator. In decimal numbers greater than ane (such every bit iii.75), the fractional office of the number is expressed past the digits to the correct of the decimal (with a value of 0.75 in this instance). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, ${\displaystyle 3{\tfrac {75}{100}}}$ .

Decimal fractions tin besides be expressed using scientific note 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 betoken 7 places to the left.

Decimal fractions with infinitely many digits to the correct of the decimal separator represent an infinite series. For example, ane / iii = 0.333... represents the infinite series three/10 + 3/100 + 3/1000 + ....

Another kind of fraction is the percent (from Latin: percent, significant "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same fashion, due east.m. 311% equals 311/100, and −27% equals −27/100.

The related concept of permille or parts per thousand (ppt) has an implied denominator of g, while the more than general parts-per notation, every bit in 75 parts per one thousand thousand (ppm), means that the proportion is 75/ane,000,000.

Whether common fractions or decimal fractions are used is often a matter of gustatory modality and context. Common fractions are used virtually oftentimes when the denominator is relatively minor. By mental calculation, it is easier to multiply xvi past 3/16 than to do the same calculation using the fraction'southward decimal equivalent (0.1875). And information technology is more accurate to multiply 15 past i/3, for instance, than it is to multiply fifteen by any decimal approximation of 1 third. Monetary values are normally expressed as decimal fractions with denominator 100, i.e., with two decimals, for example $3.75. However, as noted in a higher place, in pre-decimal British currency, shillings and pence were often given the form (only not the significant) of a fraction, as, for case, "three/vi" (read "three and six") meaning 3 shillings and half-dozen pence, and having no relationship to the fraction 3/6.

Mixed numbers [edit]

A mixed numeral (also chosen 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: ${\displaystyle 2{\tfrac {3}{16}}}$ inches, for example. Scientific measurements most invariably use decimal notation rather than mixed numbers. The sum can be implied without the utilise of a visible operator such as the advisable "+". For example, in referring to 2 unabridged cakes and iii quarters of another cake, the numerals denoting the integer part and the fractional part of the cakes tin can be written next to each other equally ${\displaystyle two{\tfrac {3}{4}}}$ instead of the unambiguous notation ${\displaystyle 2+{\tfrac {3}{four}}.}$ Negative mixed numerals, as in ${\displaystyle -two{\tfrac {three}{4}}}$ , are treated like ${\displaystyle \scriptstyle -\left(2+{\frac {3}{4}}\right).}$ Whatever 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 annotation in algebra where side by side symbols, without an explicit infix operator, denote a product. In the expression ${\displaystyle 2x}$ , the "understood" operation is multiplication. If 10 is replaced past, for example, the fraction ${\displaystyle {\tfrac {three}{four}}}$ , the "understood" multiplication needs to be replaced past explicit multiplication, to avoid the advent 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 ${\displaystyle \quad ii\times {\frac {b}{c}},\quad }$ or ${\displaystyle \quad two\left({\frac {b}{c}}\right),\;\ldots }$

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

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

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 aboriginal Egyptians expressed all fractions except ${\displaystyle {\tfrac {i}{two}}}$ , ${\displaystyle {\tfrac {2}{three}}}$ and ${\displaystyle {\tfrac {iii}{4}}}$ in this way. Every positive rational number can exist expanded as an Egyptian fraction. For example, ${\displaystyle {\tfrac {5}{7}}}$ can be written as ${\displaystyle {\tfrac {i}{ii}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.}$ Any positive rational number can be written equally a sum of unit fractions in infinitely many ways. Two ways to write ${\displaystyle {\tfrac {13}{17}}}$ are ${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}$ and ${\displaystyle {\tfrac {one}{iii}}+{\tfrac {1}{4}}+{\tfrac {1}{half dozen}}+{\tfrac {1}{68}}}$ .

Circuitous and compound fractions [edit]

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

${\displaystyle {\frac {\tfrac {1}{ii}}{\tfrac {1}{three}}}={\tfrac {1}{ii}}\times {\tfrac {iii}{1}}={\tfrac {3}{2}}}$

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

${\displaystyle {\frac {\tfrac {iii}{2}}{5}}={\tfrac {3}{2}}\times {\tfrac {1}{5}}={\tfrac {iii}{x}}}$

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

If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, considering of multiple possible interpretations, eastward.g. every bit

${\displaystyle v/(10/(xx/twoscore))={\frac {5}{10/{\tfrac {20}{twoscore}}}}={\frac {i}{4}}\quad }$ or as ${\displaystyle \quad (v/ten)/(20/40)={\frac {\tfrac {5}{x}}{\tfrac {xx}{40}}}=1}$

A compound fraction is a fraction of a fraction, or whatsoever number of fractions connected with the word of,^{[18] [19]} corresponding to multiplication of fractions. To reduce a chemical compound fraction to a simple fraction, just carry out the multiplication (run across the section on multiplication). For example, ${\displaystyle {\tfrac {3}{4}}}$ of ${\displaystyle {\tfrac {five}{7}}}$ is a chemical compound fraction, corresponding to ${\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}$ . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other. (For example, the compound fraction ${\displaystyle {\tfrac {3}{4}}\times {\tfrac {five}{vii}}}$ is equivalent to the circuitous fraction ${\displaystyle {\tfrac {3/four}{seven/five}}}$ .)

Nevertheless, "circuitous fraction" and "compound fraction" may both be considered outdated^[20] and now used in no well-defined mode, partly fifty-fifty 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 meaning of "consisting of parts".

Arithmetic with fractions [edit]

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the dominion confronting partitioning by zip.

Equivalent fractions [edit]

Multiplying the numerator and denominator of a fraction by the same (non-nil) number results in a fraction that is equivalent to the original fraction. This is true considering for any not-zilch number ${\displaystyle n}$ , the fraction ${\displaystyle {\tfrac {north}{due north}}}$ equals ${\displaystyle 1}$ . Therefore, multiplying by ${\displaystyle {\tfrac {n}{n}}}$ is the same as multiplying by 1, and any number multiplied past ane has the same value as the original number. By mode of an example, kickoff with the fraction ${\displaystyle {\tfrac {i}{2}}}$ . When the numerator and denominator are both multiplied by 2, the result is ${\displaystyle {\tfrac {ii}{four}}}$ , which has the same value (0.v) as ${\displaystyle {\tfrac {ane}{ii}}}$ . To picture show this visually, imagine cutting a cake into iv pieces; 2 of the pieces together ( ${\displaystyle {\tfrac {2}{4}}}$ ) make upwardly one-half the cake ( ${\displaystyle {\tfrac {ane}{2}}}$ ).

Simplifying (reducing) fractions [edit]

Dividing the numerator and denominator of a fraction by the aforementioned non-zip number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction ${\displaystyle {\tfrac {a}{b}}}$ are divisible by ${\displaystyle c,}$ then they tin can be written as ${\displaystyle a=cd}$ and ${\displaystyle b=ce,}$ and the fraction becomes ${\displaystyle {\tfrac {cd}{ce}}}$ , which tin be reduced by dividing both the numerator and denominator by ${\displaystyle c}$ to give the reduced fraction ${\displaystyle {\tfrac {d}{due east}}.}$

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

If the numerator and the denominator exercise non share any factor greater than 1, the fraction is already reduced to its everyman terms, and information technology is said to be irreducible, reduced, or in simplest terms. For instance, ${\displaystyle {\tfrac {three}{9}}}$ is not in lowest terms because both 3 and 9 can exist exactly divided by 3. In contrast, ${\displaystyle {\tfrac {3}{8}}}$ is in everyman terms—the only positive integer that goes into both iii and 8 evenly is ane.

Using these rules, we tin can show that ${\displaystyle {\tfrac {5}{10}}={\tfrac {1}{two}}={\tfrac {x}{20}}={\tfrac {50}{100}}}$ , for case.

As another example, since the greatest common divisor of 63 and 462 is 21, the fraction ${\displaystyle {\tfrac {63}{462}}}$ tin exist reduced to lowest terms by dividing the numerator and denominator by 21:

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

The Euclidean algorithm gives a method for finding the greatest common divisor of whatever two integers.

Comparing fractions [edit]

Comparing fractions with the same positive denominator yields the same result equally comparison the numerators:

${\displaystyle {\tfrac {3}{4}}>{\tfrac {ii}{4}}}$ because 3 > 2, and the equal denominators ${\displaystyle 4}$ are positive.

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

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

If two positive fractions have the same numerator, then the fraction with the 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 2 positive fractions have the same numerator, they represent the same number of parts, just in the fraction with the smaller denominator, the parts are larger.

One fashion to compare fractions with different numerators and denominators is to find 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 tin can be compared. It is not necessary to make up one's mind the value of the mutual denominator to compare fractions – one tin can but compare advertisement and bc, without evaluating bd, due east.thousand., comparing ${\displaystyle {\tfrac {2}{3}}}$  ? ${\displaystyle {\tfrac {1}{ii}}}$ gives ${\displaystyle {\tfrac {four}{six}}>{\tfrac {three}{6}}}$ .

For the more laborious question ${\displaystyle {\tfrac {5}{18}}}$  ? ${\displaystyle {\tfrac {four}{17}},}$ multiply top and bottom of each fraction by the denominator of the other fraction, to go a mutual denominator, yielding ${\displaystyle {\tfrac {v\times 17}{18\times 17}}}$  ? ${\displaystyle {\tfrac {xviii\times four}{18\times 17}}}$ . Information technology is not necessary to summate ${\displaystyle 18\times 17}$ – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is ${\displaystyle {\tfrac {5}{18}}>{\tfrac {4}{17}}}$ .

Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than aught, it follows that any negative fraction is less than any positive fraction. This allows, together with the to a higher place rules, to compare all possible fractions.

Addition [edit]

The first rule of improver is that just like quantities can be added; for example, diverse quantities of quarters. Different quantities, such every bit adding thirds to quarters, must first be converted to similar quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are v quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

${\displaystyle {\tfrac {2}{iv}}+{\tfrac {three}{4}}={\tfrac {5}{4}}=1{\tfrac {i}{4}}}$ .

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

Adding different quantities [edit]

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. Information technology is piece of cake to piece of work out the called type of fraction to convert to; simply multiply together the ii 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:

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

Consider adding the post-obit two quantities:

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

Beginning, convert ${\displaystyle {\tfrac {iii}{5}}}$ into fifteenths by multiplying both the numerator and denominator by three: ${\displaystyle {\tfrac {three}{5}}\times {\tfrac {3}{3}}={\tfrac {nine}{15}}}$ . Since ${\displaystyle {\tfrac {3}{3}}}$ equals ane, multiplication by ${\displaystyle {\tfrac {3}{3}}}$ does not change the value of the fraction.

Second, convert ${\displaystyle {\tfrac {2}{3}}}$ into fifteenths by multiplying both the numerator and denominator by v: ${\displaystyle {\tfrac {2}{iii}}\times {\tfrac {5}{5}}={\tfrac {10}{15}}}$ .

Now it can be seen that:

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

is equivalent to:

${\displaystyle {\frac {ix}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {iv}{xv}}}$

This method can exist expressed algebraically:

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

This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the unmarried denominators contain a common factor, a smaller denominator than the product of these can exist used. For instance, when adding ${\displaystyle {\tfrac {three}{iv}}}$ and ${\displaystyle {\tfrac {5}{6}}}$ the single denominators have a common gene ${\displaystyle two,}$ and therefore, instead of the denominator 24 (four × half dozen), the halved denominator 12 may be used, non merely reducing the denominator in the outcome, but too the factors in the numerator.

${\displaystyle {\brainstorm{aligned}{\frac {3}{4}}+{\frac {5}{6}}&={\frac {3\cdot six}{iv\cdot vi}}+{\frac {4\cdot five}{iv\cdot 6}}={\frac {xviii}{24}}+{\frac {xx}{24}}&={\frac {19}{12}}\\&={\frac {3\cdot 3}{4\cdot three}}+{\frac {two\cdot five}{2\cdot half-dozen}}={\frac {9}{12}}+{\frac {10}{12}}&={\frac {19}{12}}\end{aligned}}}$

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

Subtraction [edit]

The procedure for subtracting fractions is, in essence, the same every bit that of adding them: discover a common denominator, and alter each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will take that denominator, and its numerator volition exist the result of subtracting the numerators of the original fractions. For instance,

${\displaystyle {\tfrac {2}{iii}}-{\tfrac {1}{two}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {ane}{6}}}$

Multiplication [edit]

Multiplying a fraction past some other fraction [edit]

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

${\displaystyle {\frac {two}{3}}\times {\frac {three}{4}}={\frac {vi}{12}}}$

To explain the procedure, consider 1 3rd of i quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make upwardly a whole, twelve of these small-scale, equal slices make upwards a whole. Therefore, a third of a quarter is a twelfth. At present consider the numerators. The first fraction, two thirds, is twice equally large as i third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, iii quarters, is three times as large equally ane quarter, so 2 thirds of 3 quarters is three times equally large as two 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 example:

${\displaystyle {\frac {2}{3}}\times {\frac {3}{4}}={\frac {{\cancel {2}}^{~i}}{{\cancel {3}}^{~i}}}\times {\frac {{\abolish {3}}^{~1}}{{\cancel {4}}^{~2}}}={\frac {1}{1}}\times {\frac {1}{ii}}={\frac {1}{2}}}$

A two is a mutual factor in both the numerator of the left fraction and the denominator of the correct and is divided out of both. Iii is a mutual factor of the left denominator and right numerator and is divided out of both.

Multiplying a fraction by a whole number [edit]

Since a whole number can be rewritten every bit itself divided by 1, normal fraction multiplication rules can still use.

${\displaystyle 6\times {\tfrac {3}{iv}}={\tfrac {half dozen}{1}}\times {\tfrac {3}{four}}={\tfrac {18}{4}}}$

This method works considering the fraction 6/1 means vi equal parts, each i of which is a whole.

Multiplying mixed numbers [edit]

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

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

In other words, ${\displaystyle 2{\tfrac {3}{4}}}$ is the same as ${\displaystyle {\tfrac {8}{four}}+{\tfrac {3}{four}}}$ , making 11 quarters in total (because 2 cakes, each dissever into quarters makes viii quarters full) and 33 quarters is ${\displaystyle eight{\tfrac {one}{iv}}}$ , since eight cakes, each made of quarters, is 32 quarters in total.

Sectionalization [edit]

To divide a fraction by a whole number, you lot may either divide the numerator past the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, ${\displaystyle {\tfrac {ten}{3}}\div v}$ equals ${\displaystyle {\tfrac {2}{3}}}$ and also equals ${\displaystyle {\tfrac {10}{3\cdot 5}}={\tfrac {10}{15}}}$ , which reduces to ${\displaystyle {\tfrac {2}{3}}}$ . To divide a number past a fraction, multiply that number by the reciprocal of that fraction. Thus, ${\displaystyle {\tfrac {i}{2}}\div {\tfrac {3}{iv}}={\tfrac {1}{ii}}\times {\tfrac {4}{3}}={\tfrac {1\cdot four}{2\cdot 3}}={\tfrac {2}{3}}}$ .

Converting between decimals and fractions [edit]

To alter a mutual fraction to a decimal, exercise a long sectionalisation of the decimal representations of the numerator past the denominator (this is idiomatically as well phrased every bit "divide the denominator into the numerator"), and round the reply to the desired accuracy. For case, to change ane / four to a decimal, divide 1.00 past 4 (" 4 into 1.00"), to obtain 0.25. To modify ane / 3 to a decimal, separate 1.000... past iii (" 3 into 1.000..."), and terminate when the desired accuracy is obtained, east.1000., at 4 decimals with 0.3333. The fraction 1 / 4 can be written exactly with two decimal digits, while the fraction i / three 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 one followed by as many zeroes as at that place are digits to the right of the decimal indicate, and write in the numerator all the digits of the original decimal, but 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 convert repeating decimals into fractions.

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

0.v = v/9

0.62 = 62/99

0.264 = 264/999

0.6291 = 6291/9999

If leading zeros precede the pattern, the nines are suffixed past the same number of abaft zeros:

0.0v = v/90

0.000392 = 392/999000

0.0012 = 12/9900

If a non-repeating set of decimals precede the pattern (such as 0.1523987), one may write the number as the sum of the non-repeating and repeating parts, respectively:

0.1523 + 0.0000987

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

1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra tin can be used, such as below:

1. Let x = the repeating decimal:

   x = 0.1523987

2. Multiply both sides past the power of ten only great enough (in this example 10⁴) to move the decimal betoken just before the repeating role of the decimal number:

   10,00010 = one,523.987

3. Multiply both sides by the power of 10 (in this instance 10³) that is the same as the number of places that echo:

   10,000,000x = ane,523,987.987

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

   10,000,000x − 10,000x = i,523,987.987 − 1,523.987

5. Continue the subtraction performance to clear the repeating decimal:

   9,990,000x = ane,523,987 − one,523

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

6. Divide both sides by 9,990,000 to represent x as a fraction

   ten = 1522464 / 9990000

Fractions in abstract mathematics [edit]

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

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

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

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

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

These definitions agree in every case with the definitions given above; merely the annotation is different. Alternatively, instead of defining subtraction and sectionalization as operations, the "inverse" fractions with respect to improver and multiplication might be divers equally:

${\displaystyle {\begin{aligned}-(a,b)&=(-a,b)&&{\text{additive changed fractions,}}\\&&&{\text{with }}(0,b){\text{ every bit additive unities, and}}\\(a,b)^{-1}&=(b,a)&&{\text{multiplicative changed fractions, for }}a\neq 0,\\&&&{\text{with }}(b,b){\text{ as multiplicative unities}}.\end{aligned}}}$

Furthermore, the relation, specified as

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

is an equivalence relation of fractions. Each fraction from one equivalence form may exist considered equally a representative for the whole class, and each whole class may exist considered as one abstract fraction. This equivalence is preserved by the to a higher place defined operations, i.east., the results of operating on fractions are contained of the option of representatives from their equivalence class. Formally, for improver 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 case of fractions of integers, the fractions a / b with a and b coprime and b > 0 are oftentimes taken as uniquely determined representatives for their equivalent fractions, which are considered 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 case a fraction is an element of the field of fractions of R. For instance, polynomials in ane indeterminate, with coefficients from some integral domain D, are themselves an integral domain, telephone call it P. And then for a and b elements of P, the generated field of fractions is the field of rational fractions (also known every bit the field of rational functions).

Algebraic fractions [edit]

An algebraic fraction is the indicated quotient of two algebraic expressions. Every bit with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are ${\displaystyle {\frac {3x}{x^{2}+2x-3}}}$ and ${\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}}$ . Algebraic fractions are field of study to the same field properties as arithmetic fractions.

If the numerator and the denominator are polynomials, equally in ${\displaystyle {\frac {3x}{x^{ii}+2x-3}}}$ , the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is 1 that is not rational, every bit, for example, one that contains the variable under a fractional exponent or root, as in ${\displaystyle {\frac {\sqrt {x+2}}{ten^{2}-three}}}$ .

The terminology used to depict algebraic fractions is similar to that used for ordinary fractions. For case, an algebraic fraction is in lowest terms if the but factors common to the numerator and the denominator are 1 and −1. An algebraic fraction whose numerator or denominator, or both, contain a fraction, such as ${\displaystyle {\frac {one+{\tfrac {1}{x}}}{ane-{\tfrac {i}{ten}}}}}$ , 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 merely rather an integral domain. Similarly, the rational fractions with coefficients in a field form the field of fractions of polynomials with coefficient in that field. Considering the rational fractions with real coefficients, radical expressions representing numbers, such every bit ${\displaystyle \textstyle {\sqrt {2}}/2,}$ are also rational fractions, as are a transcendental numbers such as ${\textstyle \pi /two,}$ since all of ${\displaystyle {\sqrt {two}},\pi ,}$ and ${\displaystyle 2}$ are existent numbers, and thus considered equally coefficients. These same numbers, however, are non 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 ${\displaystyle {\frac {2x}{x^{ii}-1}}}$ can be decomposed every bit the sum of two fractions: ${\displaystyle {\frac {1}{10+ane}}+{\frac {1}{ten-1}}.}$ This is useful for the ciphering of antiderivatives of rational functions (see partial fraction decomposition for more than).

Radical expressions [edit]

A fraction may also contain radicals in the numerator or the denominator. If the denominator contains radicals, information technology can exist helpful to rationalize information technology (compare Simplified form of a radical expression), peculiarly if further operations, such equally calculation or comparison that fraction to another, are to be carried out. It is also more user-friendly if partition is to be done manually. When the denominator is a monomial square root, it tin be rationalized by multiplying both the height and the lesser of the fraction past the denominator:

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

The process of rationalization of binomial denominators involves multiplying the summit and the bottom of a fraction past the conjugate of the denominator so that the denominator becomes a rational number. For example:

${\displaystyle {\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-ii{\sqrt {five}}}}\cdot {\frac {3+2{\sqrt {five}}}{3+two{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {v}})^{ii}}}={\frac {3(iii+two{\sqrt {five}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{eleven}}}$

${\displaystyle {\frac {3}{3+2{\sqrt {five}}}}={\frac {3}{3+2{\sqrt {v}}}}\cdot {\frac {3-two{\sqrt {five}}}{iii-2{\sqrt {5}}}}={\frac {3(three-2{\sqrt {five}})}{{3}^{2}-(two{\sqrt {5}})^{ii}}}={\frac {three(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{xi}}}$

Even if this process results in the numerator being irrational, similar in the examples above, the process may still facilitate subsequent manipulations past reducing the number of irrationals one has to work with in the denominator.

Typographical variations [edit]

In computer displays and typography, simple fractions are sometimes printed as a single character, east.yard. ½ (1 one-half). See the article on Number Forms for information on doing this in Unicode.

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

• Special fractions: fractions that are presented as a single grapheme with a slanted bar, with roughly the same height and width equally 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 exist an result, especially for pocket-size-sized fonts. These are not used in modern mathematical notation, simply in other contexts.
• Case fractions: similar to special fractions, these are rendered as a unmarried typographical character, merely with a horizontal bar, thus making them upright. An case would exist ${\displaystyle {\tfrac {1}{ii}}}$ , simply rendered with the same height every bit other characters. Some sources include all rendering of fractions as case fractions if they take only 1 typographical infinite, regardless of the direction of the bar.^[26]
• Shilling or solidus fractions: 1/2, so chosen considering this notation was used for pre-decimal British currency (£sd), equally in "2/6" for a half crown, meaning ii shillings and half dozen pence. While the note "two shillings and six pence" did not represent a fraction, the frontwards slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. Information technology is also used for fractions within fractions (complex fractions) or within exponents to increase legibility. Fractions written this fashion, also known as piece fractions,^[27] are written all on i typographical line, simply take 3 or more typographical spaces.
• Built-up fractions: ${\displaystyle {\frac {1}{ii}}}$ . This note uses two or more than lines of ordinary text and results in a variation in spacing between lines when included inside other text. While large and legible, these can exist disruptive, especially for simple fractions or inside circuitous fractions.

History [edit]

The earliest fractions were reciprocals of integers: aboriginal symbols representing i part of ii, ane part of three, ane role of 4, and so on.^[28] The Egyptians used Egyptian fractions c.  thousand BC. About 4000 years ago, Egyptians divided with fractions using slightly unlike methods. They used least common multiples with unit fractions. Their methods gave the aforementioned answer every bit modern methods.^[29] The Egyptians as well had a dissimilar annotation for dyadic fractions in the Akhmim Wooden Tablet and several Rhind Mathematical Papyrus problems.

The Greeks used unit fractions and (later) connected fractions. Followers of the Greek philosopher Pythagoras (c.  530 BC) discovered that the square root of two cannot exist expressed equally 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 Republic of india wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, and operations with fractions.

A modern expression of fractions known every bit bhinnarasi seems to accept originated in India in the work of Aryabhatta (c.  Advert 500),^{[ commendation needed ]} Brahmagupta (c.  628), and Bhaskara (c.  1150).^[30] Their works form fractions by placing the numerators (Sanskrit: amsa) over the denominators ( cheda ), simply without a bar between them.^[xxx] In Sanskrit literature, fractions were e'er expressed equally an addition to or subtraction from an integer.^{[ commendation needed ]} The integer was written on one line and the fraction in its ii parts on the adjacent line. If the fraction was marked past a pocket-size circle ⟨०⟩ or cantankerous ⟨+⟩, it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, Bhaskara I writes:^[31]

६  १  २

१  १  १_०

४  ५  ९

which is the equivalent of

6  i  2

1  i  −one

four  5  9

and would be written in mod notation as 6 ane / 4 , i 1 / 5 , and 2 − 1 / ix (i.e., 1 8 / ix ).

The horizontal fraction bar is first attested in the piece of work of Al-Hassār (fl.  1200),^[30] a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence. In his word he writes: "for case, if you are told to write three-fifths and a third of a 5th, write thus, ${\displaystyle {\frac {3\quad 1}{5\quad 3}}}$ ". ^[32] The same fractional notation—with the fraction given before the integer^[xxx]—appears soon after 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 equally a common computational practice can be 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), then settled in the Northern Netherlands. Information technology is truthful that decimal fractions were used by the Chinese many centuries before Stevin and that the Farsi astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).^[35]

While the Persian mathematician Jamshīd al-Kāshī claimed to accept discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, equally decimal fractions were offset used five centuries earlier him past the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early on as the tenth century.^{[36] [n ii]}

In formal pedagogy [edit]

Pedagogical tools [edit]

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

Documents for teachers [edit]

Several states in the United States accept adopted learning trajectories from the Mutual Cadre State Standards Initiative's guidelines for mathematics instruction. 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 course ${\displaystyle a}$ / ${\displaystyle b}$ where ${\displaystyle a}$ is a whole number and ${\displaystyle b}$ is a positive whole number. (The word fraction in these standards ever refers to a not-negative number.)"^[38] The document itself too refers to negative fractions.

Come across also [edit]

• Cantankerous multiplication
• 0.999...
• Multiple
• FRACTRAN

Number systems

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

Real ${\displaystyle :\;\mathbb {R} }$ Rational ${\displaystyle :\;\mathbb {Q} }$ Integer ${\displaystyle :\;\mathbb {Z} }$ Natural ${\displaystyle :\;\mathbb {N} }$ Zero: 0 One: one 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 every bit Bringhurst mistakenly distinguish the slash ⟨/⟩ as the virgule and the fraction slash ⟨⁄⟩ equally the solidus,^{[half dozen]} although in fact both are synonyms for the standard slash.^{[7] [8]}
2. ^ While at that place is some disagreement amid history of mathematics scholars as to the primacy of al-Uqlidisi'southward contribution, there is no question every bit to his major contribution to the concept of decimal fractions.^[37]

References [edit]

1. ^ H. Wu, "The Mis-Education of Mathematics Teachers", Notices of the American Mathematical Society, Volume 58, Effect 03 (March 2011), p. 374. Archived 2017-08-xx at the Wayback Automobile.
2. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Lexicon of Mathematical Terms Used in English . Mathematical Association of America. ISBN978-0-88385-511-9.
3. ^ "Fractions". www.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.five: Use the Virgule with Words and Dates, the Solidus with Divide-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, n.^ane ". Oxford English Dictionary (1st ed.). Oxford: Oxford Academy Press. 1913.
9. ^ Weisstein, Eric Due west. "Common Fraction". MathWorld.
10. ^ ^{a b} David E. Smith (1 June 1958). History of Mathematics. Courier Corporation. p. 219. ISBN978-0-486-20430-vii.
11. ^ "Earth Wide Words: Vulgar fractions". World Broad Words. Archived from the original on 2014-10-thirty. Retrieved 2014-10-30 .
12. ^ Weisstein, Eric W. "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's Arithmetick: Or, the Ground of Arts: Educational activity the Perfect Work and Exercise of Arithmetick ... Made by Mr. Robert Record ... Afterward Augmented by Mr. John Dee. And Since Enlarged with a Third Part of Rules of Practise ... By John Mellis. And At present Diligently Perused, Corrected ... and Enlarged ; with an Appendix of Figurative Numbers ... with Tables of Board and Timber Measure ... the Get-go Calculated past R. C. But Corrected, and the Latter ... Calculated past Ro. Hartwell ... James Flesher, and are to be sold past Edward Dod. pp. 266–.
15. ^ Laurel (31 March 2004). "Math Forum – Ask Dr. Math: Can Negative Fractions Besides Be Proper or Improper?". Archived from the original on ix November 2014. Retrieved 2014-10-30 .
16. ^ "New England Compact Math Resources". Archived from the original on 2012-04-xv. Retrieved 2011-12-31 .
17. ^ Greer, A. (1986). New comprehensive mathematics for 'O' level (2nd ed., reprinted ed.). Cheltenham: Thornes. p. five. ISBN978-0-85950-159-0. Archived from the original on 2019-01-19. 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 lexicon.
20. ^ "complex fraction". Collins English Lexicon. Archived from the original on 2017-12-01. Retrieved 29 August 2022.
21. ^ "Circuitous fraction definition and meaning". Collins English Dictionary. 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 2018-03-14. Retrieved 2018-03-13 .
23. ^ Schoenborn, Barry; Simkins, Bradley (2010). "eight. 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 2014-10-21. Retrieved 2012-08-15 .
25. ^ Galen, Leslie Blackwell (March 2004). "Putting Fractions in Their Place" (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. ^ "congenital 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 (6th ed.). Philadelphia: Saunders College Pub. ISBN978-0-03-029558-4.
29. ^ Milo Gardner (December xix, 2005). "Math History". Archived from the original on December 19, 2005. Retrieved 2006-01-18 . See for examples and an explanation.
30. ^ ^{a b c d} Miller, Jeff (22 December 2014). "Earliest Uses of Various Mathematical Symbols". Archived from the original on 20 February 2016. Retrieved xv February 2016.
31. ^ Filliozat, Pierre-Sylvain (2004). "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature". In Chemla, Karine; Cohen, Robert South.; Renn, Jürgen; et al. (eds.). History of Science, History of Text. Boston Series in the Philosophy of Science. Vol. 238. Dordrecht: Springer Netherlands. p. 152. doi:10.1007/ane-4020-2321-9_7. ISBN978-1-4020-2320-0.
32. ^ Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. La Salle, Illinois: Open Court 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 Academy Printing. 1986. ISBN978-0-691-02397-7.
35. ^ Dice 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, People's republic of china, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN978-0-691-11485-9.
37. ^ "MacTutor'south al-Uqlidisi biography". Archived 2011-11-fifteen at the Wayback Machine. Retrieved 2011-11-22.
38. ^ "Mutual Core State Standards for Mathematics" (PDF). Common Core Land Standards Initiative. 2010. p. 85. Archived (PDF) from the original on 2013-10-19. Retrieved 2013-ten-ten .

External links [edit]

Wikimedia Commons has media related to Fractions.

Look up denominator in Wiktionary, the free dictionary.

Look 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.

6 Percent In Decimal Form,

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

Posted by: davisonated.blogspot.com

Share this post