Fraction (mathematics)

For other uses, see Fraction (disambiguation).

A cake with one quarter removed. The remaining three quarters are shown.

In mathematics, a fraction (from the Latin fractus, broken) is a concept of a proportional relation between an object part and the object whole. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations

A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities. A ratio is an expression which compares quantities relative to each other ^[1]

A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. Thus ³⁄₄ represents three divided by four, in decimals 0. 75, as a percentage 75%. The three equal parts of the cake are 75% of the whole cake.

Each fraction consists of a denominator (bottom) and a numerator (top), representing (respectively) the number of equal parts that an object is divided into, and the number of those parts indicated for the particular fraction. Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator Fractions are rational numbers, which means that the denominator and the numerator are integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

For example, the fraction ³⁄₄ could be used to represent three equal parts of a whole object, were it divided into four equal parts. The worked-example effect is a learning effect predicted by cognitive load theory (Sweller 1988 Because it is impossible to divide something into zero equal parts, zero can never be the denominator of a fraction (see division by zero). In A fraction with equal numerator and denominator is equal to one (e. g. ⁵⁄₅ = 1) and the fraction form is rarely, if ever, given as a final result.

In higher mathematics, a fraction is viewed as an element of a field of fractions. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients

Terminology

Historically, any number that did not represent a whole was called a "fraction". The numbers that we now call "decimals" were originally called "decimal fractions"; the numbers we now call "fractions" were called "vulgar fractions", the word "vulgar" meaning "commonplace".

Writing fractions

The numerator and denominator of a fraction may be separated by a slanting line called a solidus or slash, for example ³⁄₄, or may be written above and below a horizontal line called a vinculum, thus: $\tfrac{3}{4}$ . The solidus ( ⁄) is a punctuation mark that is not found on standard keyboards The slash ( /) is a punctuation mark It is also called a virgule, diagonal, stroke, forward slash, oblique dash, A vinculum is a horizontal line placed over a Mathematical expression, used to indicate that it is to be considered a group

The solidus may be omitted from the slanting style (e. g. ³₄) where space is short and the meaning is obvious from context, for example in road signs in some countries. Most countries post signage known as traffic signs or road signs, at the side of Roads to

Usage

Fractions are used most often when the denominator is relatively small. It is easier to multiply 32 by ³⁄₁₆ than to do the same calculation using the fraction's decimal equivalent (0. 1875). It is also more accurate to multiply 15 by ¹⁄₃, for example, than it is to multiply 15 by a decimal approximation of one third. To change a fraction to a decimal, divide the numerator by the denominator, and round off to the desired accuracy.

The word is also used in related expressions, such as continued fraction and algebraic fraction—see Special cases below. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In

Forms of fractions

Vulgar, proper, and improper fractions

A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator), for example, $\tfrac{1}{3}$ , $\tfrac{3}{4}$ and $\tfrac{4}{3}$ . In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

A vulgar fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1 (e. g. $\tfrac{4}{9}$ )—but an improper fraction (US, British or Australian) or top-heavy fraction (British only) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e. g. $\tfrac{9}{7}$ ). ^[2]

Mixed numbers

A mixed number is the sum of a whole number and a proper fraction. For instance, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: $2+\tfrac{3}{4}=2\tfrac{3}{4}$ .

An improper fraction can be thought of as another way to write a mixed number; in the " $2\tfrac{3}{4}$ " example above, imagine that the two entire cakes are each divided into quarters. Each entire cake contributes $\tfrac{4}{4}$ to the total, so $\tfrac{4}{4}+\tfrac{4}{4}+\tfrac{3}{4}=\tfrac{11}{4}$ is another way of writing $2\tfrac{3}{4}$ .

A mixed number can be converted to an improper fraction in three steps:

Multiply the whole part by the denominator of the fractional part.
Add the numerator of the fractional part to that product.
The resulting sum is the numerator of the new (improper) fraction, and the new denominator is the same as that of the fractional part of the mixed number.

Similarly, an improper fraction can be converted to a mixed number:

Divide the numerator by the denominator.
The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
The new denominator is the same as that of the original improper fraction.

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a new fraction that is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. This is true because for any number $n$ , multiplying by $\tfrac{n}{n}$ is really multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider 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) as $\tfrac{1}{2}$ . To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together ( $\tfrac{2}{4}$ ) make up half the cake ( $\tfrac{1}{2}$ ).

For example: $\tfrac{1}{3}$ , $\tfrac{2}{6}$ , $\tfrac{3}{9}$ and $\tfrac{100}{300}$ are all equivalent fractions.

Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. this is called reducing or simplifying the fraction. A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in its lowest or simplest terms. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without For instance, $\tfrac{3}{9}$ is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, $\tfrac{3}{8}$ is in lowest terms—the only number that is a factor of both 3 and 8 is 1.

Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator reversed. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which The reciprocal of $\tfrac{3}{7}$ , for instance, is $\tfrac{7}{3}$ .

Because any number divided by 1 results in the same number, it is possible to write any whole number as a fraction by using 1 as the denominator: 17 = $\tfrac{17}{1}$ (1 is sometimes referred to as the "invisible denominator"). Therefore, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be $\tfrac{1}{17}$ .

Complex fractions

A complex fraction (or compound fraction) is a fraction in which the numerator and denominator contain a fraction. For example, $\cfrac{\tfrac{1}{2}}{\tfrac{1}{3}}$ is a complex fraction. To simplify a complex fraction, divide the numerator by the denominator, as with any other fraction: $\cfrac{\tfrac{1}{2}}{\tfrac{1}{3}}=\tfrac{3}{2}$ .

Arithmetic with fractions

Fractions, like whole numbers, obey the commutative, associative, and distributive laws, and the rule against division by zero. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

Comparing fractions

Comparing fractions with the same denominator only requires comparing the numerators.

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

3 > 2

In order to compare fractions with different denominators, these are converted to a common denominator: to compare $\tfrac{a}{b}$ and $\tfrac{c}{d}$ , these are converted to $\tfrac{ad}{bd}$ and $\tfrac{bc}{bd}$ , where bd is the product of the denominators, and then the numerators ad and bc are compared.

$\tfrac{2}{3}$ ? $\tfrac{1}{2}$ gives $\tfrac{4}{6}>\tfrac{3}{6}$

This method is also known as the "cross-multiply" method which can be explained by multiplying the top and bottom numbers crosswise. The product of the denominators is used as a common (but not necessary the least common) denominator.

$\tfrac{5}{18}$ ? $\tfrac{4}{17}$

Multiply 17 by 5 and 18 by 4. Place the products of the equations on top of the denominators. The highest number identifies the largest fraction. Therefore $\tfrac{5}{18}>\tfrac{4}{17}$ as 17 × 5 = 85 is greater than 18 × 4 = 72.

In order to work with smaller numbers, the least common denominator is used instead of the product. In Mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the Least common multiple of the Denominators The fractions are converted to fractions with the least common denominator, and then the numerators are compared. ^[3]

$\tfrac{5}{6}$ ? $\tfrac{3}{4}$ gives $\tfrac{10}{12}>\tfrac{9}{12}$

Some standards-based mathematics texts such as Connected Mathematics omit instruction of least common denominators entirely. Connected Mathematics is a comprehensive problem-centered curriculum designed for all students in grades 6-8 based on the NCTM standards That text presents the use of "fraction strips"^[4] (a strip of paper folded into fractions) or "benchmark fractions" such as one-half against which a fraction such as two-fifths may be compared. While such methods may be useful to build conceptual understanding, they are controversial as they are not effective beyond the elementary school level, and such texts are often supplemented by teachers with the standard method.

Addition

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

$\tfrac24+\tfrac34=\tfrac54=1\tfrac14$ .

If $\tfrac12$ of a cake is to be added to $\tfrac14$ of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters.

Adding unlike quantities

To add fractions containing unlike quantities (e. g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction.

For adding quarters to thirds, both types of fraction are converted to $\tfrac14\times\tfrac13=\tfrac1{12}$ (twelfths).

Consider adding the following two quantities:

$\tfrac34+\tfrac23$

First, convert $\tfrac34$ into twelfths by multiplying both the numerator and denominator by three: $\tfrac34\times\tfrac33=\tfrac9{12}$ . Note that $\tfrac33$ is equivalent to 1, which shows that $\tfrac34$ is equivalent to the resulting $\tfrac9{12}$

Secondly, convert $\tfrac23$ into twelfths by multiplying both the numerator and denominator by four: $\tfrac23\times\tfrac44=\tfrac8{12}$ . Note that $\tfrac44$ is equivalent to 1, which shows that $\tfrac23$ is equivalent to the resulting $\tfrac8{12}$

Now it can be seen that:

$\tfrac34+\tfrac23$

is equivalent to:

$\tfrac9{12}+\tfrac8{12}=\tfrac{17}{12}=1\tfrac5{12}$

This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add $\tfrac{3}{4}$ and $\tfrac{5}{12}$ the denominator 48 can be used (the product of 4 and 12), but the smaller denominator 12 may also be used, being the least common multiple of 4 and 12. In Arithmetic and Number theory, the least common multiple or lowest common multiple ( lcm) or smallest common multiple of two

$\tfrac34+\tfrac{5}{12}=\tfrac{9}{12}+\tfrac{5}{12}=\tfrac{14}{12}=\tfrac76=1\tfrac16$

Subtraction

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

$\tfrac23-\tfrac12=\tfrac46-\tfrac36=\tfrac16$

Multiplication

When multiplying or dividing, it may be possible to choose to cancel down crosswise multiples that share a common factor. In Mathematics, a multiple of an Integer is the product of that integer with another integer For example:

^{2 1}⁄_{7 1} X ^{7 1}⁄_{8 4} = ¹⁄₁ X ¹⁄₄. The following will explain how to complete this equation.

Whole numbers

Considering the cake example above, if you have a quarter of the cake and you multiply the amount by three, then you end up with three quarters. We can write this numerically as follows:

$\textstyle{3 \times {1 \over 4} = {3 \over 4}}\,\!$

As another example, suppose that five people work for three hours out of a seven hour day (ie. for three sevenths of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 sevenths of a day is a whole day and 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of a day. Numerically:

$\textstyle{5 \times {3 \over 7} = {15 \over 7} = 2{1 \over 7}} \,\!$

Fractions

Considering the cake example above, if you have a quarter of the cake and you multiply the amount by a third, then you end up with a twelfth of the cake. In other words, a third of a quarter (or a third times a quarter) is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts (or twelfths). We can write this numerically as follows:

$\textstyle{{1 \over 3} \times {1 \over 4} = {1 \over 12}}\,\!$

As another example, suppose that five people do an equal amount of work that totals three hours out of a seven hour day. Each person will have done a fifth of the work, so they will have worked for a fifth of three sevenths of a day. Numerically:

$\textstyle{{1 \over 5} \times {3 \over 7} = {3 \over 35}}\,\!$

General rule

You may have noticed that when we multiply fractions, we multiply the two numerators (the top numbers) to make the new numerator, and multiply the two denominators (the bottom numbers) to make the new denominator. For example:

$\textstyle{{5 \over 6} \times {7 \over 8} = {5 \times 7 \over 6 \times 8} = {35 \over 48}}\,\!$

Mixed numbers

When multiplying mixed numbers, it's best to convert the whole part of the mixed number into a fraction. For example:

$\textstyle{3 \times 2{3 \over 4} = 3 \times \left ({{8 \over 4} + {3 \over 4}} \right ) = 3 \times {11 \over 4} = {33 \over 4} = 8{1 \over 4}}\,\!$

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

Division

To divide by a fraction, simply multiply by the reciprocal of that fraction.

$\textstyle{5 \div {1 \over 2} = 5 \times {2 \over 1} = 5 \times 2 = 10}$

$\textstyle{{2 \over 3} \div {2 \over 5} = {2 \over 3} \times {5 \over 2} = {10 \over 6} = {5 \over 3}}$

To understand why this works, consider the following:

6 inches divided by 3 inches = 2 means that we can divide 6 inches into two 3 inch parts.

6 miles divided by 3 miles = 2 means that we can divide 6 miles into two 3 mile parts.

6 half dollars divided by 3 half dollars = 2 means that we can divide 6 half dollars into two stacks of 3 half dollars each.

6/2 divided by 3/2 = 2 means that we can divide 6/2 into two parts, each 3/2 in size.

Thus, if fractions have the same denominator, to divide we just divide the numerators.

But what if fractions have different denominators?

Then, we could get a common denominator, and divide the numerators, as follows:

$\textstyle{{a \over b} \div {c \over d} = {ad \over bd} \div {bc \over bd} = {ad \over bc}.}$

But this takes too long. Instead, we learn the rule "invert and multiply", which gives the same answer.

$\textstyle{{a \over b} \div {c \over d} = {a \over b} \times {d \over c} = {ad \over bc}.}$

Here is a mathematical proof that to divide we invert and multiply.

Theorem

$\textstyle{{a \over b} \div {c \over d} = {ad \over bc}.}$

Proof

We know that division is defined to be the inverse of multiplication. That is,

$\textstyle{m \div n = q}$

if and only if

$\textstyle{n \times q = m}.$

In the expression we want to prove, multiply the quotient by the divisor $\textstyle{{c \over d} \times {ad \over bc} = {acd \over bcd}}$ which reduces to $\textstyle{{a \over b}}$ .

Therefore,

$\textstyle{{a \over b} \div {c \over d} = {ad \over bc}.}$

About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. A unit fraction is a Rational number written as a fraction where the Numerator is one and the Denominator is a positive Integer Their methods gave the same answer that our modern methods give. ^[5]

Converting repeating decimals to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, lack the same kind of precision that regular fractions (as they are explained in this article) have. Sometimes an infinite number of decimals is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions.

For repeating patterns where the repeating pattern begins immediately after the decimal point, a simple division of the pattern by the same number of nines as numbers it has will suffice. For example (the pattern is highlighted in bold):

0. 555555555555… = 5/9

0. 626262626262… = 62/99

0. 264264264264… = 264/999

0. 629162916291… = 6291/9999

In case zeros precede the pattern, the nines are suffixed by the same number of zeros:

0. 0555… = 5/90

0. 000392392392… = 392/999000

0. 00121212… = 12/9900

In case a non-repeating set of decimals precede the pattern (such as 0. 1523987987987…), we must equate it as the sum of the non-repeating and repeating parts:

0. 1523 + 0. 0000987987987…

Then, convert both of these to fractions. Since the first part is not repeating, it is not converted according to the pattern given above:

1523/10000 + 987/9990000

We add these fractions by expressing both with a common divisor. . .

1521477/9990000 + 987/9990000

And add them.

1522464/9990000

Finally, we simplify it:

31718/208125

Special cases

A unit fraction is a vulgar fraction with a numerator of 1, e. A unit fraction is a Rational number written as a fraction where the Numerator is one and the Denominator is a positive Integer g. $\tfrac{1}{7}$ .

An Egyptian fraction is the sum of distinct unit fractions, e. An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} g. $\tfrac{1}{2}+\tfrac{1}{3}$ .

A dyadic fraction is a vulgar fraction in which the denominator is a power of two, e. In Mathematics, a dyadic fraction or dyadic rational is a Rational number whose Denominator is a Power of two, i In Mathematics, a power of two is any of the Integer powers of the number two; in other words two multiplied by itself a certain g. $\tfrac{1}{8}$ .

An expression that has the form of a fraction but actually represents division by or into an irrational number is sometimes called an "irrational fraction". A common example is $\textstyle{\frac{\pi}{2}}$ , the radian measure of a right angle.

Rational numbers are the quotient field of integers. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients Rational functions are functions evaluated in the form of a fraction, where the numerator and denominator are polynomials. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In These rational expressions are the quotient field of the polynomials (over some integral domain). In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such

A continued fraction is an expression such as $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}}$ , where the a_i are integers. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} This is not an element of a quotient field.

The term partial fraction is used in algebra, when decomposing rational expressions (a fraction with an algebraic expression in the denominator). In Algebra, the partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator The goal is to write the rational expression as the sum of other rational expressions with denominators of lesser degree. For example, the rational expression $\textstyle{2x \over (x^2-1)}$ can be rewritten as the sum of two fractions: $\textstyle{1 \over (x+1)}$ and $\textstyle{1 \over (x-1)}$ .

Pedagogical tools

In primary schools, fractions have been demonstrated through Cuisenaire rods. See also Primary education A primary school (from French école primaire) is an institution where children receive the first stage of Compulsory Cuisenaire rods are a versatile Mathematical manipulative used in Elementary school as well as other levels of learning and even with adults

Parents of children learning fractions should also be aware that arithmetic is often taught very differently with reform mathematics. Reform mathematics is one name for mathematics instruction based on recommendations originally published in 1989 by the National Council of Teachers of Mathematics (NCTM Many texts do not give instruction of standard methods which may use the least common denominator, to compare or add fractions. Some introduce newly developed concepts such as "fraction strips"^[6] and benchmark fractions (1/2, 1/4, 3/4 and 1/10)^[7] which are unfamiliar to parents or mathematicians. Some are concerned that such methods will not prepare students for mathematics in college or high school. If this is the case, parents may ask their schools to supplement their children's learning with standard methods or switch to texts which give instruction in traditional methods. Fraction arithmetic is normally taught and mastered from late elementary to middle or junior high school. However, some texts such as the Connected Mathematics do not discuss division of fractions at all even through 8th grade in CMP^[8]

History

See also: history of irrational numbers. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction

The earliest known use of decimal fractions is ca. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 2800 BC as Ancient Indus Valley units of measurement. The 28th century BC is a Century which lasted from the year 2800 BC to 2701 BC The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin The Egyptians used Egyptian fractions ca. The history of Egypt is the longest continuous history as a unified state of any country in the world 1000 BC. The Greeks used unit fractions and later continued fractions and followers of the Greek philosopher Pythagoras, ca. The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. 530 BC, discovered that the square root of two cannot be expressed as a fraction. Events and trends 539 BC — Babylon is conquered by Cyrus, defeating Nabonidus; noted in such documents as that of Africanus The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, operations with fractions. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. This article is about the history of South Asia prior to the Partition of British India in 1947 In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object

References

^ (Gellert, W. A Non-Fractional number is simply a number that isn't normally written as a Fraction, the most common example being whole numbers In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions (1977). The VNR Concise Encyclopedia of Mathematics.
^ World Wide Words: Vulgar fractions
^ Visual Fractions tutorial
^ Fraction strip references
^ Milo Gardner (December 19, 2005). Events 324 - Licinius abdicates his position as Roman Emperor. Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. Math History. Retrieved on 2006-01-18. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 350 - Generallus Magnentius deposes Roman Emperor Constans and proclaims himself Emperor See for examples and an explanation.
^ NCTM Making and Investigating Fraction Strips
^ http://www.keypress.com/x18176.xml Empower Mathematics: Using Benchmarks
^ Connected Mathematics Program: A Critique

External links

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Contents

Terminology

Writing fractions

Usage

Forms of fractions

Vulgar, proper, and improper fractions

Mixed numbers

Equivalent fractions

Reciprocals and the "invisible denominator"

Complex fractions

Arithmetic with fractions

Comparing fractions

Addition

Adding unlike quantities

Subtraction

Multiplication

Whole numbers

Fractions

General rule

Mixed numbers

Division

Converting repeating decimals to fractions

Special cases

Pedagogical tools

History

See also

References

External links