A fraction (from the Latin Latin or sometimes Roman is an Italic language originally spoken in Latium and Ancient Rome. Although often considered a dead language, in view of the fact that it has no native, fluent speakers, Latin continues to be taught in schools and has been, and currently is, used in the process of new word production in modern languages from many fractus, broken) is a number A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels , that can represent part of a whole The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6. The earliest fractions were reciprocals In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a⁄b is b⁄a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one of integers The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6: ancient symbols representing one part of two, one part of three, one part of four, and so on.^[1] A much later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator, the numerator representing a number of equal parts and the denominator telling how many of those parts make up a whole. An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

A still later development was the fraction, now called simply a decimal The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations, in which the denominator is a power of ten, determined by the number of digits A digit is a symbol used in numerals (combinations of symbols, e.g. "37"), to represent numbers, (integers or real numbers) in positional numeral systems. The name "digit" comes from the fact that the 10 digits (ancient Latin digita meaning fingers) of the hands correspond to the 10 symbols of the common base 10 number system, to the right of a decimal separator The decimal separator or decimal point or decimal comma is a symbol used to mark the boundary between the integral and the fractional parts of a decimal number in a positional numeral system, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator The decimal separator or decimal point or decimal comma is a symbol used to mark the boundary between the integral and the fractional parts of a decimal number in a positional numeral system). Thus for 0.75 the numerator is 75 and the denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal.

A third kind of fraction still in common use is the percentage In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, "%", or the abbreviation "pct". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or 0.45, in which the denominator is always 100. Thus 75% means 75/100.

Other uses for fractions are to represent ratios In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Example: For every Spoon of sugar, you need 2 spoons of flour , and to represent division In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication. Thus the fraction 3/4 is also used to represent the ratio 3:4 (three to four) and the division 3 ÷ 4 (three divided by four).

In mathematics, the set of all (vulgar) fractions is called the set of rational numbers In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient, and is represented by the symbol Q.

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".

The word is also used in related expressions, such as continued fraction or any analogously defined longer expression, where a0 is an integer and all the other numbers ai are positive integers. If an infinite sequence of such integers is given, the sequence of these expressions defines an infinite continued fraction, which is often written as and algebraic fraction In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions—see Special cases below.

Writing fractions

A common or vulgar fraction is usually written as a pair of numbers, the top number called the numerator and the bottom number called the denominator. A line usually separates the numerator and denominator. If the line is slanting it is called a solidus The solidus is a punctuation mark that is not found on standard keyboards. It may also be called a shilling mark or in-line fraction bar or a forward-slash. Its Unicode encoding is U+2044 or forward slash The slash is a sign, "/", used as punctuation mark and for various other purposes. It is often called a forward slash and many other alternative names, for example ¾. If the line is horizontal, it is called a vinculum A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered a group. Vinculum is Latin for "bond", "fetter", "chain", or "tie", which is roughly suggestive of some of the uses of the symbol or, informally, a "fraction bar", thus: .

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 Traffic signs or road signs are signs erected at the side of roads to provide information to road users. With increasing speed[citation needed] of transport, the tendency is for countries to adopt pictorial signs or otherwise simplify and standardize signs, to facilitate international travel where language differences can create barriers and in in some countries.

In computer displays and typography Typography is the art and technique of arranging type, type design, and modifying type glyphs. Type glyphs are created and modified using a variety of illustration techniques. The arrangement of type involves the selection of typefaces, point size, line length, leading , adjusting the spaces between groups of letters (tracking) and adjusting the, some fractions are printed as a single character. These are:

• ¼ (one fourth or one quarter)
• ½ (one half)
• ¾ (three fourths or three quarters)
• ⅓ (one third)
• ⅔ (two thirds)
• ⅕ (one fifth)
• ⅖ (two fifths)
• ⅗ (three fifths)
• ⅘ (four fifths)
• ⅙ (one sixth)
• ⅚ (five sixths)
• ⅛ (one eighth)
• ⅜ (three eighths)
• ⅝ (five eighths)
• ⅞ (seven eighths)

More formally, scientific publishing distinguishes four ways to set fractions, together with guidelines on use:^[2]

• case fractions: – these are generally used only for simple fractions;
• special fractions: ½ – these are not generally used in formal scientific publishing, but are used in other contexts;
• shilling fractions: 1/2 – so called because this notation was used for pre-decimal British currency (£sd £sd was the popular name for the pre-decimal currencies (sterling) used in the Kingdom of England, later the United Kingdom and ultimately in much of the British Empire. This abbreviation meant "pounds, shillings, and pence", and was usually pronounced that way, having originated from the Latin words "librae, solidi, denarii"), as in 2/6 for a half crown The half crown was a denomination of British money worth two shillings and sixpence, being one-eighth of a pound and half of a crown. The half crown was first issued in 1549, in the reign of Edward VI. No half crowns were issued in the reign of Mary, but from the reign of Elizabeth I half crowns were issued in every reign except Edward VIII, until, meaning two shillings and six pence. This setting is particularly recommend for fractions inline (rather than displayed), to avoid uneven lines, and for fractions within fractions (complex fractions) or within exponents to increase legibility.
• built-up fractions: – while large and legible, these can be disruptive, particularly for simple fractions, or within complex fractions.

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 ⅓, 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.

Forms of fractions

Vulgar, proper, and improper fractions

A vulgar fraction (or common fraction) is a rational number In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient written as one integer The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 6 (the numerator) divided In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication by a non-zero integer (the denominator) such as .

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; a vulgar fraction is said to be an improper fraction (US, British or Australian) or top-heavy fraction (British, occasionally North America) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. ).^[3]

Mixed numbers

A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+"; for example, 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: .

An improper fraction can be thought of as another way to write a mixed number; consider the example below.

We can imagine that the two entire cakes are each divided into quarters, so that the denominator for the whole cakes is the same as the denominator for the parts. Then each whole cake contributes to the total, so is another way of writing .

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

1. Multiply the whole part by the denominator of the fractional part.
2. Add the numerator of the fractional part to that product.
3. The resulting sum is the numerator of the new (improper) fraction, with the 'new' denominator remaining precisely the same as for the original fractional part of the mixed number.

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

1. Divide the numerator by the denominator.
2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
3. 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, the results of the new fraction is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. That is, they retain the same integrity - the same balance or proportion. This is true because for any number n, multiplying by is really multiplying by one, and any number multiplied by one has the same value as the original number. For instance, consider the fraction : when the numerator and denominator are both multiplied by 2, the result is , which has the same value (0.5) as . To picture this visually, imagine cutting the example cake into four pieces; two of the pieces together () make up half the cake ().

For example: , , and 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 mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder in common (other than 1) is said to be irreducible or in its lowest or simplest terms. For instance, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only number that is a factor of both 3 and 8 is 1.

Any fraction can be fully reduced to its lowest terms by dividing both the numerator and denominator by their greatest common divisor In mathematics, the greatest common divisor , also known as the greatest common denominator, greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the greatest common divisor of 63 and 462 is 21, therefore, the fraction can be fully reduced by dividing the numerator and denominator by 21:

In order to find the greatest common divisor, the Euclidean algorithm In mathematics, the Euclidean algorithm[a] is an efficient method for computing the greatest common divisor (GCD), also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements may be used.

Reciprocals and the "invisible denominator"

The reciprocal In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a⁄b is b⁄a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of , for instance, is .

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 = (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 .

Complex fractions

A complex fraction (or compound fraction) is a fraction in which the numerator or denominator contains a fraction. For example, and are complex fractions. To simplify a complex fraction, divide the numerator by the denominator, as with any other fraction (see the section on division for more details):

Arithmetic with fractions

Fractions, like whole numbers, obey the commutative In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the, associative In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such, and distributive laws, and the rule against division by zero.

Comparing fractions

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

because 3>2.

One way to compare fractions with different denominators is to find a common denominator. To compare and , these are converted to and . Then bd is a common denominator and the numerators ad and bc can be compared.

? gives

As a short cut, known as "cross multiplying", you can just compare ad and bc, without computing the denominator.

?

Multiply 17 by 5 and multiply 18 by 4. Since 85 is greater than 72, .

Another method of comparing fractions is this: if two fractions have the same numerator, then the fraction with the smaller denominator is the larger fraction. The reasoning is that since, in the first fraction, fewer equal pieces are needed to make up a whole, each piece must be larger.

Also note that every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, so every negative fraction is less than any positive fraction.

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:

.

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 chosen 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 (twelfths).

Consider adding the following two quantities:

First, convert into twelfths by multiplying both the numerator and denominator by three: . Note that is equivalent to 1, which shows that is equivalent to the resulting .

Secondly, convert into twelfths by multiplying both the numerator and denominator by four: . Note that is equivalent to 1, which shows that is equivalent to the resulting .

Now it can be seen that:

is equivalent to:

This method can be expressed algebraically:

And for expressions consisting of the addition of three fractions:

This method always works, but sometimes there is a smaller denominator that can be used (a least common denominator). For example, to add and 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 In arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive of 4 and 12.

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,

Multiplication

Multiplying by a whole number

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:

As another example, suppose that five people work for three hours out of a seven hour day (i.e. 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:

Multiplying by a fraction

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:

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:

In general, 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:

When multiplying (or dividing), it may be possible to choose to cancel down crosswise multiples In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantity a such as integer, real number, or complex number, b is a multiple of a if b = na for some integer n. The n is also called coefficient or multiplier. Additionally, if a is not zero, this is equivalent to saying that b / a is an integer with (often simply called, 'cancelling tops and bottom lines') that share a common factor.^[4] For example:

2⁄7 × ⅞ = 2 1⁄7 1 × 7 1⁄8 4 = 1⁄1 × ¼ = ¼

A two is a common factor In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder in both the numerator of the left fraction and the denominator of the right so is divided out of both. A seven is a common factor of the left denominator and right numerator.

Mixed numbers

When multiplying mixed numbers, it's best to convert the mixed number into an improper fraction. For example:

In other words, is the same as , making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total) and 33 quarters is , since 8 cakes, each made of quarters, is 32 quarters in total)

Division

Division by a fraction is done by multiplying the dividend by the reciprocal of the divisor, in accordance with the identity

A proof for the identity, from fundamental principles, can be given as follows:

About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. 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

Rationalization

A fraction may need to be rationalized if the denominator contains irrational numbers In mathematics, an irrational number is any real number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. It can be proven that irrational numbers are precisely those real, imaginary numbers or complex numbers A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the, in order to make it easier to work with. When the denominator is a monomial, it can be rationalized by multiplying top and the bottom of the fraction by the denominator:

The process of rationalization of binomial In elementary algebra, a binomial is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon. It is the simplest kind of polynomial involves multiplying the top and the bottom of a fraction by the conjugate In algebra, a conjugate of an element in a quadratic extension field of a field K is its image under the unique non-identity automorphism of the extended field that fixes K. If the extension is generated by a square root of an element r of K, then the conjugate of is for , and in particular in the case of the field C of complex numbers as an of the denominator so that the denominator becomes a rational number. For example:

Even if this process results in the numerator being irrational or complex, like in the examples above, the process may still facilitate subsequent manipulations by reducing the number of irrationals one has to work with in the denominator, or by making the denominator real in the case of a complex expression.

Special cases

A unit fraction is a vulgar fraction with a numerator of 1, e.g. .

An Egyptian fraction An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above is the sum of distinct unit fractions, e.g. . This term derives from the fact that the ancient Egyptians expressed all fractions except , and in this manner.

A dyadic fraction is a vulgar fraction in which the denominator is a power of two 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 number of times. Note that one is a power of two. Written in binary, a power of two always has the form 100...0, just like a power of ten in the decimal system, e.g. .

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 , the radian measure of a right angle.

Rational numbers are the quotient field In mathematics, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0. The field of fractions of the ring R is sometimes denoted by Quot or Frac(R) of integers. Rational functions In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions are functions evaluated in the form of a fraction, where the numerator and denominator are polynomials. These rational expressions are the quotient field of the polynomials In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (over some integral domain In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity).

A continued fraction or any analogously defined longer expression, where a0 is an integer and all the other numbers ai are positive integers. If an infinite sequence of such integers is given, the sequence of these expressions defines an infinite continued fraction, which is often written as is an expression such as

where the a_i are integers. This is not an element of a quotient field.

The term partial fraction In algebra, the partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function. The outcome of a full partial fraction expansion expresses that function as a sum of fractions, where: is used in algebra, when decomposing rational expressions (a fraction with an algebraic expression in 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 can be rewritten as the sum of two fractions: and . This is useful for calculating certain integrals in calculus.

Pedagogical tools

In primary schools A primary school is an institution in which children receive the first stage of compulsory education known as primary or elementary education. Primary school is the preferred term in the United Kingdom and many Commonwealth Nations, and in most publications of the United Nations Educational, Scientific, and Cultural Organization (UNESCO). In some, fractions have been demonstrated through Cuisenaire rods, fraction bars, 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.

History

The earliest known use of fractions is ca. 2800 BC as Ancient Indus Valley units of measurement.^{[citation needed]} The Egyptians There is evidence of rock carvings along the Nile terraces and in the desert oases. In the 10th millennium BC, a culture of hunter-gatherers and fishers replaced a grain-grinding culture. Climate changes and/or overgrazing around 8000 BC began to desiccate the pastoral lands of Egypt, forming the Sahara. Early tribal peoples migrated to the Nile used Egyptian fractions ca. 1000 BC. The Greeks used unit fractions and later continued fractions and followers of the Greek philosopher Pythagoras, ca. 530 BC, discovered that the square root of two cannot be expressed as a fraction. In 150 BC Jain mathematicians in India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, operations with fractions.

In Sanskrit literature, fractions, or rational numbers were always expressed by an integer followed by a fraction. When the integer is written on a line, the fraction is placed below it and is itself written on two lines, the numerator called amsa part on the first line, the denominator called cheda “divisor” on the second below. If the fraction is written without any particular additional sign, one understands that it is added to the integer above it. If it is marked by a small circle or a cross (the shape of the “plus” sign in the West) placed on its right, one understands that it is subtracted from the integer. For example, Bhaskara I writes^[6]

६ १ २ १०
४ ५ ९

That is,

6 1 2
1 1 1०
4 5 9

to denote 6+1/4, 1+1/5, and 2–1/9

Al-Hassār, a Muslim mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appears soon after in the work of Fibonacci in the 13th century.^{[citation needed]}

In discussing the origins of decimal fractions, Dirk Jan Struik states that (p. 7):^[7]

"The introduction of decimal fractions as 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. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).^[8]"

While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.^[9]

See also

• Rational number

References

1. ^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0030295580, "The Egyptians endeavored to avoid some of the computational difficulties encountered with fractions by representing all fractions, except 2/3, as the sum of so-called unit fractions. ... Thus, we find 2/7 expressed as 1/4 + 1/28." The book has a picture of the symbols the Egyptians used for unit fractions. "One fourth" looks like a blacked in square with an ellipse over it, 2/3 like an ellipse with an upside down U crossing it.
2. ^ (Galen 2004)
3. ^ World Wide Words: Vulgar fractions
4. ^ BBC GCSE Bitsize
5. ^ Milo Gardner (December 19, 2005). "Math History". http://egyptianmath.blogspot.com. Retrieved 2006-01-18. See for examples and an explanation.
6. ^ (Filliozat 2004, p. 152)
7. ^ D.J. Struik, A Source Book in Mathematics 1200-1800 (Princeton University Press, New Jersey, 1986). ISBN 0-691-02397-2
8. ^ P. Luckey, Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī (Steiner, Wiesbaden, 1951).
9. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859.

• Galen, Leslie Blackwell (March 2004), "Putting Fractions in Their Place", American Mathematical Monthly 111 (3), http://www.integretechpub.com/research/papers/monthly238-242.pdf

Categories: Fractions | Elementary arithmetic | Numbers | Division

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