Fraction (mathematics)
Overview
A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists of a numerator and a denominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole.
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.
A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists of a numerator and a denominator—the numerator representing a number of equal parts and the denominator indicating 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 equal a whole. The picture to the right illustrates 3/4 of a cake.
Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10^{−2} respectively, all of which are equivalent to 1/100). An integer
(e.g. the number 7) has an implied denominator of one, meaning that the number can be expressed as a fraction like 7/1.
Other uses for fractions are to represent ratio
s and to represent division
. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four). In mathematics the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers and is represented by the symbol Q. The word fraction is also used to describe continued fraction
s, algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational number
s, such as √2/2 (see square root of 2
) and π/4 (see proof that π is irrational).
written as an ordered pair
of integer
s, called the numerator and denominator, separated by a line. The denominator cannot be zero. Two notations are widely used. One, as in the example 2/5, uses a slanting line to separate the numerator and denominator. The other, as in the example , has the numerator above a horizontal line and the denominator below the line. The slanting line is called a solidus
or forward slash
, the horizontal line is called a vinculum or, informally, a "fraction bar".
, simple fractions are sometimes printed as a single character, e.g. ½ (one half).
Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:
of the entire fraction is less than 1. A vulgar fraction is said to be an improper fraction (U.S., 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. ).
This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". (This often causes confusion when improperly taught.) In algebra, is not a mixed number. Instead, multiplication is understood. .
An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:
Similarly, an improper fraction can be converted to a mixed number as follows:
of a fraction. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is .
If, in a complex fraction, there is no clear way to tell which fraction line takes precedence, then the expression is improperly formed, and meaningless.
s to the right of a decimal separator
, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part
of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, .
Decimal fractions can also be expressed using scientific notation
with negative exponents, such as 6.023×10^{−7}, which represents 0.0000006023. The 10^{−7} represents a denominator of 10^{7}. Dividing by 10^{7} moves the decimal point 7 steps to the left.
Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, 1/3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .
Another kind of fraction is the percentage
, in which the implied denominator is always 100. Thus 75% means 75/100. Related concepts are the permille
, with 1000 as implied denominator, and the more general parts-per notation
, as in 75 parts per million, meaning that the proportion is 75/1,000,000.
Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation
, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pense were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pense, and having no relationship to the fraction 3/6.
is a vulgar fraction with a numerator of 1, e.g. . Unit fractions can also be expressed using negative exponents, as in 2^{−1} which represents 1/2, and 2^{−2} which represents 1/(2^{2}) or 1/4.
An Egyptian fraction 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. Every positive rational number can be expanded as an Egyptian fraction, but the representation is in general not unique. For example, .
A dyadic fraction
is a vulgar fraction in which the denominator is a power of two
, e.g. .
.
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 simple fraction in which the numerator and denominator are coprime
[that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible
, in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.
Using these rules, we can show that = = = .
A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor
. For example, as the greatest common divisor of 63 and 462 is 21, the fraction can be reduced to lowest terms by dividing the numerator and denominator by 21:
The Euclidean algorithm
gives a method for finding the greatest common divisor of any two positive integers.
because 3>2.
If two fractions have the same numerator, then the fraction with the smaller denominator is the larger fraction. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. The fraction with the smaller denominator represents these fewer but larger pieces.
One way to compare fractions with different numerators and 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
It is not necessary to determine the value of the common denominator to compare fractions. This short cut is known as "cross multiplying" – you can just compare ad and bc, without computing the denominator.
?
Multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator: ?
The denominators are now the same, but it is not necessary to calculate their value – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), .
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.
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:.
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
of 4 and 12.
Why does this work? First, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths.
A short cut for multiplying fractions is called "cancellation". In effect, we reduce the answer to lowest terms during multiplication. For example:
A two is a common factor
in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.
This method works because the fraction 6/1 means six equal parts, each one of which is a whole.
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.
Two fractions can be compared using the rule if and only if . Two fractions and are equivalent if and only if .
To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits in the original decimal, omitting the decimal point. Thus 12.3456 = 123456/10000.
s is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions.
The preferred way to indicate a repeating decimal is to place a bar over the digits that repeat, for example 0. = 0.789789789… 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:
In case leading zero
s precede the pattern, the nines are suffixed by the same number of trailing zero
s:
In case a non-repeating set of decimals precede the pattern (such as 0.1523), we can write it as the sum of the non-repeating and repeating parts, respectively:
Then, convert the repeating part to a fraction:
where a_{i} are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficient
s a_{i} can be determined by applying the Euclidean algorithm
to (a,b).
of two algebraic expression
s. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions.
If the numerator and the denominator are polynomial
s, as in , the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent, as in .
The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factor common to the numerator and the denominator is 1. An algebraic fraction whose numerator or denominator, or both, contains a fraction, such as , is called a complex fraction.
Rational numbers are the quotient field of integers. Rational expressions are the quotient field of the polynomial
s (over some integral domain). Since a coefficient
is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction. Another example (over the reals) is , the radian measure of a right angle.
The term partial fraction
is used when decomposing rational expressions. 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 in many areas such as integral calculus and differential equations.
, a cube root, etc., it can be helpful to rationalize
it, especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a monomial
square root, it can be rationalized by multiplying top and the bottom of the fraction by the denominator:
The process of rationalization of binomial
denominators involves multiplying the top and the bottom of a fraction by the conjugate
of the denominator so that the denominator becomes a rational number. For example:
Even if this process results in the numerator being irrational, 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.
, fraction bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboard
s, counters and computer software.
of integer
s: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians
used Egyptian fractions ca. 1000 BC. About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fraction
s. Their methods gave the same answer as modern methods.
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
६ १ २
१ १ १_{०}
४ ५ ९
That is,
6 1 2
1 1 1_{०}
4 5 9
to denote 6+1/4, 1+1/5, and 2–1/9
Al-Hassār, a Muslim mathematician from Fez
, Morocco
specializing in Islamic inheritance jurisprudence
during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, ." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.
In discussing the origins of decimal fractions, Dirk Jan Struik
states that (p. 7):
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 Baghdad
i mathematician Abu'l-Hasan al-Uqlidisi
as early as the 10th century.
A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists of a numerator and a denominator—the numerator representing a number of equal parts and the denominator indicating 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 equal a whole. The picture to the right illustrates 3/4 of a cake.
Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10^{−2} respectively, all of which are equivalent to 1/100). An integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
(e.g. the number 7) has an implied denominator of one, meaning that the number can be expressed as a fraction like 7/1.
Other uses for fractions are to represent ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s and to represent division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
. Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four). In mathematics the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers and is represented by the symbol Q. The word fraction is also used to describe continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s, algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s, such as √2/2 (see square root of 2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
) and π/4 (see proof that π is irrational).
Common, vulgar, or simple fractions
A common fraction (or vulgar fraction or simple fraction) is a rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction 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...
written as an ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, called the numerator and denominator, separated by a line. The denominator cannot be zero. Two notations are widely used. One, as in the example 2/5, uses a slanting line to separate the numerator and denominator. The other, as in the example , has the numerator above a horizontal line and the denominator below the line. The slanting line is called a solidus
Solidus (punctuation)
The solidus is a punctuation mark used to indicate fractions including fractional currency. It may also be called a shilling mark, an in-line fraction bar, or a fraction slash....
or forward slash
Slash (punctuation)
The slash is a sign used as a punctuation mark and for various other purposes. It is now often called a forward slash , and many other alternative names.-History:...
, the horizontal line is called a vinculum or, informally, a "fraction bar".
Writing simple fractions
In computer displays and typographyTypography
Typography is the art and technique of arranging type in order to make language visible. The arrangement of type involves the selection of typefaces, point size, line length, leading , adjusting the spaces between groups of letters and adjusting the space between pairs of letters...
, simple fractions are sometimes printed as a single character, e.g. ½ (one half).
Scientific publishing distinguishes four ways to set fractions, together with guidelines on use:
- case fractions: (these are generally used only for simple fractions);
- special fractions: ½ (these are not used in modern mathematical notation, but in other contexts);
- shilling fractions: 1/2 (so called because this notation was used for pre-decimal British currency (£sd£sd£sd was the popular name for the pre-decimal currencies used in the Kingdom of England, later the United Kingdom, and ultimately in much of the British Empire...
), as in 2/6 for a half crownHalf crown (British coin)The half crown was a denomination of British money worth half of a crown, equivalent to two and a half shillings , or one-eighth of a pound. The half crown was first issued in 1549, in the reign of Edward VI...
, meaning two shillings and six pence, particularly recommended 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).
Proper and improper common fractions
A common 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 valueAbsolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of the entire fraction is less than 1. A vulgar fraction is said to be an improper fraction (U.S., 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. ).
Mixed numbers
A mixed number (also called a mixed numeral or mixed fraction) 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: .This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". (This often causes confusion when improperly taught.) In algebra, is not a mixed number. Instead, multiplication is understood. .
An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:
- Write the mixed number as a sum .
- Convert the whole number to an improper fraction with the same denominator as the fractional part, .
- Add the fractions. The resulting sum is the improper fraction. In the example, .
Similarly, an improper fraction can be converted to a mixed number as follows:
- Divide the numerator by the denominator. In the example, , divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.
- The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.
- The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus .
Reciprocals and the "invisible denominator"
The reciprocal of a fraction is another fraction with the numerator and denominator reversed. The reciprocal of , for instance, is . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverseMultiplicative inverse
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...
of a fraction. Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as , where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is .
Complex fractions
In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, and are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:If, in a complex fraction, there is no clear way to tell which fraction line takes precedence, then the expression is improperly formed, and meaningless.
Compound fractions
A compound fraction is a fraction of a fraction, or any mumber of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication). For example, of is a compound fraction, corresponding to . The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other.Decimal fractions and percentages
A decimal fraction is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digitNumerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
s to the right of a decimal separator
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale (for examples, see decimal separator). Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, viz. 100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part
Fractional part
All real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, .
Decimal fractions can also be expressed using scientific notation
Scientific notation
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians, doctors, and engineers.In scientific...
with negative exponents, such as 6.023×10^{−7}, which represents 0.0000006023. The 10^{−7} represents a denominator of 10^{7}. Dividing by 10^{7} moves the decimal point 7 steps to the left.
Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, 1/3 = 0.333... represents the infinite series 3/10 + 3/100 + 3/1000 + ... .
Another kind of fraction is the percentage
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% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...
, in which the implied denominator is always 100. Thus 75% means 75/100. Related concepts are the permille
Permille
A per mil or per mille is a tenth of a percent or one part per thousand. It is written with the sign ‰ , which looks like a percent sign with an extra zero at the end...
, with 1000 as implied denominator, and the more general parts-per notation
Parts-per notation
In science and engineering, the parts-per notation is a set of pseudo units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they are pure numbers with no associated units of measurement...
, as in 75 parts per million, meaning that the proportion is 75/1,000,000.
Whether common fractions or decimal fractions are used is often a matter of taste and context. Common fractions are used most often when the denominator is relatively small. By mental calculation
Mental calculation
Mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation , or in a competition context...
, it is easier to multiply 16 by 3/16 than to do the same calculation using the fraction's decimal equivalent (0.1875). And it is more accurate to multiply 15 by 1/3, for example, than it is to multiply 15 by any decimal approximation of one third. Monetary values are commonly expressed as decimal fractions, for example $3.75. However, as noted above, in pre-decimal British currency, shillings and pense were often given the form (but not the meaning) of a fraction, as, for example 3/6 (read "three and six") meaning 3 shillings and 6 pense, and having no relationship to the fraction 3/6.
Special cases
A unit fractionUnit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...
is a vulgar fraction with a numerator of 1, e.g. . Unit fractions can also be expressed using negative exponents, as in 2^{−1} which represents 1/2, and 2^{−2} which represents 1/(2^{2}) or 1/4.
An Egyptian fraction 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. Every positive rational number can be expanded as an Egyptian fraction, but the representation is in general not unique. For example, .
A dyadic fraction
Dyadic rational
In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator is a power of two, i.e., a number of the form a/2b where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3...
is a vulgar fraction in which the denominator is a power of two
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
, e.g. .
Arithmetic with fractions
Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zeroDivision by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
.
Equivalent fractions
Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number , the fraction . Therefore, multiplying by is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with 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 a cake into four pieces; two of the pieces together () make up half the cake ().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 simple fraction in which the numerator and denominator are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
[that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible
Irreducible fraction
An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
, in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.
Using these rules, we can show that = = = .
A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
. For example, as the greatest common divisor of 63 and 462 is 21, the fraction can be reduced to lowest terms by dividing the numerator and denominator by 21:
The Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
gives a method for finding the greatest common divisor of any two positive integers.
Comparing fractions
Comparing fractions with the same denominator only requires comparing the numerators.because 3>2.
If two fractions have the same numerator, then the fraction with the smaller denominator is the larger fraction. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. The fraction with the smaller denominator represents these fewer but larger pieces.
One way to compare fractions with different numerators and 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
It is not necessary to determine the value of the common denominator to compare fractions. This short cut is known as "cross multiplying" – you can just compare ad and bc, without computing the denominator.
?
Multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator: ?
The denominators are now the same, but it is not necessary to calculate their value – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), .
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
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
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,Multiplying a fraction by another fraction
To multiply fractions, multiply the numerators and multiply the denominators. Thus:Why does this work? First, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two thirds of one quarter. Thus two thirds times three quarters is six twelfths.
A short cut for multiplying fractions is called "cancellation". In effect, we reduce the answer to lowest terms during multiplication. For example:
A two is a common factor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
in both the numerator of the left fraction and the denominator of the right and is divided out of both. Three is a common factor of the left denominator and right numerator and is divided out of both.
Multiplying a fraction by a whole number
Place the whole number over one and multiply.This method works because the fraction 6/1 means six equal parts, each one of which is a whole.
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
To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, equals and also equals , which reduces to . To divide by a fraction, multiply by the reciprocal of that fraction. Thus, .Formulas for the arithmetic of fractions
Let be positive integers.Two fractions can be compared using the rule if and only if . Two fractions and are equivalent if and only if .
Converting between decimals and fractions
To change a common fraction to a decimal, divide the denominator into the numerator. Round the answer to the desired accuracy. For example, to change 1/4 to a decimal, divide 4 into 1.00, to obtain 0.25. To change 1/3 to a decimal, divide 3 into 1.0000..., and stop when the desired accuracy is obtained. Note that 1/4 can be written exactly with two decimal digits, while 1/3 cannot be written exactly with any finite number of decimal digits.To change a decimal to a fraction, write in the denominator a 1 followed by as many zeroes as there are digits to the right of the decimal point, and write in the numerator all the digits in the original decimal, omitting the decimal point. Thus 12.3456 = 123456/10000.
Converting repeating decimals to fractions
Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite number of repeating decimalRepeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...
s is required to convey the same kind of precision. Thus, it is often useful to convert repeating decimals into fractions.
The preferred way to indicate a repeating decimal is to place a bar over the digits that repeat, for example 0. = 0.789789789… 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:
- 0. = 5/9
- 0. = 62/99
- 0. = 264/999
- 0. = 6291/9999
In case leading zero
Leading zero
A leading zero is any 0 digits, that lead a number string in a positional notation. For example, James Bond's famous identifier, 007, has two leading zeros. Leading zeros occupy most significant digits, which could be left blank or omitted for the same numeric value...
s precede the pattern, the nines are suffixed by the same number of trailing zero
Trailing zero
In mathematics, trailing zeros are a sequence of 0s in the decimal representation of a number, after which no other digits follow....
s:
- 0.0 = 5/90
- 0.000 = 392/999000
- 0.00 = 12/9900
In case a non-repeating set of decimals precede the pattern (such as 0.1523), we can write it as the sum of the non-repeating and repeating parts, respectively:
- 0.1523 + 0.0000
Then, convert the repeating part to a fraction:
- 0.1523 + 987/9990000
Continued fractions
A continued fraction is an expression such aswhere a_{i} are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s a_{i} can be determined by applying the Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
to (a,b).
Algebraic fractions
An algebraic fraction is the indicated quotientQuotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
of two algebraic expression
Algebraic expression
In mathematics, an algebraic expression is an expression that contains only algebraic numbers, variables and algebraic operations. Algebraic operations are addition, subtraction, multiplication, division and exponentiation with integral or fractional exponents...
s. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions.
If the numerator and the denominator are polynomial
Polynomial
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 integer exponents...
s, as in , the algebraic fraction is called a rational fraction (or rational expression). An irrational fraction is one that contains the variable under a fractional exponent, as in .
The terminology used to describe algebraic fractions is similar to that used for ordinary fractions. For example, an algebraic fraction is in lowest terms if the only factor common to the numerator and the denominator is 1. An algebraic fraction whose numerator or denominator, or both, contains a fraction, such as , is called a complex fraction.
Rational numbers are the quotient field of integers. Rational expressions are the quotient field of the polynomial
Polynomial
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 integer exponents...
s (over some integral domain). Since a coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
is a polynomial of degree zero, a radical expression such as √2/2 is a rational fraction. Another example (over the reals) is , the radian measure of a right angle.
The term partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
is used when decomposing rational expressions. 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 in many areas such as integral calculus and differential equations.
Rationalization of denominators
If the denominator contains an square rootSquare root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
, a cube root, etc., it can be helpful to rationalize
Rationalisation (mathematics)
In elementary algebra, root rationalisation is a process by which surds in the denominator of an irrational fraction are eliminated.These surds may be monomials or binomials involving square roots, in simple examples...
it, especially if further operations, such as adding or comparing that fraction to another, are to be carried out. It is also more convenient if division is to be done manually. When the denominator is a monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
square root, it can be rationalized by multiplying top and the bottom of the fraction by the denominator:
The process of rationalization of binomial
Binomial
In algebra, a binomial is a polynomial with two terms —the sum of two monomials—often bound by parenthesis or brackets when operated upon...
denominators involves multiplying the top and the bottom of a fraction by the conjugate
Conjugate (algebra)
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...
of the denominator so that the denominator becomes a rational number. For example:
Even if this process results in the numerator being irrational, 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.
Pedagogical tools
In primary schools, fractions have been demonstrated through Cuisenaire rodsCuisenaire rods
Cuisenaire rods give students a hands-on elementary school way to learn elementary math concepts, such as the four basic arithmetic operations and working with fractions....
, fraction bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboard
Geoboard
A geoboard is a mathematical manipulative often used to explore basic concepts in plane geometry such as perimeter, area or the characteristics of triangles and other polygons...
s, counters and computer software.
History
The earliest fractions were reciprocalsMultiplicative inverse
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...
of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s: ancient symbols representing one part of two, one part of three, one part of four, and so on. The Egyptians
History of Egypt
Egyptian history can be roughly divided into the following periods:*Prehistoric Egypt*Ancient Egypt**Early Dynastic Period of Egypt: 31st to 27th centuries BC**Old Kingdom of Egypt: 27th to 22nd centuries BC...
used Egyptian fractions ca. 1000 BC. About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fraction
Unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...
s. Their methods gave the same answer as modern methods.
The Greeks
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
used unit fractions and later continued fractions and followers of the Greek philosopher Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
, ca. 530 BC, discovered that the square root of two cannot be expressed as a fraction. In 150 BC Jain mathematicians in India
History of India
The history of India begins with evidence of human activity of Homo sapiens as long as 75,000 years ago, or with earlier hominids including Homo erectus from about 500,000 years ago. The Indus Valley Civilization, which spread and flourished in the northwestern part of the Indian subcontinent from...
wrote the "Sthananga Sutra
Sthananga Sutra
As per the Śvetāmbara belief, Sthananga Sutra forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala. This is the reason why, under the leadership of Devardhigani Ksamasramana, the eleven Angas of the Svetambara canon were...
", which contains work on the theory of numbers, arithmetical operations, operations with fractions.
In Sanskrit literature
Sanskrit literature
Literature in Sanskrit begins with the Vedas, and continues with the Sanskrit Epics of Iron Age India; the golden age of Classical Sanskrit literature dates to late Antiquity . Literary production saw a late bloom in the 11th century before declining after 1100 AD...
, 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
Bhaskara I
Bhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
writes
६ १ २
१ १ १_{०}
४ ५ ९
That is,
6 1 2
1 1 1_{०}
4 5 9
to denote 6+1/4, 1+1/5, and 2–1/9
Al-Hassār, a Muslim mathematician from Fez
Fez
Fez may refer to:*Fez , a brimless felt hat, once widespread in the Ottoman Empire*Fes, a city in Morocco**FEZ, the IATA code of Fes-Saïss Airport*Free Economic Zone*Fez , a painting by an American artist...
, Morocco
Morocco
Morocco , officially the Kingdom of Morocco , is a country located in North Africa. It has a population of more than 32 million and an area of 710,850 km², and also primarily administers the disputed region of the Western Sahara...
specializing in Islamic inheritance jurisprudence
Islamic inheritance jurisprudence
Islamic Inheritance jurisprudence is a field of Islamic Jurisprudence that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called Mīrāth, and its branch of Islamic law is technically known as ʿulm al-farāʾiḍ...
during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, ." This same fractional notation appears soon after in the work of Leonardo Fibonacci in the 13th century.
In discussing the origins of decimal fractions, Dirk Jan Struik
Dirk Jan Struik
Dirk Jan Struik was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.- Life :...
states that (p. 7):
"The introduction of decimal fractions as a common computational practice can be dated back to the FlemishFlemish RegionThe Flemish Region is one of the three official regions of the Kingdom of Belgium—alongside the Walloon Region and the Brussels-Capital Region. Colloquially, it is usually simply referred to as Flanders, of which it is the institutional iteration within the context of the Belgian political system...
pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon StevinSimon StevinSimon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...
(1548-1620), then settled in the Northern NetherlandsNetherlandsThe Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
. It is true that decimal fractions were used by the ChineseChinese mathematicsMathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....
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)."
While the Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
mathematician Jamshīd al-Kāshī
Jamshid al-Kashi
Ghiyāth al-Dīn Jamshīd Masʾūd al-Kāshī was a Persian astronomer and mathematician.-Biography:...
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 Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...
i mathematician Abu'l-Hasan al-Uqlidisi
Abu'l-Hasan al-Uqlidisi
Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician who was active in Damascus and Baghdad. As his surname indicates, he was a copyist of Euclid's works. He wrote the earliest surviving book on the positional use of the Arabic numerals, Kitab al-Fusul fi al-Hisab al-Hindi around 952...
as early as the 10th century.