Fraction (mathematics)

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  (Redirected from Denominator) For other uses, see Fraction. A cake with one quarter removed. The remaining three quarters are shown. A cake with one quarter removed. The remaining three quarters are shown.

In mathematics, a fraction (from the Latin fractus, broken) is a concept of a proportional relation between an object part and the object whole.

A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities.

A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus 34 represents three divided by four, in decimals 0.75, as a percentage 75%. The three equal parts of the cake are 75% of the whole cake.

Each fraction consists of a denominator (bottom) and a numerator (top), representing (respectively) the number of equal parts that an object is divided into, and the number of those parts indicated for the particular fraction. Fractions are rational numbers, which means that the denominator and the numerator are integers.

For example, the fraction 34 could be used to represent three equal parts of a whole object, were it divided into four equal parts. Because it is impossible to divide something into zero equal parts, zero can never be the denominator of a fraction (see division by zero). A fraction with equal numerator and denominator is equal to one (e.g. 55 = 1) and the fraction form is rarely, if ever, given as a final result.

In higher mathematics, a fraction is viewed as an element of a field of fractions.

Contents

Terminology

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

Writing fractions

\tfrac{3}{4}

The solidus may be omitted from the slanting style (e.g. 34) where space is short and the meaning is obvious from context, for example in road signs in some countries.

Usage

Fractions are used most often when the denominator is relatively small. It is easier to multiply 32 by 316 than to do the same calculation using the fraction's decimal equivalent (0.1875). It is also more accurate to multiply 15 by 13, for example, than it is to multiply 15 by a decimal approximation of one third. To change a fraction to a decimal, divide the numerator by the denominator, and round off to the desired accuracy.

The word is also used in related expressions, such as continued fraction and algebraic fraction—see Special cases below.

Forms of fractions

Vulgar, proper, and improper fractions

A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator).

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; but an improper fraction (US, British or Australian) or top-heavy fraction (British only) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. 97).

Mixed numbers

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

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, and the new denominator is the same as that of the fractional part of the mixed number.

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

  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

\tfrac{n}{n}\tfrac{1}{3}\tfrac{3}{9}

Reciprocals and the "invisible denominator"

\tfrac{3}{7}\tfrac{17}{1}

Complex fractions

\cfrac{\tfrac{1}{2}}{\tfrac{1}{3}}
\cfrac{\tfrac{1}{2}}{\tfrac{1}{3}}=\tfrac{1}{2}\times\tfrac{3}{1}=\tfrac{3}{2}.

Indeterminate Form

\tfrac{0}{0}

Arithmetic with fractions

Fractions, like whole numbers, obey the commutative, associative, and distributive laws, and the rule against division by zero.

Comparing fractions

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

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

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

\tfrac{5}{18}
\tfrac{5}{18}>\tfrac{4}{17}

In order to work with smaller numbers, the least common denominator is used instead of the product. The fractions are converted to fractions with the least common denominator, and then the numerators are compared.

\tfrac{5}{6}

Some standards-based mathematics texts such as Connected Mathematics omit instruction of least common denominators entirely. That text presents the use of "fraction strips" (a strip of paper folded into fractions) or "benchmark fractions" such as one-half against which a fraction such as two-fifths may be compared. While such methods may be useful to build conceptual understanding, they are controversial as they are not effective beyond the elementary school level, and such texts are often supplemented by teachers with the standard method.

Addition

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

\tfrac24+\tfrac34=\tfrac54=1\tfrac14
If  of a cake is to be added to  of a cake, the pieces need to be converted into comparable quantities, such as cake-eighths or cake-quarters. \tfrac12

Adding unlike quantities

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

\tfrac14\times\tfrac13=\tfrac1{12}

Consider adding the following two quantities:

\tfrac34+\tfrac23
\tfrac34\tfrac23

Now it can be seen that:

\tfrac34+\tfrac23

is equivalent to:

\tfrac9{12}+\tfrac8{12}=\tfrac{17}{12}=1\tfrac5{12}
\tfrac{3}{4}
\tfrac34+\tfrac{5}{12}=\tfrac{9}{12}+\tfrac{5}{12}=\tfrac{14}{12}=\tfrac76=1\tfrac16

Subtraction

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

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

Multiplication

When multiplying or dividing, it may be possible to choose to cancel down crosswise multiples that share a common factor. For example:

2 17 1 X 7 18 4 = 11 X 14. The following will explain how to complete this equation.


Whole numbers

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

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

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

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

Fractions

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

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

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

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

General rule

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

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

Mixed numbers

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

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

Division

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

\textstyle{5 \div {1 \over 2} = 5 \times {2 \over 1} = 5 \times 2 = 10}
\textstyle{{2 \over 3} \div {2 \over 5} = {2 \over 3} \times {5 \over 2} = {10 \over 6} = {5 \over 3}}

To understand why this works, consider the following:

6 inches divided by 3 inches = 2 means that we can divide 6 inches into two 3 inch parts.
6 miles divided by 3 miles = 2 means that we can divide 6 miles into two 3 mile parts.
6 half dollars divided by 3 half dollars = 2 means that we can divide 6 half dollars into two stacks of 3 half dollars each.
6/2 divided by 3/2 = 2 means that we can divide 6/2 into two parts, each 3/2 in size.

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

But what if fractions have different denominators?

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

\textstyle{{a \over b} \div {c \over d} = {ad \over bd} \div {bc \over bd} = {ad \over bc}.}
But this takes too long. Instead, we learn the rule "invert and multiply", which gives the same answer.
\textstyle{{a \over b} \div {c \over d} = {a \over b} \times {d \over c} = {ad \over bc}.}

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

\textstyle{{a \over b} \div {c \over d} = {ad \over bc}.}
We know that division is defined to be the inverse of multiplication. That is,
\textstyle{m \div n = q}
if and only if
\textstyle{n \times q = m}.
In the expression we want to prove, multiply the quotient by the divisor 
<img  alt=
Therefore,
\textstyle{{a \over b} \div {c \over d} = {ad \over bc}.}

About 4,000 years ago Egyptians divided with fractions using slightly different methods. They used least common multiples with unit fractions. Their methods gave the same answer that our modern methods give.

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

Rationalization of a fraction is a technique of estimation which has found later use in working with complex numbers. A fraction may need to be rationalized if the denominator contains any irrational numbers, imaginary numbers or complex numbers. The process of rationalization involves multiplying the top and the bottom of a fraction by its conjugate so that the denominator becomes a rational number. Even if this process results in the numerator being irrational or complex, the process may still facilitate algebra by reducing the number of irrationals one has to work with or putting the expression into a + bi form.

Special cases

\tfrac{1}{7}\tfrac{1}{2}+\tfrac{1}{3}\tfrac{1}{8}\textstyle{\frac{\pi}{2}}

Rational numbers are the quotient field of integers. Rational 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 (over some integral domain).

a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}} \textstyle{2x \over (x^2-1)}

Pedagogical tools

In primary schools, fractions have been demonstrated through Cuisenaire rods.

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

See also the external links below.

History

See also: History of irrational numbers

The earliest known use of decimal fractions is ca. 2800 BC as Ancient Indus Valley units of measurement.[citation needed] The Egyptians 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.

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.