Fraction (mathematics)

In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. For example, each part of a cake split into four equal parts is called a quarter (and represented numerically as ); two quarters is half the cake, and eight quarters would make two cakes.

Mathematically, a fraction is a quotient of numbers, like , or more generally, an element of a field of fractions.

In our cake example above, where a quarter is represented numerically as , the bottom number, called the denominator, is the total number of equal parts making up the cake as a whole, and the top number, called the numerator, is the number of these parts we have. For example, the fraction represents three quarters.

The numerator and denominator may be separated by a slanting line called a solidus or slash, for example , or may be written above and below a horizontal line called a vinculum, like so: $\textstyle\frac{3}{4}$ .

The numerator and denominator are the "terms" of the fraction. The word "numerator" is related to the word "enumerate," meaning to "tell how many"; thus the numerator tells us how many parts we have in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the denominator tells us what kind of parts we have (halves, thirds, fourths, etc.). Note that because it is impossible to divide something into zero equal parts, zero can never be the denominator of a fraction.

The word is also used in related expressions, like 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), for example as opposed to 1.

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 if the absolute value of the numerator is greater than or equal to the absolute value of the denominator.

Mixed numbers

A mixed number is the sum of a whole number and a proper fraction. For instance, you could have two entire cakes and three quarters of another cake. The whole and fractional parts of the number are written right next to each other: 2 + = 2.

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

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

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

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

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

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a new fraction that is said to be equivalent to the original fraction. The word equivalent means that the two fractions have the same value. This is true because for any number n, multiplying by 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 as . To see this, imagine cutting the example cake into four pieces; two of the pieces together () make up half the cake ().

We can say, for example, that , , , 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. We call this reducing the fraction. A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be irreducible or in lowest terms. For instance, is not in lowest terms because both 3 and 9 can be evenly divided by 3. In contrast, is in lowest terms — the only number that's a factor of both 3 and 8 is 1.

Reciprocals and the "invisible denominator"

The reciprocal of a fraction is another fraction with the numerator and denominator swapped. 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 = . (We sometimes call the number 1 the "invisible denominator.") Therefore, we can say that, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be .

Arithmetic with fractions

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

Addition and subtraction

Adding fractions

Adding fractions can be tricky. The first rule of addition is that you can only add like quantities, and so, while it is easy to add halves and halves or thirds and thirds, you cannot add halves and thirds unless you find a way to make them like quantities.

The quickest way to add fractions is to multiply the denominators, and then change both fractions to equal fractions over that denominator. For example,

\textstyle{\frac12+\frac23=\frac36+\frac46=\frac76}

This always works, but sometimes there is a smaller denominator that will also work (a least common denominator). For example, to add + , we can use the denominator 48, but we could also use the smaller denominator 12, which is the least common multiple of 4 and 12.

\textstyle{\frac34+\frac{5}{12}=\frac{9}{12}+\frac{5}{12}=\frac{14}{12}=\frac76}

Subtracting fractions

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,

\textstyle{\frac23-\frac12=\frac46-\frac36=\frac16}

Multiplication and division

Multiplication

=By whole numbers

= If you consider the cake example above, if you have a quarter of the cake, and you multiple 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 seventh 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 seventh of a day is a whole day, 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of day. Numerically:

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

=By fractions

= If you consider 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

As another example, suppose that five people do an equal amount 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:

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=General rule

= You may have noticed that when we multiply fractions, we simply multiply the two numerators (the top numbers), and multiply the two denominators) (the bottom numbers). For example:

\textstyle

=By 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 ( \right ) = 3 \times {11 \over 4} = {33 \over 4} = 8{1 \over 4}}

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

Division

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

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

\textstyle

To understand why this works, consider that $\textstyle{\frac a b \div \frac c d = \frac{acd}{bcd} \div \frac c d = (\frac{ad}{bc} \times \frac c d) \div \frac c d = \frac{ad}{bc} \times (\frac c d \div \frac c d) = \frac{ad}{bc} \times 1 = \frac a b \times \frac d c}$ . Therefore, $\textstyle{\frac a b \div \frac c d = \frac a b \times \frac d c}$ .

About 4,000 years ago Egyptians divided with fractions using slightly different methods, using least common multiples with unit fractions. See for examples and an expanation.

Special cases

A unit fraction is a vulgar fraction with a numerator of 1 (¹/₇).

An Egyptian fraction is the sum of distinct unit fractions (¹/₃+¹/₅).

A decimal fraction is a vulgar fraction where the denominator is a power of 10 (⁴/₁₀₀). A dyadic fraction is a vulgar fraction in which the denominator is a power of two (¹/₈).

A compound fraction is a fraction where the numerator or denominator (or both) contain fractions, $\frac{\,\frac{2}{3}\,}{\,\frac{1}{5}\,}$ , these can be simplified to give vulgar fractions.

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" (an oxymoron). A common example is π/2, the radian measure of a right angle.

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 continued fraction is an expression such as $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}}$ , where the a_i are integers. This is not an element of a quotient field.

The term partial fraction is used in algebra, when decomposing rational functions. The goal of the method of partial fractions is to write rational functions as sums of other rational functions with denominators of lesser degree.

Pedagogical tools

In Primary Schools, fractions have been demonstrated through Cuisenaire rods.

History

See also: Egyptian fractions, history of irrational numbers.

The earliest known use of decimal fractions is ca. 2800 BC as Ancient Indus Valley units of measurement. 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.

References

External links

Fractions | Mathematical disambiguation | Elementary arithmetic | Numbers

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