Encyclopedia
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 ; 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: .
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 . 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 is a rational number written as one integer divided by a non-zero integer , 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 or
top heavy 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 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 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 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 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 = . Therefore, we can say that, except for zero, every fraction or whole number has a reciprocal. The reciprocal of 17 would be .
Complex Fractions
A complex fraction is a fraction in which either the numerator or denominator or both contain a fraction. For example, is a complex fraction. To solve a complex fraction, simply divide the numerator by the denominator, as in any other fraction.
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
The first rule of addition is that you can only add like quantities, for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below.
Adding like quantitiesImagine you have a pocket containing two quarter, and another pocket containing three quarters; in total, you have five quarters. Since four quarters is equivalent to one , this can be represented as follows:
.
Adding unlike quantitiesTo add fractions containing unlike quantities , it is necessary to convert all amounts to like quantities. It's easily to work out the type of fraction that we need to convert to, simply by multiplying together the two denominators of each fraction.
Suppose we are adding quarters to thirds, then we will convert both types of fraction to .
Consider adding the following two quantities:
.
First we convert into twelfths by multiplying both the numerator and denominator by three: . Note that is equivalent to , which shows that is equivalent to the resulting
Secondly, we convert into twelfths by multiplying both the numerator and denominator by four: . Note that is equivalent to , which shows that is equivalent to the resulting
Now we can see that:
is equivalent to:
.
This always works, but sometimes there is a smaller denominator that will also work . 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.
.
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,
.
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:
As another example, suppose that five people work for three hours out of a seven hour day . In total, they will have worked for 15 hours , 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:
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 , is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts . We can write this numerically as follows:
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:
General rule
You may have noticed that when we multiply fractions, we simply multiply the two
numerators , and multiply the two
denominators) . For example:
By mixed numbers
When multiplying mixed numbers, it's best to convert the whole part of the mixed number into a fraction. For example:
In other words, is the same as , making 11 quarters in total . And 33 quarters is 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.
To understand why this works, consider that . Therefore, .
About 4,000 years ago Egyptians divided with fractions using slightly
different methods, using least common multiples with unit fractions.
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 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 most repeating patterns, a simple division of the pattern by the same amount of nines as numbers it has will suffice. For example :
- 0.555… = 5/9
- 0.264264264… = 264/999
- 0.629162916291… = 6291/9999
In case zeros precede the pattern, the nines are suffixed by the same amount 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 , we must equate it as the sum of the non-repeating and repeating parts:
- 0.1523 + 0.0000987987987…
Then, convert both of these to fractions. Since the first part is not repeating, it is not converted according to the pattern given above:
- 1523/10000 + 987/9990000
We add these fractions by expressing both with a common divisor...
- 1521477/9990000 + 987/9990000
... and add them.
- 1522464/9990000
Finally, we simplify it:
- 31718/208125
Special cases
A
unit fraction is a vulgar fraction with a numerator of 1 .
An
Egyptian fraction is the sum of distinct unit fractions .
A
decimal fraction is a vulgar fraction where the denominator is a
power of 10 . A
dyadic fraction is a vulgar fraction in which the denominator is a power of two .
A
compound fraction is a fraction where the numerator or denominator contain fractions, , 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" . A common example is p/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 .
A
continued fraction is an expression such as , where the
ai 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.
See also the external links below.
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.
See also
- Rational number
- For other meanings of the word 'fraction', see fraction
References
External links
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