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Fraction (mathematics)

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Image:Cake-quarters.jpg thumb|right|300px|A cake divided into four equal quarters. Each fraction of the cake is represented numerically as ¹⁄₄. It can be seen that two quarters (2 x {{Fraction|1|4}} = {{Fraction|2|4}}) is equivalent to half ({{Fraction|1|2}}) the cake. 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 {{Fraction|1|4}}); two quarters is half the cake, and eight quarters would make two cakes. Mathematically, a fraction is a quotient of numbers, like {{Fraction|3|4}}, or more generally, an element of a field of fractions. In our cake example above, where a quarter is represented numerically as {{Fraction|1|4}}, 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 {{Fraction|3|4}} represents three quarters. The numerator and denominator may be separated by a slanting line, or may be written above and below a horizontal line. 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 ''rational function algebraic fraction, see #Special cases Special cases below.

Forms of fractions

Proper and improper fractions
If the numerator and denominator of a fraction are both Negative and non-negative numbers positive, then the fraction is a '''proper fraction''' if the numerator is less than the denominator, but an '''improper fraction''' otherwise. If either the numerator or denominator (or both) are negative, their absolute values should be compared to determine whether the fraction is proper or improper.

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 + {{Fraction|3|4}} = 2{{Fraction|3|4}}. An improper fraction can be thought of as another way to write a mixed number; in the "2{{Fraction|3|4}}" example above, imagine that the two entire cakes are each divided into quarters. Each entire cake contributes {{Fraction|4|4}} to the total, so {{Fraction|4|4}} + {{Fraction|4|4}} + {{Fraction|3|4}} = {{Fraction|11|4}} is another way of writing 2{{Fraction|3|4}}. 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.^{#fn_1 1} The word '''equivalent''' means that the two fractions have the same value. For instance, consider the fraction {{Fraction|1|2}}. When the numerator and denominator are both multiplied by 2, the result is {{Fraction|2|4}}, which has the same value as {{Fraction|1|2}}. To see this, imagine cutting the example cake into four pieces; two of the pieces together ({{Fraction|2|4}}) make up half the cake ({{Fraction|1|2}}). We can say, for example, that {{Fraction|1|3}}, {{Fraction|2|6}}, {{Fraction|3|9}}, and {{Fraction|100|300}} 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 divisor factors in common (other than 1) is said to be '''irreducible''' or '''in lowest terms.''' For instance, {{Fraction|3|9}} is not in lowest terms because both 3 and 9 can be evenly divided by 3. In contrast, {{Fraction|3|8}} ''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 {{Fraction|3|7}}, for instance, is {{Fraction|7|3}}. 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 = {{Fraction|17|1}}. (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 {{Fraction|1|17}}.

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 1/2 + 2/3 = 3/6 + 4/6 = 7/6. This always works, but sometimes there is a smaller denominator that will also work (a least common denominator). For example, to add 3/4 + 5/12, 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. 3/4 + 5/12 = 9/12 + 5/12 = 14/12. (This answer can be reduced to 7/6.)

=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, 2/3 - 1/2 = 2/2 × 2/3 - 3/3 × 1/2 = 4/6 - 3/6 = 1/6.

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: :

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: :

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: :

{1 \over 3} \times {1 \over 4} = {1 \over 12}

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: :

{1 \over 5} \times {3 \over 7} = {3 \over 35}

==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: :

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

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

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}

In other words,

2{3 \over 4}

is the same as

\left ({{8 \over 4} + {3 \over 4}} \right )

, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total). And 33 quarters is

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. :

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

{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 that a/b divided by c/d equals acd/bcd divided by c/d which equals ad/bc multiplied by c/d divided by c/d. Since any number divided by itself is 1, we get ad/bc. About 4,000 years ago Egyptians divided with fractions using slightly different methods. Egyptians used the least common multiple technique to divide unit fractions. Examples can be found at http://egyptianmath.blogspot.com

Special cases
A '''vulgar fraction''' (or '''common fraction''') is a rational number written as one integer (the ''numerator'') division (mathematics) divided by a non-zero integer (the ''denominator''), for example ⁴/₃ as opposed to 1¹/₃. The line that separates the numerator and the denominator is called the vinculum if it is horizontal, a solidus if it is slanting. 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 exponentiation 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{2}{3}\Bigg /\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 might be 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. See also the external links below.

History
:''See also: Egyptian fractions, Irrational number#History history of irrational numbers.'' The earliest known use of decimal decimal fractions is ca. 2800 BC as Ancient Indus Valley units of measurement. The History of Egypt Egyptians used Egyptian fractions ca. 1000 BC. The Ancient Greece 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 Ancient India India wrote the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, operations with fractions.

Footnotes
* #fn_1_back 1.The reason belongs more in the section on multiplying fractions, but in essence, it's this: when I multiply the numerator and denominator of a fraction by 2, for example, I'm really multiplying the entire fraction by {{Fraction|2|2}}. But {{Fraction|2|2}} = 1, and multiplying any number by 1 results in the same number.

See also
*For basic arithmetic with fractions, see vulgar fraction *For other meanings of the word 'fraction', see fraction (disambiguation)

External links

- Curricula for Creating Fractions
- Curricula for Teaching about Fractions
- Teaching Fractions: New Methods, New Resources
- Worksheets: Identifying Fractions
- Worksheets: Improper Fractions to Mixed Numbers
- Curricula for Teaching about Equivalent Fractions
- Free online quizzes about Fractions
- ''Endless Examples & Exercises'' for fractions Category:Fractions * Category:Mathematical disambiguation Category:Elementary arithmetic Category:Numbers ca:Fracció cs:Zlomek da:Brøk de:Bruchrechnung es:Fracción eo:Frakcio (matematiko) fr:Fraction ko:분수 it:Frazione (matematica) he:שבר (מתמטיקה) nl:Breuk (rekenen) ja:分数 no:Brøk pl:Ułamek pt:Fração sl:Ulomek sv:Fraktion zh:分數

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