Fraction (mathematics)

Categories: Mathematical disambiguation | Elementary arithmetic | Numbers | Fractions

Image:Cake-quarters.jpg
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 1⁄4 = 2⁄4) is equivalent to half (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 1⁄4); two quarters is half the cake, and eight quarters would make two cakes.

Mathematically, a fraction is a quotient of numbers, like 3⁄4, or more generally, an element of a quotient field.

In our cake example above, where a quarter is represented numerically as 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 3⁄4 represents three quarters.

The numerator and denominator are the terms of the fraction. The word "numerator" is related to the word "enumerate." To enumerate means to "tell how many"; thus the numerator tells us how many fractional 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.).

The word is also used in related expressions, like continued fraction, see Special cases below.

Contents

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Arithmetic

There are four basic arithmetic operations, which in order of simplicity for fractions, includes (1) Multiplication (2) Addition (3) Subtraction (4) Division.

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Multiplication

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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:

<math>3 \times {1 \over 4} = {3 \over 4}</math>

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:

<math>5 \times {3 \over 7} = {15 \over 7} = 2{1 \over 7}</math>
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By fractions

If you consider the cake example above, if you have a quarter of the cake, and you multiple 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:

<math>{1 \over 3} \times {1 \over 4} = {1 \over 12}</math>

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:

<math>{1 \over 5} \times {3 \over 7} = {3 \over 35}</math>
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General rule

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

<math>{5 \over 6} \times {7 \over 8} = {5 \times 7 \over 6 \times 8} = {35 \over 48}</math>
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By mixed whole number/fractions

If we are multiplying fractions that include a whole number component, then it is best to convert the whole number into a fraction. For example:

<math>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}</math>

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

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Commutativity

It is also worth recalling that multiplication is commutative which just means that the order of the numbers we are multiplying does not matter. In other words, three lots of a quarters is equivalent to a quarter of three; numerically:

<math>3 \times {1 \over 4} = {1 \over 4} \times 3 = {3 \over 4}</math>

Note that when talking, we say "three times a quarter", but "a quarter of three", the implication being that in the latter example, we are talking about a fractional part of a larger number.

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Special cases

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Particular vulgar fractions

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Other fractions

Let us end with the only example on this page where the "fraction" is not an element of a quotient field:

  • A continued fraction is an expression such as <math>a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}} </math>, where the ai are integers.

The term partial fraction is used in algebra, when decomposing rational functions. However, a partial fraction is an expression of a particular decomposition, and so is more than just an element of a quotient field.

The term irrational fraction is sometimes used to indicate a magnitude whose quotient with another fixed magnitude is irrational, e.g. "1 is an irrational fraction of 2π". "Fraction", in this sense, simply means "a part of the whole", not a strict ratio in the mathematical sense. Taking the latter meaning, the term is an oxymoron.

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Pedagogical tools

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

See also the external links below.

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See also

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External links

ca:Fracció cs:Zlomek da:Brøk de:Bruchrechnung es:Fracción eo:Frakcio (matematiko) fr:Fraction he:שבר (מתמטיקה) it:Frazione (matematica) ja:分数 nl:Breuk (rekenen) no:Brøk pl:Ułamek pt:Fração sl:Ulomek sv:Fraktion zh:分數