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- or 3 ÷ 4
- or 3/4
- or 3⁄4
In this article, we will use the last of these notations. The first quantity, the number "on top of the fraction", is called the numerator, and the other number is called the denominator. The denominator can never be zero because division by zero is not defined. All vulgar fractions are rational numbers and, by definition, all rational numbers can be expressed as vulgar fractions.
To understand the meaning of any vulgar fraction consider some unit (e.g. a cake) divided into an equal number of parts (or slices). The number of slices into which the cake is divided is the denominator. The number of slices in consideration is the numerator. So: Were I to eat 2 slices of a cake divided into 8 equal slices then I would have eaten 2⁄8 (or two eighths) of the cake. Note that had the cake been divided into 4 slices and had I eaten one of those then I would have eaten the same amount of cake as before. Hence, 2⁄8 = 1⁄4. Had I eaten 1 and a half full cakes I would have eaten 12 of the one eighth of a cake slices or 12⁄8. If the cakes been divided into quarters I would have eaten 6⁄4 cakes. 12⁄8 = 6⁄4 = 3⁄2 = (1 + 1⁄2) cakes.
Several rules for calculation with fractions are useful:
If both the numerator and the denominator of a fraction are multiplied or divided by the same non-zero number, then the fraction does not change its value. For instance, 4⁄6 = 2⁄3 and 1⁄x = x / x2.
To add or subtract two fractions, you first need to change the two fractions so that they have a common denominator, for example the lowest common multiple of the denominators which is called the lowest common denominator; then you can add or subtract the numerators. For instance, 2⁄3 + 1⁄4 = 8⁄12 + 3⁄12 = 11⁄12.
To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. For instance, 2⁄3 × 1⁄4 = (2×1) / (3×4) = 2⁄12 = 1⁄6. It is helpful to read "2⁄3 × 1⁄4" as "two thirds of one quarter". If I took 2⁄3 of a cake and gave 1⁄4 of that part away, the part I gave away would be equivalent to 1⁄6 of a full cake.
Reciprocal of fractions. To take the reciprocal of fractions, simply swap the numerator and the denominator, so the reciprocal of 2⁄3 is 3⁄2. If the numerator is 1, i.e. the fraction is a unit fraction, then the reciprocal is an integer, namely the denominator, so the reciprocal of 1⁄3 is 3⁄1 or 3.
Dividing fractions. As dividing is the same as multiplying by the reciprocal, to divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. For instance, (2⁄3) / (4⁄5) = 2⁄3 × 5⁄4 = (2×5) / (3×4) = 10⁄12 = 5⁄6.
Other ways of writing fractions
Any rational number can be written as a vulgar fraction. If a fraction is greater than 1—i.e., the numerator is greater than the denominator—then it is known as an improper fraction. An example is 11⁄4, which is a little less than 3.
A fraction greater than 1 can also be written as a mixed number, i.e. as the sum of a positive integer and a fraction between 0 and 1 (sometimes called a proper fraction). For example
- 23⁄4 = 11⁄4.
This notation has the advantage that one can readily tell the approximate size of the fraction; it is rather dangerous however, because 23⁄4 risks being understood as 2×3⁄4, which would equal 3⁄2, rather than 2+3⁄4 . To indicate multiplication between an integer and a fraction, the fraction is instead put inside parentheses: 2 (3⁄4) = 2 × 3⁄4.
Numbers which are not integers can be written as decimals. For example
- 2.75 = 11⁄4.
The mark between the integer is a decimal point, though in many countries it is represented by a comma. If a number is a decimal fraction (i.e. its denominator only has 2 and/or 5 as its prime factors) then it can be written as a finite decimal. If it is rational then it can be written as recurring decimal, and if it is irrational, then as a non-recurring decimal with an infinite number of digits after the decimal point.
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