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For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a×b=a×c, then the multiplicative term a can be canceled out if a≠0, resulting in the equivalent expression b=c; this is equivalent to dividing through by a.
By the commutative law, the middle two terms cancel: = leaving (+) = The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring.
We can also write negative fractions, which represent the opposite of a positive fraction. For example, if 1 / 2 represents a half-dollar profit, then − 1 / 2 represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), − 1 / 2 ...
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32 / 13 = 2 / 1 and its inverse are the only solutions in base 4 with two digits. [2]An example of anomalous cancellation with more than two digits is 165 / 462 = 15 / 42 , and an example with different numbers of digits is 98 / 392 = 8 / 32 .
An irrational fraction is one that contains the variable under a fractional exponent. [12] An example of an irrational fraction is / / /. The process of transforming an irrational fraction to a rational fraction is known as rationalization.
A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that the last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented ...
Where numerator and denominator can each be expressed in one word, a fraction is usually spelled out (e.g. a two-thirds majority; moved one-quarter mile); use figures if a fraction appears with a unit symbol (e.g. 1 ⁄ 4 mi (markup: {{frac|1|4}} mi), not a quarter of a mi or one-quarter mi).