Primary
Fraction ○꠹｜Definition｜1st｜20251119205401-00-⌔
Fraction
A fraction (from Latin: fractus, “broken”) represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A simple fraction (examples: 1/2 and 17/3 ) consists of an integer numerator, displayed above a line (or before a slash like 1 ⁄ 2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4 , the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
Fractions can be used to represent ratios and division.1 Thus the fraction 3/4 can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).
Negative fractions 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 , −1/2 and 1/−2 all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive, −1/−2 represents positive one-half.
In mathematics a rational number is a number that can be represented by a fraction of the form a/b , where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol or Q, which stands for quotient. The term fraction and the notation a/b can also be used for mathematical expressions that do not represent a rational number (for example ), or even do not represent any number (for example the rational fraction ).
A rational number, expressed as where p and q are coprime integers and is in base , has a terminating representation in base if and only if q divides a power of b, or ,for some and some integer > 0.
By cross multiplying, the equality is equivalent to . Because q doesn’t divide p, it must divide , and the expansion will not continue.
Printed 2026-06-28.
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Link to original Footnotes
H. Wu, “The Mis-Education of Mathematics Teachers”, Notices of the American Mathematical Society, Volume 58, Issue 03 (March 2011), p. 374. Archived 2017-08-20 at the Wayback Machine. ↩
Secondary
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