Fractions name equal parts—and every rational number
From Latin fractus, 'broken', a fraction tells how many equal pieces of a whole you have: one-half, three-quarters. The same a/b notation writes ratios, divisions, and the entire set of rational numbers, Q. Even the names are telling: numerator means counter, and denominator means namer.
A fraction names some number of equal parts, such as one-half, three-quarters, or eight-fifths. The same symbol 3/4 can stand for the ratio 3:4 of part to whole and for the division of three by four. Negative fractions express the opposite quantity: if 1/2 means a half-dollar profit, −1/2 means a half-dollar loss, and the sign rules for division make −1/2, (−1)/2, and 1/(−2) the same number.
The top number is the numerator, Latin for 'counter', and the bottom is the denominator, or 'namer', which says what kind of part is being counted; 8/5 means eight parts of the kind called fifths. The bar between them may be horizontal, slanted, or diagonal. English reads the denominator as an ordinal, pluralized when the numerator is not 1, as in two-fifths, though any fraction can also be read as 'two over five'. Hyphens matter: 'two-fifths' is a single quantity, while 'two fifths' suggests two separate fifths.
A simple, common, or vulgar fraction is a ratio a/b of integers with b not zero; the term was coined to set it apart from the sexagesimal fractions used in astronomy. A unit fraction has 1 on top, like 1/7, and a dyadic fraction has a power of two underneath, like 1/8.
Every rational number can be written as a/b with integers a and b and b nonzero, and the set of them is labelled Q, for quotient. The notation reaches beyond those numbers, though: √2/2 looks like a fraction but is not rational, and the rational fraction 1/x is not a number at all. A fraction p/q in lowest terms has a terminating expansion in base b exactly when q divides some power of b.
Source: Fraction
