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

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

Special cases

A vulgar fraction is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator). The line that separates the numerator and the denominator is called the vinculum. Rational numbers are the quotient field of integers.

Particular vulgar fractions

irreducible fraction: a vulgar fraction "in lowest terms", where the numerator is an integer, the denominator is a positive integer, and the highest common factor of the numerator and the denominator is 1;
proper fraction: a vulgar fraction with (absolute) value between 0 and 1;
improper fraction: a vulgar fraction with a (absolute) value greater than 1;
unit fraction: a vulgar fraction with a numerator of 1;
Egyptian fraction: the sum of distinct unit fractions;
decimal fraction: a vulgar fraction where the denominator is a power of 10;
dyadic fraction: a vulgar fraction in which the denominator is a power of two.

Other fractions

A mixed fraction: A mixed fraction is an integer plus a proper fraction.
A compound fraction is a fraction where the numerator or denominator (or both) contain fractions.
Rational functions are the quotient field of polynomials (over some integral domain).

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 $a_{0}+{\frac {1}{a_{1}+{\frac {1}{a_{2}+...}}}}$ , where the a_i 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.

Pedagogical tools

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

External links

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 === Other fractions ===
-Fractions which are rational numbers and could be written as vulgar fractions include:
 *A '''[[mixed fraction]]''': A mixed fraction is an integer plus a proper fraction.
 *A '''compound fraction''' is a fraction where the numerator or denominator (or both) contain fractions.
+*[[Rational function]]s are the [[quotient field]] of [[polynomial]]s (over some [[integral domain]]).
-Fractions which are '''not''' necessarily rational numbers include:
-* '''[[Partial fraction]]s''', used to decompose [[rational function]]s.
-* [[Rational function]]s are the [[quotient field]] of [[polynomial]]s (over some [[integral domain]]).
 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 ''a<sub>i</sub>'' are integers.
+The term '''[[partial fraction]]''' is used in algebra, when decomposing [[rational function]]s. 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 2&pi;". "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]].