fraccions

Fraccions are mathematical expressions that represent a part of a whole or a ratio between two quantities. They are written as a/b, where the numerator a is divided by the denominator b, which must be a nonzero integer. When the numerator and denominator are integers, a fraction denotes a rational number.

Fractions can be categorized as proper (numerator smaller than the denominator), improper (numerator equal to or larger than the denominator), or mixed numbers that combine an integer and a proper fraction. Two fractions are equivalent if they represent the same value; this can be checked by cross-multiplication. Simplification, or reduction, expresses a fraction in lowest terms by dividing the numerator and denominator by their greatest common divisor.

Arithmetic with fractions involves several rules. Addition and subtraction require a common denominator. Multiplication multiplies the numerators together and the denominators together. Division is performed by multiplying by the reciprocal of the divisor. After any operation, the result should be simplified if possible.

Fractions can be converted to decimals by performing the division a ÷ b, and to percentages by multiplying by 100. Finite decimal expansions occur for fractions whose denominators have only 2 and/or 5 as prime factors; others yield repeating decimals. Fractions are widely used in measurements, cooking, probability, statistics, and algebra, and they underpin many mathematical methods for solving equations and comparing quantities.

Historically, fractions have deep roots in various ancient cultures and have been formalized within modern mathematics to support precise arithmetic and analysis. In many languages, including Catalan, the term for fraction is fracció or fraccions, reflecting its fundamental role in mathematics.