Bruchform

Bruchform, or fraction form, is a way to express a rational number as a ratio of two integers, written as a/b with a and b integers and b not equal to zero. The value represents the quotient of the numerator and denominator. Fractions express exact values and are fundamental in algebra, arithmetic, and various applications.

Equivalence and simplification: Fractions a/b and c/d are equivalent if ad = bc. They can be simplified by dividing numerator and denominator by their greatest common divisor, yielding the fraction in lowest terms. Fractions can also be written as mixed numbers, such as 11/4 = 2 3/4.

Conversions: To convert a mixed number to an improper fraction, multiply the whole part by the denominator and add the numerator. To convert back, divide the numerator by the denominator; the quotient is the whole part and the remainder becomes the new numerator.

Operations: Addition and subtraction require a common denominator: a/b ± c/d = (ad ± bc) / bd. Multiplication uses straightforward multiplication: (a/b)·(c/d) = (ac)/(bd). Division by a fraction is multiplication by its reciprocal: (a/b) ÷ (c/d) = (a/b)·(d/c) provided c ≠ 0.

Notation and conventions: Negative fractions typically place the minus sign in the numerator or the denominator. The denominator cannot be zero. Fractions are often preferred over decimals in contexts requiring exact values, such as algebra and theoretical mathematics, while decimals may be convenient for measurements and approximate computations.

History and context: Fractions have ancient origins and have developed as a standard notation for ratios, proportions, and exact arithmetic across cultures and eras.