käänteisluvut

Käänteisluvut, or reciprocal numbers, refer to pairs of numbers that multiply to one. For a real number a ≠ 0, its reciprocal is denoted 1/a or a⁻¹. The reciprocal of a positive number remains positive, while the reciprocal of a negative number is negative. For fractions, the reciprocal is obtained by swapping numerator and denominator: the reciprocal of p/q (with q ≠ 0) is q/p. Reciprocal operations are foundational in algebra, particularly in solving equations, simplifying expressions, and working with proportions. Inverse functions and multiplicative identities crucially rely on reciprocals; for instance, the product a · a⁻¹ equals the multiplicative identity, 1. Reciprocals also appear in geometry, such as when determining the slope of a perpendicular line, where slopes multiply to –1. In trigonometry, reciprocal identities involve cosecant, secant, and cotangent functions, which are defined as 1/sin, 1/cos, and 1/tan, respectively. The concept extends to complex numbers, where the reciprocal of a nonzero complex number z is given by 1/z = conjugate(z)/|z|². In calculus, limits and derivatives occasionally involve reciprocals, exemplified by the derivative of 1/x being –1/x². Within the broader framework of field theory, the existence of multiplicative inverses for all nonzero elements is a defining property of fields. When working with zero, no reciprocal exists because no number multiplied by zero yields one, illustrating the special role of zero as a singular element in multiplicative structures.