Multiplicative inverse
The reciprocal (also the reciprocal value or reciprocal) of a number other than is, in arithmetic, that number which, when multiplied by , gives the number ; it is notated as or .
Properties
The closer a number is to , the farther its reciprocal is from . The number itself has no reciprocal and is not a reciprocal. The reciprocal function described by (see figure) has a pole there. The reciprocal of a positive number is positive, the reciprocal of a negative number is negative. This finds its geometric expression in the fact that the graph decomposes into two hyperbolic branches, which lie in the first and third quadrants, respectively. The reciprocal function is an involution, that is, the reciprocal of the reciprocal of is again If a quantity is inversely proportional to a quantity then it is proportional to the reciprocal of
The reciprocal of a fraction, that is, the reciprocal of a quotient with is obtained by interchanging the numerator and denominator:
From this follows the calculation rule for dividing by a fraction: Dividing by a fraction is done by multiplying by its reciprocal. See also fraction calculation.
The inverse of a natural number is called a root fraction.
Also, to every complex number different from with real numbers there is a reciprocal With the absolute value of and the complex number zconjugate to holds:
The graph of the reciprocal function is a hyperbola.
Examples
- The reciprocal of is again .
- The reciprocal of is .
- The reciprocal of is .
- The reciprocal of the fraction is .
- The inverse of the complex number is .