# Reciprocal

The reciprocal value (also the reciprocal value or the reciprocal ) of a different number is in arithmetic the number which, when multiplied, results in the number ; it is noted as or . ${\ displaystyle 0}$ ${\ displaystyle x}$${\ displaystyle x}$ ${\ displaystyle 1}$${\ displaystyle {\ tfrac {1} {x}}}$${\ displaystyle x ^ {- 1}}$

## properties

The graph of the reciprocal function is a hyperbola .

The closer a number is to, the further away its reciprocal is from . The number itself has no reciprocal value and is also not a reciprocal value. 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 splits into two hyperbolic branches, which lie in the first and third quadrant. The reciprocal function is an involution , i.e. H. 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${\ displaystyle 0}$${\ displaystyle 0}$${\ displaystyle 0}$${\ displaystyle y = f (x) = {\ tfrac {1} {x}}}$${\ displaystyle x}$${\ displaystyle x.}$${\ displaystyle y}$ ${\ displaystyle x,}$${\ displaystyle x.}$

The downside break of a break , which is the reciprocal of a quotient with one obtained by the numerator and denominator interchanged: ${\ displaystyle {\ tfrac {a} {b}}}$${\ displaystyle a, b \ neq 0,}$

${\ displaystyle {\ frac {1} {\ frac {a} {b}}} = {\ frac {b} {a}}}$

From this follows the calculation rule for dividing by a fraction: A fraction is divided by multiplying by its reciprocal value. See also fractions .

The reciprocal of a natural number is called a stem fraction . ${\ displaystyle {\ tfrac {1} {n}}}$ ${\ displaystyle n}$

Also at any of various complex number with real numbers , there is a reciprocal With the absolute value of and to conjugate complex number applies: ${\ displaystyle 0}$ ${\ displaystyle z = a + b \ mathrm {i}}$${\ displaystyle a, b}$${\ displaystyle {\ tfrac {1} {z}}.}$ ${\ displaystyle | z | = {\ sqrt {a ^ {2} + b ^ {2}}}}$${\ displaystyle z}$${\ displaystyle z}$ ${\ displaystyle {\ overline {z}} = from \ mathrm {i}}$

${\ displaystyle {\ frac {1} {a + b \ mathrm {i}}} = {\ frac {1} {z}} = {\ frac {\ overline {z}} {z {\ overline {z} }}} = {\ frac {\ overline {z}} {| z | ^ {2}}} = {\ frac {ab \ mathrm {i}} {a ^ {2} + b ^ {2}}} = {\ frac {a} {a ^ {2} + b ^ {2}}} - {\ frac {b} {a ^ {2} + b ^ {2}}} \ mathrm {i}}$

## Examples

• The reciprocal of 1 is again 1.
• The reciprocal of 0.001 is 1000.
• The reciprocal of is${\ displaystyle 2}$${\ displaystyle {\ tfrac {1} {2}} = 0 {,} 5.}$
• The reciprocal of the fraction is${\ displaystyle {\ tfrac {2} {5}}}$${\ displaystyle {\ tfrac {5} {2}} = 2 {\ tfrac {1} {2}} = 2 {,} 5.}$
• The reciprocal of the complex number is .${\ displaystyle 3 + 4 \ mathrm {i}}$${\ displaystyle {\ tfrac {1} {3 + 4 \ mathrm {i}}} = {\ tfrac {3} {25}} - {\ tfrac {4} {25}} \ mathrm {i}}$

## generalization

A generalization of the reciprocal is the multiplicative inverse to a unit of a unitary ring . It is also defined by the property where denotes the one element of the ring. ${\ displaystyle x ^ {- 1}}$ ${\ displaystyle x}$${\ displaystyle x ^ {- 1} \ cdot \ x = x \ cdot \ x ^ {- 1} = 1}$${\ displaystyle 1}$

If it is z. If, for example, a ring of matrices is involved, the unit element is not the number but the unit matrix . Matrices for which there is no inverse matrix are called singular . ${\ displaystyle 1,}$