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Reciprocal (multiplicative inverse) — definition, properties and examples

The reciprocal (multiplicative inverse) of a nonzero number is 1 divided by that number. This article explains definitions, arithmetic rules, examples, and generalizations in algebra and other systems.

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The reciprocal, also called the multiplicative inverse, of a nonzero number x is the number that when multiplied by x gives 1. In elementary notation the reciprocal of x is written as 1/x or x-1. The operation is undefined for zero because no number times zero equals 1. For a short technical overview see multiplicative inverse.

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Reciprocal function
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Basic properties and simple examples

Two numbers are reciprocals of each other precisely when their product equals 1. Examples commonly used in arithmetic include:

  • 2.5 and 0.4 are reciprocals because 2.5 × 0.4 = 1.
  • -0.2 and -5 are reciprocals because -0.2 × -5 = 1.
  • 1 and -1 are their own reciprocals: 1 × 1 = 1 and (-1) × (-1) = 1.

To find the reciprocal of a fraction, interchange numerator and denominator: the reciprocal of a/b (with a ≠ 0) is b/a. Whole numbers are treated as fractions with denominator 1, so the reciprocal of 8 is 1/8.

Rules for calculation and algebraic notation

Common rules involving reciprocals are used throughout arithmetic and algebra. Division by a number is the same as multiplication by its reciprocal: a ÷ b = a × (1/b) when b ≠ 0. Notationally, x-1 denotes the reciprocal of x in algebraic expressions. The reciprocal operation interacts predictably with signs and rationality: the reciprocal of a nonzero rational number is rational, and the reciprocal of a nonzero real number is another real number (except for zero, which has none).

Generalizations and context

The idea of a multiplicative inverse extends beyond ordinary numbers. In abstract algebra, elements that possess a multiplicative inverse in a ring or field are called units. Matrix inversion plays a similar role for square matrices: an invertible matrix has a multiplicative inverse under matrix multiplication, although this inverse is not obtained by taking elementwise reciprocals. The same inverse concept appears in complex numbers, rational functions, and other algebraic structures.

History, uses and notable facts

The reciprocal is one of the earliest arithmetic concepts used for division and proportional reasoning; it remains fundamental in solving equations, converting units, simplifying expressions, and in calculus where derivatives and integrals often involve reciprocal functions. Useful facts to remember: zero has no reciprocal; reciprocals preserve the property of being rational or irrational (a nonzero rational has a rational reciprocal, and a nonzero irrational has an irrational reciprocal); and reciprocals are central when converting division into multiplication to simplify calculations.

Further reading

Properties

The closer a number is to {\displaystyle 0}, the farther its reciprocal is from {\displaystyle 0}. The number {\displaystyle 0}itself has no reciprocal and is not a reciprocal. The reciprocal y=f(x)={\tfrac 1x}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 xis again x.If a quantity yis inversely proportional to a quantity x,then it is proportional to the reciprocal of x.

The reciprocal of a fraction, that is, the reciprocal of a quotient {\tfrac ab} with a,b\neq 0, is obtained by interchanging the numerator and denominator:

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

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 {\tfrac 1n}of a natural number nis called a root fraction.

Also, to every complex number {\displaystyle 0}different from z=a+b{\mathrm i}with real numbers a,bthere is a reciprocal {\tfrac {1}{z}}.With the absolute value |z|={\sqrt {a^{2}+b^{2}}}of zand the complex number zzconjugate to \overline {z}=a-b{\mathrm i}holds:

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

The graph of the reciprocal function is a hyperbola.
The graph of the reciprocal function is a hyperbola.

Examples

  • The reciprocal of 1is again 1.
  • The reciprocal of {\displaystyle 0{,}001}is 1000.
  • The reciprocal of 2is {\displaystyle {\tfrac {1}{2}}=0{,}5}.
  • The reciprocal of the fraction {\tfrac {2}{5}}is {\displaystyle {\tfrac {5}{2}}=2{\tfrac {1}{2}}=2{,}5}.
  • The inverse of the complex number 3+4{\mathrm i}is {\tfrac {1}{3+4{\mathrm i}}}={\tfrac {3}{25}}-{\tfrac {4}{25}}{\mathrm i}.

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