Reciprocal¶

Definition¶

The reciprocal of a nonzero rational number $x\text{,}$ written $x^{-1}\text{,}$ is the unique rational number satisfying

If $x=a/b$ with $a,b\in\mathbb{Z}\setminus\{0\}\text{,}$ then

The reciprocal of $0$ is undefined. The same definition applies to nonzero real numbers.

Interpretation¶

The reciprocal $x^{-1}$ undoes multiplication by $x$: multiplying by $x$ and then by $x^{-1}$ returns every number to where it started, because $x\,x^{-1}=1$$1$. It plays the same role for multiplication as the negation $-x$ plays for addition, and it makes division possible: $a/b=a\,b^{-1}$$a{b}^{-1}$.

The curve $y=1/x$ pairs each nonzero $x$ with its reciprocal. The points $(2,\tfrac{1}{2})$ and $(\tfrac{1}{2},2)$ show that taking the reciprocal twice returns the original number, and $(-1,-1)$ shows that a reciprocal has the same sign as $x$. There is no point above $x=0$$0$, because $0\cdot y=0\ne1$ for every $y$.

A common slip is to treat reciprocals like sums: $(x+y)^{-1}$ is not $x^{-1}+y^{-1}$$y^{-1}$, since $x=y=1$ gives $\tfrac{1}{2}\ne2$$2$. In Math II the reciprocal returns as an example of an inverse element.

Key Results¶

• The reciprocal of a nonzero number is unique.
• $(x^{-1})^{-1}=x$ for every $x\neq0\text{.}$
• $(xy)^{-1}=x^{-1}y^{-1}$ when $x$ and $y$ are nonzero.
• For positive numbers, taking reciprocals reverses order: if $0<x<y\text{,}$ then $x^{-1}>y^{-1}\text{.}$
• Zero has no reciprocal.

Examples¶

• The reciprocal of $2$ is $1/2\text{.}$
• The reciprocal of $-3/4$ is $-4/3\text{.}$
• The reciprocal of $1/5$ is $5\text{.}$

Prerequisites¶

• Rational Number
• Multiplication and Product

Related Concepts¶

• Rational Function
• Exponentiation and Power

Sources¶

• Math I: Calculus · Lecture 2

Aliases: multiplicative inverse Symbols: element of, integers

Last updated: 20 October 2026 03:02