Definition:Reciprocal

Definition

Let $X$ be a number or an expression such that $X$ is not equal to, nor evaluates to, zero.

Then $\dfrac 1 X$ is called the reciprocal of $X$.

The real function $f: \R_{\ne 0} \to \R$ defined as:

$\forall x \in \R_{\ne 0}: \map f x = \dfrac 1 x$

is called the reciprocal function.

Examples

Reciprocal of $2$

The reciprocal of $2$ is $\dfrac 1 2$.

Reciprocal of $\pi$

$\dfrac 1 \pi \approx 0 \cdotp 31830 \, 98861 \, 83790 \, 67153 \, 77675 \, 26745 \, 02872 \, 40689 \, 19291 \, 480 \ldots$

Reciprocal of $e$

$\dfrac 1 e \approx 0 \cdotp 36787 \, 94411 \, 71442 \, 32159 \, 55237 \, 70161 \, 46086 \, 74458 \, 11131 \, 031 \ldots$

Reciprocal of $1 + x$

The reciprocal of $1 + x$ is $\dfrac 1 {1 + x}$.

This is defined whenever $x \ne -1$.

Warning

Note the domain of the function $f: \R \setminus \set 0 \to \R$.

That is, $\dfrac 1 0$ is not defined.

Also see

• Definition:Multiplicative Inverse of Number
• Definition:Harmonic Numbers
• Definition:Natural Logarithm

• Results about reciprocals can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): Glossary
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Glossary
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): reciprocal: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): reciprocal: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): reciprocal