Reciprocal Calculator

Calculate the reciprocal (1/x) of any number, fraction, or decimal with step-by-step verification.

Enter a Number

Enter any non-zero number. For fractions, enter numerator and denominator separately.

Result

Reciprocal of x
--
Input (x)--
Reciprocal (1/x)--
As Fraction--
As Decimal--
Verification: x * (1/x)--

Step-by-Step Solution

Reciprocal of x = 1/x

What is a Reciprocal?

The reciprocal of a number x is defined as 1/x, or equivalently, the number that when multiplied by x gives 1. Every non-zero number has a reciprocal. The reciprocal is also called the multiplicative inverse.

Reciprocal Rules

Integer Reciprocal

The reciprocal of an integer n is 1/n.

Reciprocal of 5 = 1/5 = 0.2

Fraction Reciprocal

Flip the numerator and denominator.

Reciprocal of 3/4 = 4/3

Decimal Reciprocal

Divide 1 by the decimal number.

Reciprocal of 0.25 = 1/0.25 = 4

Zero has no Reciprocal

Division by zero is undefined, so 0 has no reciprocal.

1/0 = undefined

Properties of Reciprocals

  • The reciprocal of the reciprocal of x is x: 1/(1/x) = x
  • The product of a number and its reciprocal is always 1: x * (1/x) = 1
  • The reciprocal of a negative number is negative: 1/(-x) = -1/x
  • The reciprocal of 1 is 1, and the reciprocal of -1 is -1.

Applications

Reciprocals are used in division (dividing by a number is the same as multiplying by its reciprocal), solving equations, physics (resistance in parallel circuits), and many other mathematical contexts. Understanding reciprocals is essential for working with fractions and algebraic expressions.