Multiplicative inverse (Reciprocal)

Compute the multiplicative inverse (reciprocal) 1/x of a real number

Compute reciprocal

What is the reciprocal?

The reciprocal of a number x is 1/x. Multiplying a number by its reciprocal always yields 1: x · (1/x) = 1.

1 ÷ 8 = ?
Enter number
Any real number (except 0)
Calculation result
1/x =
Calculation: Reciprocal of x is 1 divided by x

Reciprocal Info

Properties

Reciprocal: Multiplicative inverse of a number

1/x x⁻¹ reciprocal

Rule: x · (1/x) = 1 for all x ≠ 0
Warning: Division by 0 is not defined

Quick examples
Reciprocal of 4: 1/4 = 0.25
Reciprocal of 0.5: 1/0.5 = 2
Reciprocal of -2: 1/(-2) = -0.5
Reciprocal of 0: not defined!


Formulas and properties of the reciprocal

Definition
\[\text{Reciprocal of } x = \frac{1}{x} = x^{-1}\]

Basic definition of the multiplicative inverse

Fundamental property
\[x \cdot \frac{1}{x} = 1 \quad (x \neq 0)\]

A number times its reciprocal is always 1

Reciprocal of reciprocal
\[\frac{1}{\frac{1}{x}} = x \quad (x \neq 0)\]

The reciprocal of the reciprocal is the original number

Reciprocal of negative numbers
\[\frac{1}{-x} = -\frac{1}{x} \quad (x \neq 0)\]

Sign rules for the reciprocal

Fraction rules
\[\frac{1}{\frac{a}{b}} = \frac{b}{a} \quad (a,b \neq 0)\]

Reciprocal of a fraction by swapping numerator and denominator

Undefined case
\[\frac{1}{0} = \text{not defined}\]

Division by zero is mathematically not allowed

Example calculations with reciprocals

Example 1: Positive number
x = 4
Reciprocal: 1/4 = 0.25
Verification: 4 × 0.25 = 1 ✓
\[\frac{1}{4} = 0{,}25\]
Example 2: Decimal number
x = 0.5
Reciprocal: 1/0.5 = 2
Verification: 0.5 × 2 = 1 ✓
\[\frac{1}{0{,}5} = \frac{1}{\frac{1}{2}} = 2\]
Example 3: Negative number
x = -2
Reciprocal: 1/(-2) = -0.5
Verification: (-2) × (-0.5) = 1 ✓
\[\frac{1}{-2} = -\frac{1}{2} = -0{,}5\]
Example 4: Zero (Undefined)
x = 0
Problem: Division by 0 not allowed
Result: Reciprocal does not exist
\[\frac{1}{0} = \text{undefined}\]
Practical applications
Fractions
Solving equations
Ratios
Physical units

The reciprocal is a fundamental concept in many areas of mathematics

Applications of the reciprocal

The reciprocal is a fundamental concept with many applications:

Fractions & algebra
  • Division as multiplication by reciprocal
  • Simplify and manipulate fractions
  • Solve equations (x/a = b → x = a·b)
  • Ratio calculations
Physics & engineering
  • Electrical resistance (R) and conductance (G = 1/R)
  • Frequency (f) and period (T = 1/f)
  • Speed and time
  • Optics: focal length and diopter
Economics & statistics
  • Price-performance ratios
  • Productivity measures
  • Unit conversions
  • Normalization and scaling
Mathematical foundations
  • Multiplicative inverses in fields
  • Matrix inversion
  • Function inversion
  • Complex numbers and conjugation

The reciprocal: Foundation of division and algebra

The reciprocal or multiplicative inverse of a number x is the number 1/x that when multiplied by x yields 1. This seemingly simple concept underpins division as "multiplication by the reciprocal" and is fundamental to algebra. The reciprocal connects arithmetic operations with algebraic structures and shows the deep relationship between multiplication and division.

Properties
  • x · (1/x) = 1 for all x ≠ 0
  • Reciprocal of reciprocal is x
  • Distributivity: 1/(ab) = (1/a)·(1/b)
  • Sign preservation rules
Significance
  • Basis of division a/b = a·(1/b)
  • Fraction arithmetic and ratios
  • Solve linear equations
  • Unit conversions
Special cases
  • Reciprocal of 0 not defined
  • Reciprocal of 1 is 1
  • Reciprocal of -1 is -1
  • For |x| < 1, |1/x| > 1
Summary

The reciprocal links elementary arithmetic with algebraic structures. The simple definition 1/x leads to fundamental insights about multiplicative structures and enables the elegant treatment of divisions as multiplications. From elementary arithmetic to abstract algebra, the reciprocal remains a central concept demonstrating how mathematical simplicity and depth go hand in hand.