What Is a Reciprocal?
A reciprocal in mathematics is one divided by the number in question — also known as the multiplicative inverse. For any non-zero number x, the reciprocal is simply 1/x.
The key property of a reciprocal is that when you multiply a number by its reciprocal, you always get 1:
- x × (1/x) = 1
- Example: 5 × 1/5 = 5 × 0.2 = 1
The name "reciprocal" comes from Latin, possibly from the phrase reque proque, meaning "back and forth." The reciprocal of x may also be written as x⁻¹ — raising a number to the power of −1 is the same as finding its reciprocal.
How to Find the Reciprocal
Reciprocal of a Number
To find the reciprocal of a whole number, simply divide 1 by that number.
- Reciprocal of 7 → 1/7 ≈ 0.142857
- Reciprocal of 4 → 1/4 = 0.25
- Reciprocal of 100 → 1/100 = 0.01
Reciprocal of a Fraction
To find the reciprocal of a fraction, swap the numerator and denominator (the top and bottom of the fraction).
- Reciprocal of 3/4 → 4/3 ≈ 1.333
- Reciprocal of 1/2 → 2/1 = 2
- Reciprocal of 5/6 → 6/5 = 1.2
Reciprocal of a Decimal
To find the reciprocal of a decimal, divide 1 by the decimal number. Our calculator also converts the result to a simplified fraction.
- Reciprocal of 3.25 → 1/3.25 = 4/13 ≈ 0.3077
- Reciprocal of 0.5 → 1/0.5 = 2
- Reciprocal of 2.5 → 1/2.5 = 2/5 = 0.4
Examples: What Is the Reciprocal of 4 and 1/2?
Example 1: Reciprocal of 4
- Select Number as the input type.
- Enter 4 in the number field.
- Click Calculate.
- Result: reciprocal = 1/4 = 0.25
Example 2: Reciprocal of 1/2
- Select Fraction as the input type.
- Enter 1 as the numerator and 2 as the denominator.
- Click Calculate.
- Result: reciprocal = 2/1 = 2
US Measurement & Metric Rate Reciprocals
Reciprocals are particularly useful for inverting rates. Our calculator supports both US (imperial) and metric unit systems:
- US units: If a car travels at 60 miles/hour, the reciprocal rate is 1/60 hours/mile — telling you how long each mile takes.
- US fuel efficiency: 30 miles/gallon → reciprocal is 1/30 gallons/mile.
- Metric units: 5 km/L (fuel efficiency) → reciprocal is 0.2 L/km (fuel consumption).
- Metric speed: 80 km/h → reciprocal is 0.0125 h/km (time per km).
Enter the numerator and denominator units manually for full flexibility with any US or metric unit combination.
Currency Exchange Rate Reciprocals
Reciprocals are directly useful for converting exchange rates. If you know the rate from currency A to currency B, the reciprocal gives you the rate from B back to A:
- 1 USD = 0.92 EUR → reciprocal: 1 EUR = 1/0.92 ≈ 1.087 USD
- 1 GBP = 1.27 USD → reciprocal: 1 USD = 1/1.27 ≈ 0.787 GBP
- 1 EUR = 160 JPY → reciprocal: 1 JPY = 1/160 ≈ 0.00625 EUR
The calculator supports 36 world currencies including USD, EUR, GBP, JPY, CAD, AUD, CHF, CNY, INR, BRL, RUB, and more.
FAQs
What is the reciprocal of 5?
0.2. Since 5 = 5/1, the reciprocal is 1/5 = 0.2.
What is the reciprocal of 2/3?
3/2 = 1.5. Taking the reciprocal of a fraction means swapping numerator and denominator.
Does 0 have a reciprocal?
No. The reciprocal of 0 would be 1/0, which is undefined. Division by zero is not allowed in mathematics.
What is the reciprocal of a negative number?
The reciprocal of a negative number is also negative. For example, the reciprocal of −4 is −1/4 = −0.25.
What is the reciprocal of 1?
1. The number 1 is its own reciprocal since 1 × 1 = 1.
How is the reciprocal used in division?
Dividing by a number is the same as multiplying by its reciprocal. For example, 8 ÷ 2 = 8 × (1/2) = 4. This is the foundation of fraction division.