Reciprocal Calculator: One Over a Number
The reciprocal of a number, also called its multiplicative inverse, is simply one divided by that number. It is the value you multiply by to get back to one, the cornerstone of dividing by fractions and of countless physics and engineering relationships where a rate is the inverse of a quantity. This tool returns the reciprocal of any number you enter, along with the product check (which is always one for valid inputs) and the reciprocal of the reciprocal, which returns your original value. Zero is rejected because division by zero is undefined.
Reciprocal formula
Reciprocal = 1 / x
Product check = x * (1 / x) = 1
Reciprocal of reciprocal = 1 / (1 / x) = x
Here x is your input number. The reciprocal is defined for every nonzero number; zero has no reciprocal because division by zero is undefined.
How the reciprocal works
- The reciprocal of a number is one divided by that number.
- A number times its reciprocal always equals one, the defining property.
- The reciprocal of a fraction swaps its numerator and denominator.
- The reciprocal shares the sign of the original number, so negatives stay negative.
- Zero has no reciprocal, so this calculator returns n/a for a zero input.
Reciprocal: frequently asked questions
What is a reciprocal?
The reciprocal of a number is one divided by that number, also called the multiplicative inverse. The reciprocal of 4 is 1/4, which equals 0.25. Multiplying any nonzero number by its reciprocal always gives exactly 1, which is the defining property of the reciprocal.
What is the reciprocal of a fraction?
To find the reciprocal of a fraction, swap the numerator and the denominator. The reciprocal of 2/3 is 3/2, which equals 1.5. As a decimal you can also compute it directly as one divided by the decimal value of the fraction. This calculator accepts decimals, so enter the fraction as a decimal first.
Why does zero have no reciprocal?
Zero has no reciprocal because division by zero is undefined: there is no number that multiplies by zero to give 1, since anything times zero is zero. This calculator returns n/a when you enter zero, reflecting that the multiplicative inverse simply does not exist for zero.
What is the reciprocal of a negative number?
The reciprocal of a negative number is also negative. The reciprocal of -5 is -1/5, which equals -0.2, because a negative times a negative gives the positive product of 1 required by the definition. The sign of a reciprocal always matches the sign of the original number.
How is the reciprocal used in real calculations?
Reciprocals turn division into multiplication, which underlies dividing by fractions, computing rates such as resistance from conductance, and converting between quantities like focal length and optical power. The reciprocal of the reciprocal returns the original number, since the inverse of an inverse undoes itself.
Official sources
- National Institute of Standards and Technology: Digital Library of Mathematical Functions: notation and algebra.
- National Aeronautics and Space Administration: Mathematics reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.