This online Reciprocal Calculator finds the reciprocal of a decimal number or a fraction (proper, improper or mixed number). The numerator and the denominator of a fraction are separated by the slash symbol (/) with no spaces in between. It is important to leave space between the whole number and the fraction part of a mixed number.
If the reciprocal of a fraction is an improper fraction you can find its mixed number representation simply by clicking the ‘Calculate’ button again. If the reciprocal of a decimal is a repeating decimal the repetend is enclosed in parentheses.
What Is a Reciprocal
A reciprocal or multiplicative inverse of a number x, denoted by 1/x or x−1, is the number such that if multiplied by x the result is multiplicative identity, 1. For example, the reciprocal of 4 is one fourth (1/4 or 0.25), and the reciprocal of 0.5 is 1 divided by 0.5, or 2.
It is easy to understand that multiplying by a number is equal to dividing by its reciprocal and vice versa. This allows us to figure out how to find the reciprocal of a fraction.
The reciprocal of a proper or improper fraction can be retrieved by swapping its numerator and denominator. That is, the reciprocal of a fraction a/b is a fraction b/a. In the case of a mixed number, you first need to convert it to an improper fraction. For example, the reciprocal of 4/5 is 5/4 or 1 1/4 or 1.25, and the reciprocal of 2 1/3 = 7/3 is 3/7.
Our Reciprocal Calculator easily finds reciprocals for fractions and mixed numbers. If a reciprocal is an improper fraction you can find its representation as a mixed number simply by clicking the ‘Calculate’ button again or vice versa.
Repeating Decimals
Since finding the reciprocal of a real number is reduced to division, the result of such a division can be either a terminating decimal or a repeating decimal. If, when dividing, you end up with a remainder of zero, then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.
A repeating decimal, also called a recurring decimal, is a number whose decimal representation eventually becomes periodic. The repeating part of a decimal expansion, the repetend, is denoted using several notational conventions, in our calculator it is enclosed in parentheses, for example:
7 / 12 = 0.583333333333… = 0.58(3),
1 / 7 = 0.142857142857… = 0.(142857).
There is no limitation on the length of the resultant decimal, for this we use arbitrary-precision decimal arithmetic.
Complex Numbers
All calculations in our Reciprocal Calculator do not involve the use of complex numbers. To find the reciprocal of a complex number, you can use our Complex Numbers Calculator by simply dividing 1 by that complex number.
Examples:
Reciprocal of 3.6 is 0.2(7)
Reciprocal of 7.0 is 0.(142857)
Reciprocal of 7 is 1/7
Reciprocal of 8/5 is 5/8
Reciprocal of 1 4/5 is 5/9
Reciprocal of 5/7 is 7/5 or 1 2/5
Related calculators
Check out our other math calculators such as Convolution Calculator and Greatest Common Divisor Calculator.