Reciprocal Calculator

What is the Reciprocal of a Number?

The reciprocal of a number, also known as its multiplicative inverse, is simply 1 divided by that number ($1/x$ or $x^{-1}$). When any non-zero number is multiplied by its reciprocal, the result is always equal to 1:

$$x \times \frac{1}{x} = 1$$

Finding the reciprocal is one of the most fundamental operations in algebra, physics, and electronics. For example, dividing by a fraction is accomplished by multiplying by its reciprocal ("keep, change, flip"). In electrical engineering, the reciprocal of electrical resistance is electrical conductance ($G = 1 / R$, measured in Siemens).

How to Find the Reciprocal of Different Numbers

1. Reciprocal of a Whole Number

Any whole number $n$ can be written as the fraction $n / 1$. Flipping numerator and denominator yields:

$$\text{Reciprocal of } n = \frac{1}{n}$$

For instance, the reciprocal of 5 is $1/5 = 0.2$. The reciprocal of 8 is $1/8 = 0.125$.

2. Reciprocal of a Fraction

For any common fraction $a / b$, swap the top and bottom values:

$$\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}$$

For example, the reciprocal of $2/3$ is $3/2 = 1.5$ (or mixed number $1\frac{1}{2}$). If you want to convert fractions to ratios, use our Ratio to Fraction Calculator.

3. Reciprocal of a Mixed Number

To find the reciprocal of a mixed number like $1\frac{3}{4}$:

1. Convert to an improper fraction: $1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4}$.
2. Invert the fraction: $\frac{4}{7}$.

4. Reciprocal of Negative Numbers and Decimals

The reciprocal of a negative number is always negative: the reciprocal of $-4$ is $-1/4 = -0.25$. For decimals, write the decimal as a fraction (e.g. $0.75 = 3/4$) and invert it ($4/3 = 1.3333$).

Why Does Zero Have No Reciprocal?

Zero is the only real number that does not have a reciprocal. Attempting to compute the reciprocal of 0 requires dividing 1 by 0 ($1/0$), which is mathematically undefined. There is no number that can be multiplied by 0 to yield 1, because anything multiplied by 0 equals 0.