# Reciprocal Calculator

The reciprocal of a fraction calculator is a free online tool that displays the reciprocal or multiplicative inverse of a number. With STUDYQUERIES’s online reciprocal calculator, you can do the calculations faster and see the multiplicative inverse of any number within a fraction of a second.

## How to Use the Reciprocal of a Fraction Calculator?

You can use a fraction calculator to calculate the reciprocal of a fraction by following the steps below:

• Step 1: Enter the fraction in the input box
• Step 2: Click “Solve” to get the multiplicative inverse
• Step 3: In the output field, you will see the reciprocal of the given fraction

## What is Reciprocal And Calculator?

The word reciprocal has its roots in Latin as the word $$\pmb{\color{red}{reciprocus}}$$ meaning $$\pmb{\color{red}{returning}}$$. It returns to the original number when you take the reciprocal of an inverted number. Multiplying the reciprocal of one number by another number gives one as a product. It is also known as the multiplicative inverse.

A reciprocal can simply be defined as the inverse of a number or value. For a real number $$n$$, reciprocal is $$\frac{1}{n}$$, such as reciprocal of $$3$$ is $$\frac{1}{3}$$. Similarly, the reciprocal of $$5$$ is $$\frac{1}{5}$$ and so on. What is the reciprocal of $$0$$? Can you tell me the reciprocal of decimal numbers? This article will explain it all.

### Definition In Maths

$$\pmb{\color{red}{According\ to\ the\ reciprocal\ definition\ in\ math,\ the\ reciprocal\ of\ a\ number\ is\ defined}}$$

$$\pmb{\color{red}{as\ the\ expression\ which\ when\ multiplied\ by\ the\ number\ gives\ the\ product\ as\ 1.}}$$

In other words, when the product of two numbers is 1, they are said to be reciprocals. Similarly, the reciprocal of a number is the division of 1 by the number.

The multiplicative inverse is a reciprocal synonym commonly used in mathematics. You may see the term in the future. They have the same meaning.

There are many other definitions as well:

• It is also known as the multiplicative inverse.
• In other words, it is like turning the number upside down.
• It can also be found by switching the numerator and denominator.
• All the numbers have reciprocal except $$0$$.
• The product of a number and its reciprocal is equal to $$1$$.
• Generally, reciprocal is written as, $$\frac{1}{x}$$ or $$x^{-1}$$ for a number $$x$$.

### How to Find the Reciprocal of a Number?

We know that the reciprocal of a number is the inverse of the given number, and we can easily find it by writing 1 over any number. We can find the reciprocal of natural numbers, integers, fractions, decimals, and mixed fractions. Let’s have a look at the examples given below.

• Natural Number: Reciprocal of $$x$$ is $$\frac{1}{x}$$, e.g.-Reciprocal of $$8$$ is $$\frac{1}{8}$$
• Integer: Reciprocal of $$x$$, $$x\neq 0$$ is $$\frac{1}{x}$$, e.g.- Reciprocal of $$-2$$ is $$\frac{-1}{2}$$
• Fraction: Reciprocal of $$\frac{x}{y}$$, $$x,y\neq 0$$, is $$\frac{y}{x}$$, e.g.- Reciprocal of $$\frac{7}{5}$$ is $$\frac{5}{7}$$
• Decimal: Reciprocal of $$(x)$$, is $$\frac{1}{x}$$, e.g.- Reciprocal of $$0.1$$ is $$\frac{1}{0.1}$$

### Reciprocal of a Negative Number

For any negative number $$-n$$, reciprocal will be its inverse with a minus sign with it. Also, for variable terms, such as $$-ax^3$$, reciprocal can be calculated, and thus, reciprocal will be $$\frac{-1}{ax^3}$$. You can find the reciprocal of any negative number or variable by following the steps below:

• Step 1: Make any negative number an improper fraction by writing one below it as the denominator.
• Step 2: Reverse the numerator and denominator.
• Step 3: Add a negative sign to the result.

For example, the reciprocal of $$-21$$ is $$\frac{-1}{21}$$.

Read Also: Inverse Property: Definition, Uses & Examples

### Reciprocal of a Decimal Number

The reciprocal of a decimal number is even easier to find. Divide one by a decimal number or write one over a decimal number to find its reciprocal.

For example, the reciprocal of $$8.9$$ is $$\frac{1}{8.9}$$

### Reciprocal of a Fraction

A fraction consists of a numerator and a denominator. Exchange the numerator and denominator of a fraction to find its reciprocal. The resulting fraction is reciprocal. Calculate the reciprocal of a fraction consisting of variables in the same way

For example, the reciprocal of $$\frac{9}{10}$$ is $$\frac{10}{9}$$ and similarly the reciprocal of $$\frac{a}{b}$$ is $$\frac{b}{a}$$.

### Reciprocal of a Mixed Number

In mathematics, a mixed fraction or mixed number is a combination of a whole number and a proper fraction. It can also be a combination of numbers or variables. The reciprocal of a mixed fraction is always a proper fraction. You can find the reciprocal of a mixed number by following these steps:

• Step 1: Convert the mixed number into an improper fraction.
• Step 2: Interchanging the numerator and denominator. That fraction is reciprocal.

For example, the reciprocal of $$2{\frac{2}{3}}$$

The improper form of $$2{\frac{2}{3}}$$ is $$\frac{8}{3}$$

The reciprocal of $$\frac{8}{3}$$ is $$\frac{3}{8}$$.

### Finding Unity

If we multiply the reciprocal of a number by the number itself, we will get the value equal to unity $$1$$. Let us see some examples here:

$$\pmb {\color{red}{{Number}\times{It’s\ reciprocal}=1}}$$

$$4 \times \frac{1}{4} = 1$$

$$13 \times \frac{1}{13} = 1$$

$$0.1 \times \frac{1}{0.1} = 1$$

$$\frac{2}{7} \times \frac{7}{2} = 1$$

From the above examples, we can see that the multiplication of a number to its reciprocal gives $$1$$. Therefore, it defines reciprocal as a value to be multiplied by another value to get $$1$$.

### Important Points To Remember

• A reciprocal number is also known as its multiplicative inverse.
• The product of a number and its reciprocal is equal to $$1$$.
• The reciprocal of a reciprocal gives the original number. For example, the reciprocal of $$7$$ is $$\frac{1}{7}$$, and the reciprocal of $$\frac{1}{7}$$ is $$7$$.
• The reciprocal of a number $$x$$ is written as $$\frac{1}{x}$$ or $$x^{-1}$$.
• You can find the reciprocal of a mixed fraction by converting it into an improper fraction and determining its reciprocal.

## FAQs

How to determine the reciprocal of a fraction?

The reciprocal of a fraction can be determined by interchanging the values of the numerator and denominator. For example, $$\frac{3}{7}$$ is a fraction. The reciprocal of $$\frac{3}{7}$$ is $$\frac{7}{3}$$.

How to determine the reciprocal of the mixed fraction?

To find the reciprocal of the mixed fraction, first, convert the mixed fraction into the improper fraction, and then take the reciprocal of the improper fraction. For example, $$2\frac{3}{7}$$ is a mixed fraction. When it is converted to an improper fraction, we get $$\frac{17}{3}$$. Hence, the reciprocal of $$\frac{17}{3}$$ is $$\frac{3}{17}$$.

What is the reciprocal of 0?

The number zero $$0$$ does not have a reciprocal. Because, if any reciprocal number is multiplied by 0, it will not give the product as 1. It will result in zero.

What is the reciprocal of infinity?

The reciprocal of infinity is zero (0). It means that $$\frac{1}{\infty}$$=0. It is noted that the reciprocal of infinity is zero exactly, which means not infinitesimal.

How do you find a reciprocal?

To find the reciprocal of a fraction, switch the numerator and the denominator (the top and bottom of the fraction, respectively). So, simply speaking, the reciprocal of $$\frac{a}{b}$$ is $$\frac{b}{a}$$. To find the reciprocal of a number, divide 1 by the number.

What is a reciprocal of 3?

$$3\times \frac{1}{3}=1$$. so the reciprocal of $$3$$ is $$\frac{1}{3}$$ (and the reciprocal of $$\frac{1}{3}$$ is $$3$$.)

What is the reciprocal of 8?

The reciprocal of $$8$$ is $$1$$ divided by $$8$$, i.e. $$\frac{1}{8}$$.

What is the reciprocal of $$\frac{2}{3}$$?

The reciprocal of $$\frac{2}{3}$$ is $$\frac{3}{2}$$. The product of $$\frac{2}{3}$$ and it’s reciprocal $$\frac{3}{2}$$ is $$1$$.

What is the reciprocal of $$\frac{3}{5}$$ as a fraction?

To find the reciprocal of a fraction, interchange the numerator and denominator. Hence, the reciprocal of $$\frac{3}{5}$$ is $$\frac{5}{3}$$.

What is the reciprocal of $$\frac{3}{4}$$ as a fraction?

The reciprocal of $$\frac{3}{4}$$ is $$\frac{4}{3}$$.