Table of Contents

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

Reciprocal Calculator

**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.

**Additional Definitions of Reciprocal**

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 \(\frac{4}{3}\).