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Reciprocal or multiplicative inverse of a number simply refers to the expression which when multiplied by the original number always gives the product as 1. For any real number ‘n’, the reciprocal will be ‘1/n’ as the product of ‘n’ and ‘1/n’ is 1. The reciprocal of a number can be easily found for any number by writing 1 over it. Here, the exchange of position of numerator and denominator takes place to obtain the reciprocal. We also can find the reciprocal of natural numbers, positive and negative integers, fractions, decimals and mixed fractions. For example, the reciprocal of the numbers 7 is 1/7, 2.8 is 1 /2.8 and so on.
Key Terms: Reciprocals, Multiplicative inverse, reciprocal of a negative number, reciprocal of a mixed number.
Reciprocal
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Reciprocal of a number refers to an expression which when multiplied by that number gives the product as 1. Therefore, if the product of the two numbers is 1, they are said to be reciprocals of each other. The reciprocals are also known as Multiplicative Inverse.
Reciprocal
Thus, on taking the reciprocal of an inverted number it always returns to the original number. Generally, for a number ‘n’ the reciprocal will be written as ‘1/n’. Here, the exchange of position of numerator and denominator takes place.
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Finding Reciprocal of a Number
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Reciprocal of a number is the multiplicative inverse of the said number and it can be easily found for any number by writing 1 over it. We also can find the reciprocal of natural numbers, positive and negative integers, fractions, decimals and mixed fractions. Here is a table of reciprocals given below-
| Type | Reciprocal | Examples |
|---|---|---|
| Natural Number (n) | 1/n | Reciprocal of 5 is 1/5 |
| Integer n, n is not equal to 0. | 1/n | Reciprocal of -5 id -1/5 |
| Fraction n/m, n,m are not equal to 0 | m/n | Reciprocal 3/5 is 5/3 |
| Decimal (n) | 1/n | Reciprocal 0.5 is 1/0.5 |
Also Read: Complex numbers and Quadratic equations
Reciprocal of Negative Number
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For any negative number -n, reciprocal will be its inverse with the minus sign intact with it. For the variable terms, such as -ax2, the reciprocal can be written as -1/ ax2. The following steps can be taken to find the reciprocal of a negative number or a variable:
- For a negative number, an improper fraction can be made by putting 1 below it as the denominator.
- Interchange the numerator and the denominator.
- Add a negative sign to the resultant.
For example, the reciprocal of -99 is -1/99.
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Reciprocal of a Fraction
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Fractions consist of a denominator and a numerator. To find the reciprocal of a fraction the numerator and denominator should be exchanged. The resultant fraction will be the required reciprocal. The reciprocal of the fractions containing variables can be computed in the same fashion.
Reciprocal of Fraction
For instance, the reciprocal of 25/33 is 33/25 and in a similar way, the reciprocal of n/m is m/n.
Reciprocal of a Decimal Number
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Reciprocal of a decimal number can be found in a similar way. To find the reciprocal of a decimal number, 1 can be written over the number or 1 can be divided by the decimal number.
For example, the reciprocal of 7.8 is 1/7.8.
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Reciprocal of a Mixed Number
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Mixed fraction is a mixture of a whole number and a proper fraction. It can also be a mixture of two variables as well. Reciprocal of mixed fractions are proper fractions as well. Here are the steps that can be taken to find the reciprocal of a mixed number:
- The mixed number has to be converted to an improper fraction.
- The numerator and denominator are to be exchanged. The resultant fraction is the required reciprocal.
For instance, let’s take a reciprocal of 5⅓:
Here, the improper form of 5⅓ is 16/3.
Now, the reciprocal of 16/3 is 3/16.
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Things to Remember
- Reciprocal of a number refers to an expression which when multiplied by that number gives the product as 1.
- The reciprocals are also known as Multiplicative Inverse.
- Reciprocal of a number ‘n’ can be written as ‘1/n’ or ‘n-1’
- Reciprocal of a mixed fraction can be obtained by converting the mixed fraction into an improper fraction first and then determining its reciprocal.
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Sample Questions
Ques: What is a reciprocal? (2 marks)
Ans: Reciprocal of a number refers to an expression which when multiplied by that number gives the product as 1. For example, the reciprocal of 5 is 1/5.
Ques: What is the reciprocal of the number ‘0’? (2 marks)
Ans: The number 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 always result in 0.
Ques: is there any difference between reciprocal and inverse? (2 marks)
Ans: Generally, the term ‘inverse’ means opposite operation, such as addition and such as addition and subtraction or multiplication and division are the inverse operations, whereas the reciprocal refers to the inverse of any value when multiplied by that number gives the product as 1.
Ques: What is the reciprocal of Infinity? (2 marks)
Ans: The reciprocal of infinity is 1/∞, which is equal to zero (0). Thus, 1/∞=0. Therefore, the reciprocal of infinity is zero exactly. It means it is not infinitesimal.
Ques: What is the Reciprocal of a Decimal Number? (2 marks)
Ans: To find the reciprocal of a decimal number, 1 can be written over the number or 1 can be divided by the decimal number.
For example, the reciprocal of 6.9 is 1/6.9
Ques: What is the reciprocal of a mixed fraction? (2 marks)
Ans: A mixed fraction is a mixture of a whole number and a proper fraction. Here, the mixed fraction has to be converted to an improper fraction first. Then numerator and denominator are to be exchanged. The resultant fraction is the required reciprocal of the given mixed fraction.
let’s take a reciprocal of 5
⅓ :Here, the improper form of 5
⅓ is 16/3.Now, the reciprocal of 16/3 is 3/16.
Ques: Find out the reciprocal of the number -54. (2 marks)
Ans: The reciprocal of -54 is -1/54.
Ques: A pizza is sliced into 8 equal pieces. Rohan eats 5 slices of the pizza and leaves the rest. Determine the reciprocal of the quantity of the pizza eaten by Rohan. (2 marks)
Ans: Since Rohan ate 5 slices out of 8, it can be said that he ate 5/8 of the pizza. Now the reciprocal of 5/8 has to be determined here. We can do it by interchanging the numerator and denominator to obtain the reciprocal. Thus, the reciprocal of 5/8 becomes 8/5.
Therefore, 8/5 is the reciprocal of the pizza eaten by Rohan.
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