Reciprocal Definition
The word reciprocal has its roots in Latin as the word “reciprocus” meaning “returning”. On taking reciprocal of an inverted number, it returns to the original number. The reciprocal of a given number when multiplied by that number gives one as a product. Thus, it is also called the multiplicative inverse.
In math, reciprocal can simply be defined as the inverse of a number or a value. For a real number n, reciprocal is 1/n, such as reciprocal of 3 is 1/3. Similarly, reciprocal of 5 is 1/5 and so on. What is the reciprocal of 0? What is the reciprocal of decimal number? Let's learn about it all in this lesson.
What is Reciprocal?
According to the reciprocal definition in math, the reciprocal of a number is defined 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 of each other. The reciprocal of a number is also defined as the division of 1 by that number.
The multiplicative inverse is a commonly-used reciprocal synonym. You may come across this term in the future. They mean the same.
How to Find the Reciprocal of a Number?
We are already familiar that the reciprocal of a number is the inverse of the given number and we can easily find it for any number by writing 1 over it. We can find the reciprocal of natural numbers, integers, fractions, decimals, as well as mixed fractions. Let's have a look at the table given below.
| Type | Reciprocal | Example |
|---|---|---|
| Natural Number x | 1/x | Reciprocal of 4 is 1/4 |
| Integer x, x≠0 | 1/x | Reciprocal of -4 is -1/4 |
| Fraction x/y, x,y≠0 | y/x | Reciprocal of 2/7 is 7/2 |
| Decimal (x) | 1/x | Reciprocal of 0.1 is 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 -ax2, reciprocal can be calculated and thus, reciprocal will be -1/ax2. The following steps can be followed to find the reciprocal of any negative number or variable:
- Step 1:For any negative number, make it an improper fraction by writing one below it as the denominator.
- Step 2: Interchange the numerator and denominator.
- Step 3: Add a negative sign to the resultant value.
For example, the reciprocal of -31 is -1/31.
Reciprocal of a Decimal Number
It's even easier to find the reciprocal of a decimal number. To find the reciprocal of a decimal number, just divide one by the decimal number or write one over the number.
For example, the reciprocal of 9.8 is 1/9.8
Reciprocal of a Fraction
All fractions consist of a numerator and a denominator. To find the reciprocal of a fraction, interchange the numerator and denominator. The resultant fraction is the required reciprocal. The reciprocal of a fraction, consisting of variables can be calculated in the same manner
For example, the reciprocal of 7/10 is 10/7 and similarly the reciprocal of a/b is b/a.
Reciprocal of a Mixed Number
A mixed fraction or a mixed number is a mixture of a whole number and a proper fraction. This can a mixture of numbers or even variables. Reciprocal of mixed fractions are always proper fractions. The following steps can be followed to find the reciprocal of a mixed number:
- Step 1: Convert the mixed number to an improper fraction.
- Step 2: Interchanging the numerator and denominator. The resultant fraction is the required reciprocal.
For example, the reciprocal of \(2\frac{1}{3}\):
- The improper form of \(3\frac{1}{2}\) is 7/2
- The reciprocal of 7/2 is 2/7.
Important Points
- The reciprocal of a number is also called its multiplicative inverse.
- The product of a number and its reciprocal is equal to 1.
- Reciprocal of a reciprocal gives the original number. For example, reciprocal of 5 is 1/5 and reciprocal of 1/5 is 5.
- Reciprocal of a number x is written as 1/x or x-1.
- Reciprocal of a mixed fraction can be obtained by converting the mixed fraction into an improper fraction and determining its reciprocal.
Solved Examples on Reciprocal
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Example 1: A pizza is sliced into 8 pieces. Tom eats 3 slices of the pizza and leaves the rest. Determine the reciprocal of the quantity of the pizza eaten by Tom.
Solution:
Since Tom ate 3 slices out of 8, it implies he ate 3/8 of the pizza. We need to determine the reciprocal of 3/8. Let's interchange the numerator and the denominator to obtain the reciprocal. The reciprocal of 3/8 is 8/3.
Therefore, 8/3 is reciprocal of the quantity of pizza eaten by Tom.
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Example 2: Daniel bought 4 apples from the market. He ate \(1\tfrac{1}{2}\) apples from the 4 he bought. Determine the reciprocal of \(1\tfrac{1}{2}\).
Solution:
To determine the reciprocal of \(1\tfrac{1}{2}\), convert the mixed fraction into an improper fraction. \(1\tfrac{1}{2}\)=3/2. Now, find the reciprocal of 3/2. Reciprocal of 3/2 is 2/3.
Therefore, 2/3 is reciprocal of \(1\tfrac{1}{2}\).
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Example 3: Find the reciprocal of -9/10. Verify your answer.
Solution:
The reciprocal of -9/10 is -10/9. To verify the answer, we will multiply -9/10 with its reciprocal and check if the product is 1, -9/10 x -10/9 = 1
Therefore, -10/9 is reciprocal of -9/10.
Practice Questions on Reciprocal
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FAQs on Reciprocal Definition
How do you Explain Reciprocal?
The reciprocal of a number is 1 divided by that number. It can also be explained as: when the product of two numbers is 1, they are said to be reciprocals of each other.
What is the Reciprocal of 1?
The reciprocal of 1 is 1. To determine the reciprocal of 1, divide 1 by 1. (1/1 = 1)
Does 0 have a Reciprocal?
No, 0 has no reciprocal. All numbers have reciprocals, except 0.
How to Find Reciprocal of a Mixed Number?
The following steps can be followed to find the reciprocal of a mixed number:
- Step 1: Convert the mixed number to an improper fraction.
- Step 2: Interchanging the numerator and denominator. The resultant fraction is the required reciprocal.
For Example, reciprocal of \(3\frac{1}{2}\) = 2/7.
What is the Difference Between Reciprocal and Inverse?
The inverse means opposite operation, such as addition and subtraction are inverse operations, whereas the reciprocal refers to the inverse of any value, also called the multiplicative inverse.
What is Another Word for Multiplicative Inverse?
Reciprocal is another word for the multiplicative inverse.
What is the Reciprocal of Infinity?
The reciprocal of infinity is zero (0). (1/∞=0)
What is the Reciprocal of a Decimal Number?
The reciprocal of a decimal is one divided by the decimal number or one over the number.
How to Find the Reciprocal of a Negative Number?
To find the reciprocal of a negative number follow the steps given below:
- Step 1: For any negative number, make it an improper fraction by writing one below it.
- Step 2: Swap or interchange the numerator and denominator.
- Step 3: Add a negative sign to the resultant value.
For example, reciprocal of -9 is -1/9.
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