In Maths,
reciprocal
is simply defined as the changed of a value or a number. If n is a existent number, so its reciprocal will be 1/northward. It means that nosotros take to catechumen the number to the upsidedown grade. For case, the reciprocal of 9 is i divided by 9, i.e. 1/ix. At present, if we multiply a number by its reciprocal, information technology gives a value equal to one. It is also called
multiplicative changed
.
The word reciprocal came from the Latin discussion “reciprocus” meaning “returning”. Hence, it returns its original value, if we take the reciprocal of an inverted number. In this article, we are going to learn the definition of reciprocal, how to detect the reciprocal of numbers, fractions and decimals with many examples.

 Definition
 Reciprocal of a Number
 Reciprocal of a Negative Number
 Reciprocal of a Fraction
 Reciprocal of a Mixed Fraction
 Reciprocal of a Decimal
 Finding Unity
 Application of Reciprocal
 Rules for Reciprocal
 Examples
 Practice Questions
 FAQs
Definition
In
Mathematics, the reciprocal of any quantity is, one divided by that quantity. For any number ‘a’, the reciprocal will be 1/a. If the given number is multiplied past its reciprocal, we get the value 1.
Case: Reciprocal of a number 7 is 1/7.
Thus, if nosotros multiply vii and 1/vii, we get 1.
I.e., 7 × (1/7) = 1
Other Definitions of Reciprocal
It has many other definitions also :
 It is also called the
multiplicative inverse
.  It is like to turning the
number upside down.  It is also constitute byinterchanging 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,
i/x or x^{one}
for a number x.
Example: The reciprocals of 3 and 8 are 1/3 and 1 /8.
It is besides expressed by the number raised to the power of negative one and tin exist found for fractions and decimal numbers also.
In maths, when you take the reciprocal twice, you lot will get the aforementioned number that you started with.
Example: The reciprocal of 4 is i/4. When you repeat this step information technology becomes
4/1 or iv.
Thus, you get the aforementioned number where you started with.
Not For Zero
We cannot use the reciprocal condition on zero, since it will return an indefinite value.
1/0 = Undefined
Therefore, we can have a reciprocal for all real numbers but not for nil.
Reciprocal of a Number
It is divers as one over that number. In other words, the reciprocal of a number is defined as the one divided by the given number.
Instance:
Find the reciprocal of 5
Solution: To find the solution, we will use 10 = ane/x
Therefore, v= 1/v
The reciprocal of a function, f(x) = f(ane/x)
Reciprocal of Negative Numbers
For any negative number x, the reciprocal can be found by writing the inverse of the given number with a minus sign forth with that (i.east) i/x. For example, the reciprocal of – 4x
^{
2
}
is written as 1/4x
^{
2
}
. Become through the following steps to detect the reciprocal of the negative number.
Step ane:
For any negative number, write the given number in the class of an improper fraction by writing the number 1 in the denominator.
Step 2:
Now, interchange the numerator and denominator values.
Pace three:
Add a minus sign () to the resultant number.
Now, Consider a negative number, 17.
Step i: Convert the number 17 in the improper fraction. (i.e) 17/1.
Step 2: Interchanging the numerator and denominator value, we get 1/17.
Step 3: Finally, adding a negative sign to the resultant number, we get one/17.
Therefore, the reciprocal of 17 is 1/17.
Reciprocal of a Fraction
The reciprocal of a fraction can be plant by interchanging the numerator and the denominator values.
Example:
Find the reciprocal of ii / 3
Solution:
To observe the solution nosotros will follow the postobit steps
The reciprocal of 2/iii is iii/2. (or)
Apply the formula, x = 1/ten,
Here, 10 = two/3
Thus, x = 1 / x = one / (two/3)
= 3/two
Therefore, the reciprocal of a fraction 2/3 is iii/2.
Reciprocal of a Mixed Fraction
In order to observe the same for a mixed fraction, catechumen it into improper fractions and perform the functioning.
Consider a mixed fraction, 4(1/2).
The first step is to convert a mixed fraction into an improper fraction.
iv(1/2) = 9/2
Now, you get the fraction and practice the same operation for finding the reciprocal by flipping the numerator and the denominator.
Therefore, the solution for ix/two is 2/9.
Reciprocal of a Decimal
The reciprocal of a decimal is the same every bit it is for a number defined by one over the number.
Example:
Notice the reciprocal of a decimal 0.75
Solution:
The reciprocal of a number, x = 1/x
Therefore, 0.75= i/0.75
An alternate method to find information technology is given below.
Consider the aforementioned example, 0.75.
First, you have to check whether the given decimal number is possible for converting into a fractional number. Hither 0.75 is written as 3/four
Now, find the reciprocal of 3/4 which gives iv/three
When you verify both the solutions, information technology results in the same.
That is, 1/0.75 = i.33 and
four/3 = i.33
Finding Unity
If we multiply the reciprocal of a number past the number itself, nosotros will become the value equal to unity (1). Allow us see some examples here:
 2 × (one/2) = ane
 3 × (ane/3) = 1
 10 × (i/x) = one
 50 × (1/fifty) = one
 100 × (1/100) = ane
From the above examples, we can see that the multiplication of a number to its reciprocal gives 1. Hence, we can say that the number is multiplied past its reciprocal, we become i.
Application of Reciprocal
The near important awarding of reciprocal is used in division performance for fractions. If we desire to divide the first fraction by the 2nd fraction, the result can be found past multiplying the first fraction with the reciprocal of the second fraction.
For example, (2/v) ÷ (vii/5)
Here, the first fraction is 2/5
The 2d fraction is 7/5
Thus, the reciprocal of the second fraction is 5/seven
Hence, (ii/5) ÷ (7/5) = (2/5) × (5/7)
(2/5) ÷ (7/v) = ii/7.
Rules for Reciprocal
The two important rules for reciprocal are:
 For a number ten, the reciprocal will exist one/x or also can be written as x^{1}. For example, if 7 is the number, so the reciprocal will exist 1/7.
 For a fraction x/y, the reciprocal will exist y/x. For example, if 3/5 is the given fraction, so its reciprocal will be 5/3.
Solved Examples
Go through the below examples:
Example 1:
Detect the reciprocal of 2 and ix
Solution:
Given that, the two integers are 2 and 9
Therefore, the reciprocal of 2 is 1/two
The reciprocal of ix is 1/9
Example 2:
Determine the reciprocal of 3 / (ii/three)
Solution:
Given number 3/(⅔) is a fraction.
3 / (2/3) can be written every bit nine/2
i.e., iii/(⅔) = 9/two
Hence, the reciprocal of 9/2 is two/9.
Instance three:
Find the reciprocal of 5/4
Solution:
Given fraction is 5/iv
The reciprocal of 5/4 is 4/5.
Case 4:
Find the reciprocal of the decimal 0.25.
Solution:
Given decimal number = 0.25.
Hence, the reciprocal of 0.25 is one/0.25.
Alternate method:
The fraction equivalent to the decimal number 0.25 is 1/4.
Hence, the reciprocal of ¼ is iv/1.
Verification:
The resultant number obtained from both methods will effect in the same value.
(i.e.) 1/0.24 = iv or 4/one.
Example 5:
Detect the reciprocal of the negative number 45.
Solution:
Given that the negative number is 45.
Hence, the reciprocal of 45 is ane/45.
Practice Questions on Reciprocals
Find the reciprocal for the following numbers:
 29
 14/15
 i.25
 80

ax
^{ ii }
Frequently Asked Questions on Reciprocal
Define reciprocal.
The reciprocal is defined as the multiplicative inverse of a number. In other words, the reciprocal of a number is defined as i divided past that number. The product of a given number and its reciprocal will always give the value 1.
How to determine the reciprocal of a fraction?
The reciprocal of a fraction tin can be determined by interchanging the values of the numerator and denominator. For example, ¾ is a fraction. The reciprocal of ¾ is 4/3.
How to determine the reciprocal of the mixed fraction?
To find the reciprocal of the mixed fraction, commencement, convert the mixed fraction into the improper fraction, and so accept the reciprocal of the improper fraction. For instance, 2 ¾ is a mixed fraction. When information technology is converted to an improper fraction, we get 11/4. Hence, the reciprocal of 11/4 is 4/11.
What is the reciprocal of 0?
The number zero (0) does non have a reciprocal. Considering, if whatsoever reciprocal number is multiplied past 0, it will non give the product as 1. Information technology volition result in cipher.
What is the reciprocal of infinity?
The reciprocal of infinity is 1/∞, which is equal to nix. It means that 1/∞=0. Information technology is noted that the reciprocal of infinity is zero exactly, which means not minute.
For more information on finding reciprocals and other Maths concepts, visit BYJU’S – The Learning App and also get various Maths related videos to understand the concept in an easy and engaging mode.
What is the Reciprocal of 17
Source: https://byjus.com/maths/reciprocal/