What is reciprocal? Explained with examples.

The reciprocal word is derived from the Latin word “reciprocus” which means returning. Hence if we take the reciprocal of the inverted term then we obtain an original value back. In Mathematics, when are say the reciprocal of a function or a number it means to take the inverse of that number or function.

If we have a number y then the reciprocal of y is 1/y. Remember that y is not equal to zero.

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Definition of Reciprocal

The reciprocal of any physical quantity is one over that quantity. The product of a number and its reciprocal are always unity. If y is a number and 1/y is its reciprocal then y * 1/y = 1.

 Generally, it is also called multiplicative inverse. The reciprocal of a number y is denoted as y^-1. By interchanging the numerator and denominator we find the reciprocal of a given number.

Kinds of reciprocal

1. Reciprocal of Zero

The rule of reciprocal does not apply to zero because it gives an undefined answer.

1/0 = Undefined

Except for 0, we can find the reciprocal of all real and complex numbers.

2. Reciprocal of a Fraction

The reciprocal of a fraction is just we have to interchange the numerator and denominator. if we have a fraction a/b then its reciprocal is b/a.

Example:

Evaluate for the reciprocal of 10/7.

Solution: We interchange its numerator and denominator

Then the reciprocal is 7/10.

Or using formula

Let y=10/7

The formula of reciprocal (y replaced with 1/y).

Put the value of y in the above formula

 1/[10/7] = 7/10 

3. Reciprocal of Mixed Fraction

For the reciprocal of a mixed fraction, we first convert it into a simple fraction by multiplying the denominator with the whole number and then adding the nominator to the result. After obtaining the simple fraction apply the formula of reciprocal.

Example:

Find the reciprocal of a mixed fraction 5(2/3).

Solution:

Step 1: Change into an Improper fraction.

5(2/3) = 17/3

Step 2: Use the formula now.

 1/[17/3] = 3/17 

The reciprocal examples can be solved easily with the help of a reciprocal calculator.

4. Reciprocal of Whole Number

When the given number is a whole number then the reciprocal of a number is obtained by writing the quantity over one. If y is a whole number, then its reciprocal is 1/y.

Example:

Find the reciprocal of a whole number 9.

Solution:

We just write it over 1.

1/9

So, 1/9 is the reciprocal of 9.

5. Reciprocal of a Negative Number

For the reciprocal of a negative number, we just take the reciprocal of a number without a negative sign then the negative sign is put along with the reciprocal of that number. For example, the inverse of –y is -1/y.

Example:

Find the reciprocal of -10.

Solution:

Step 1: First we have to write the given whole number in the improper form by putting denominator 1.

10/1

Step 2: Now interchange the numerator and denominator.

1/10.

Step 3: Add a minus sign (-) with the resultant value.

-1/10

So, the reciprocal of -10 is -1/10.

6. Reciprocal of Decimal Number

For the reciprocal of decimal numbers, we first convert it into a fraction if it is possible. Then change the nominator with the denominator. if it is not possible to convert into a fraction then we just put the decimal value over one.

Example:

Find the reciprocal of a decimal value of 1.562

Solution:

Method 1:

 By using the formula of reciprocal of a number y is 1/y.

Therefore, 1.562= 1/1.562

Now after simplification of 1/1.562 answer becomes 500/781.

Method 2:

Step 1: Change the given decimal into a fraction.

1.562=1562/1000

Step 2: Now apply the rule of reciprocal of decimal “Interchange the nominator with denominator”.

1000/1562

Step 3: Now simplify the given fraction 

500/781

So, the reciprocal of 1.562 is 500/781.

Note: we can check the reciprocal of a given number is right by multiplying the reciprocal with the original number answer always equal to 1.

7. Reciprocal of Algebraic Terms

We can also find the reciprocal of algebraic terms just by putting the given algebraic in the denominator of one.

Example:

Find the reciprocal of the algebraic term 2x^2 + 1.

Solution:

By using the rule of reciprocal

1/ (2x^2 + 1)

8. Reciprocal of a Function

The reciprocal of a function is just obtained by putting the function f(x) in the denominator of 1.

If f(x) is a function which is not equal to zero then the reciprocal of f(x) is written as 1/ f(x).

Example:

Find the reciprocal of a function f(x) = sin2x 

Solution:

Apply the rule of reciprocal of a function f(x) is 1/f(x).

→ 1/ sin2x 

Examples of Reciprocal

We go through some more examples of reciprocal.

Example 1:

Find the reciprocal of 2/[3/5]

Solution:

Step 1: Write the formula for reciprocal

1/y where y is the given number.

Step 2: Put the value of y in the formula

1/2/[3/5]

=1 * 3/(2*5)

=3/10

So, the reciprocal of 2/[3/5] is 3/10.

Example 2:

What is the reciprocal of the complex number 2+5i?

Solution:

Step 1: Apply the rule of reciprocal.

 1/[2+5i]

 The reciprocal of a complex number is similar to the real number.  

Importance of Reciprocal

Reciprocal is very useful when we divide a number with a fraction. Actually, we convert the division into multiplication by taking the reciprocal of the fraction after the division sign. We can understand it with an example.

Example:

Solve the given expression 2 / [4/3].

Solution:

Step 1: When a number is divided by a fraction, we take the reciprocal of the given fraction after the division sign.

The reciprocal of 4/3 is 3/4. 

Step 2: Multiply the given number to the reciprocal of 4/3.

2 * 3/4

Step 3: Simplify the given expression.

2 * 3/4 =3/2

Summary:

In this article, we have learned about the definition of reciprocal and various rules to find the best solution. Now you can find the reciprocal of any quantity after learning from this post.