Reciprocal – Meaning and Examples | Multiplicative Inverse

Reciprocal is the inverse of a number or a value. It is also known as the multiplicative inverse. In other words, the reciprocal of a number is defined as 1 divided by that number. The word reciprocal comes from the Latin word reciprocus, which means “returning”. Every number has a reciprocal, except for 0 as 1/0 is undefined.

To find the reciprocal of a number, you can:

• Divide 1 by the number
• Raise the number to the power of -1
• Write the original number as a fraction and “flip” it

For example, the reciprocal of 9 is 1/9.

Reciprocal Meaning in Math

Reciprocal of a number in maths is the number which when multiplied with the original number gives the product 1. It is also called the Multiplicative Inverse of the number.

The reciprocal is a number for each given number, when multiplied by the given number results in 1 i.e., multiplicative identity. It is also called the Multiplicative Inverse of the number.

Mathematically, Reciprocals can be defined as:

For a number “a”, “b” is called it’s reciprocal if and only if b = 1/a.

For example, the reciprocal of 2 is 1/2, because when you multiply 2 by 1/2, the result is 1:

2 × (1/2) = 1

And similarly, the reciprocal of 1/3 is 3, because

1/3 × 3 = 1

How to Find the Reciprocal of a Number

To find the reciprocal of any number, we can use the following steps:

Step 1: Take the number given, and check whether it is 0 or not.

Step 2: If the number given is 0, then the reciprocal of doesn’t exist as division by 0 is not defined.

Step 3: If the number given is not 0, then calculate the reciprocal by changing the numerator with the denominator and the denominator with the numerator, as all the numbers can be written in the form of a/b where a and b both are real numbers. 

Step 4: Simplify the obtained number, if necessary (for the case of fractions and decimals).

Step 5: Verify the result by multiplying both the given number and obtained number, if the result of multiplication is 1 then the obtained number is the reciprocal of the given number.

Let’s consider an example for better understanding.

Example: Find the Reciprocal of the number 3.

Solution:

Step 1: Take the number given, which is 3.

Step 2: Since the number given is not 0, we proceed to the next step.

Step 3: To calculate the reciprocal, we interchange the numerator and denominator. In this case, the reciprocal of 3 is 1/3 [as 3 can be written as 3/1].

Step 4: The reciprocal, 1/3, is already in simplified form.

Step 5: To verify the result, we multiply the given number (3) with the obtained reciprocal (1/3).

3 × (1/3) = 1

The result of the multiplication is 1, which means that the obtained number (1/3) is indeed the reciprocal of the given number (3).

Therefore, the reciprocal of 3 is 1/3.

Reciprocal of a Fraction

To find the reciprocal of fractions, we interchange the numerator and denominator with each other in any given fraction.

For Example, let’s consider the fraction 3/4. Its reciprocal can be found by interchanging the numerator and denominator, which gives us 4/3. Some other examples of Reciprocal of Fractions are:

• Reciprocal of 2/5 is 5/2.
• Reciprocal of 1/3 is 3/1 (which simplifies to 3).
• Reciprocal of 7/8 is 8/7 and so on.

Reciprocal of Mixed Fractions

To find the reciprocal of a mixed fraction, we need to convert the mixed fraction into an improper fraction.

To convert the mixed fraction into an improper fraction, multiply the whole number with the denominator and add it to the numerator. This result becomes our new numerator and the denominator remains the same.

The resulting fraction is then inverted by swapping the numerator and denominator to find the reciprocal.

For example, let’s consider the mixed fraction [Tex]2\frac{1}{3} [/Tex].

To find its reciprocal, first, we need to convert it to an improper fraction:

 [Tex]2\frac{1}{3} [/Tex]= (2 × 3 + 1) / 3 = 7/3

Next, we invert the fraction by swapping the numerator and denominator, giving us the reciprocal:

Reciprocal of  [Tex]2\frac{1}{3} [/Tex] is 3/7.

Similarly, we can find the reciprocals of other mixed fractions:

• Reciprocal of [Tex]1\frac{2}{5} [/Tex] is 5/7.
• Reciprocal of [Tex]3\frac{4}{7} [/Tex] is 7/25.
• Reciprocal of [Tex]5\frac{3}{8} [/Tex] is 8/43 and so on.

Reciprocal of a Negative Number

To find the reciprocal of the negative number, we divide 1 by the negative number and simplify it further.

For Example, let’s consider a negative number, -4. Its reciprocal can be found by dividing 1 by -4, which gives us -1/4. Similarly, we can find the reciprocals of other negative numbers:

• Reciprocal of -11 is -1/11.
• Reciprocal of -13 is -1/13.
• Reciprocal of -15 is -1/15 and so on.

Reciprocal of Decimals

Reciprocals are generally associated with whole numbers, fractions, and negative numbers but we can also find reciprocal for decimal numbers. The process of finding the reciprocal remains the same but we need to simplify our result to find the Reciprocal of Decimals.

For example, let’s consider the decimal 0.5. Its reciprocal can be found by dividing 1 by 0.5, 

1/0.5 = 10/5 = 2

Thus, 2 is the reciprocal of 0.5.

Similarly, we can find the reciprocals of other decimal numbers:

• Reciprocal of 0.25 is 4.
• Reciprocal of 0.1 is 10.
• Reciprocal of 0.333. . . is 3 and so on.

Reciprocal of 0

To find the reciprocal of a non-zero number, you just divide 1 by that number. But the reciprocal of zero (0) is undefined because dividing by zero isn’t allowed in math. So, there’s no reciprocal for zero.

Reciprocal of 1

Reciprocal means the multiplicative inverse of a number. For example, the reciprocal of 1 is 1 itself, as 1 multiplied by 1 equals 1. So, the reciprocal of 1 is 1.

Reciprocal of Algebraic Expressions

Other than numbers, we can also find the reciprocals of algebraic expressions.

Let’s discuss the reciprocals of algebraic expressions in some detail :

Reciprocal of Monomials

Monomial is an algebraic expression consisting of a single term that term can be anything such as a coefficient, a variable, or a variable raised to any random power.

To find the reciprocal for any given monomial, we perform a similar operation to the number i.e., divide the number 1 with the given monomial.

For example, let’s consider the monomial 2x³. Its reciprocal can be found by dividing 1 by itself i.e.,

Reciprocal of 2x³ is 1/(2x³).

Some other examples of reciprocals of a monomial include,

• Reciprocal of 5y² is 1/(5y²).
• Reciprocal of -4z is 1/(-4z).
• Reciprocal of -2ab is 1/(-2ab) = -1/(2ab) and so on.

Reciprocal of Polynomials

If any algebraic expression that has non-negative powers has more than one term, then it is considered a polynomial. The expression is called a polynomial and similar to the monomial reciprocal of polynomials can be found using the same procedure.

For example, let’s consider a polynomial 2x² – 3x + 7. Thus, its reciprocal is 1/(2x² – 3x + 7). 

Similarly, we can find the reciprocals of other polynomials:

• Reciprocal of 2x³ + 3x² – 4x + 1 is 1/(2x³ + 3x² – 4x + 1).
• Reciprocal of 5x² – 2xy + 3y² is 1/(5x² – 2xy + 3y²).
• Reciprocal of 3a⁴  + 2a² b – ab²  + 1 is 1/(3a⁴  + 2a² b – ab²  + 1) and so on.

Application of Reciprocals

A significant application of the reciprocal is evident in the division of fractions. When dividing the first fraction by the second fraction, the quotient can be determined by multiplying the first fraction by the reciprocal of the second fraction.

For instance, Consider (3/4) ÷ (2/3)

In this case,

• First Fraction is 3/4
• Second Fraction is 2/3

Therefore, reciprocal of the second fraction is 3/2

Consequently, (3/4) ÷ (2/3) is equivalent to (3/4) × (3/2)

Resulting in (3/4) ÷ (2/3) = 9/8

Reciprocal Examples

Problem 1: 6 is a whole number so what will be the reciprocal of 6?

Solution:

We can write 6 as 6/1 so the reciprocal of 6 is 1/6

So this way by the multiplicative inverse property we can find the reciprocal of whole number.

Problem 2: Find the reciprocal of 65. 

Solution:    

The reciprocal of number or multiplicative inverse of 65 is 

We have property a = 1/a

So, the reciprocal of number 65 is 1/65

Problem 3: What is the multiplicative inverse of 5/2? 

Solution:

To find the multiplicative inverse of number,

The multiplicative inverse of a whole number: a/b is b/a  

So, 5/2 = a/b.  

         a = 5, b = 2

So now a/b = b/a

Multiplicative inverse of 5/2 is 2/5

Problem 4: Find the multiplicative inverse or reciprocal of -2/5.

Solution:

The multiplicative inverse of (-2/5 ) is (5/ -2)

Reciprocals Practice Problems

Here are some practice problems on reciprocals for you to solve:

1. What is the reciprocal of 4/13?

2. Find the reciprocal of 7/18.

3. Check if 5/1 is reciprocal of 13/65.

4. If the reciprocal of a number is -2/19 find the original number.

5. If the reciprocal of a number is 17/6, find the original number.

6. Find the reciprocal of 0.75.

7. Find the reciprocal of -10.

8. What is the reciprocal of 0.01?

9. Determine the reciprocal of 3.14.

10. What is the reciprocal of the variable a (assuming [Tex]a \neq 0[/Tex]))?

Multiplicative Inverse – FAQs

What is Reciprocal in Maths?

For a number A, B is called reciprocal of A if and only if

B = 1/A 

OR

AB = 1

How do you find the Reciprocal of a Number?

To find the reciprocal of a number, you need to divide 1 by the given number. For example, if you want to find the reciprocal of 4, you would calculate 1/4 which can be further simplified to 0.25.

What is the Reciprocal of a Positive Rational Number?

The reciprocal of a positive rational number is simply another positive rational number obtained by inverting the original number.

Can the Reciprocal of Zero be Defined?

No, the reciprocal of zero is undefined as division by zero is not defined i.e., 1/0 is not defined.

What is the Reciprocal of a Fraction?

Reciprocal of a Fraction is the fraction obtained by interchanging the numerator and denominator of the given fraction. For example, the reciprocal of 3/4 is 4/3.

What is the Reciprocal of a Negative Number?

Reciprocal of a Negative Number is a negative number obtained by dividing 1 by the given negative number. For example, the reciprocal of -2 is -1/2.

What is the Product of a Number and its Reciprocal?

The product of a number and its reciprocal is always 1. For example, if you multiply 5 by its reciprocal (1/5), the result is 1. This property holds true for any non-zero number.

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