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The multiplicative inverse of a number is its reciprocal. In simple terms, the multiplicative inverse of a number is the number you multiply it by to get a product of 1. For example, the multiplicative inverse of 5 is 1/5 because 5 × 1/5 = 1.
The term “inverse” refers to something that counters the effect of the original. When applied to multiplication, it means multiplying a number by its multiplicative inverse will always result in 1.
In this article, we will study about multiplicative inverse of a number, its examples, properties and more.
What is the Multiplicative Inverse?
The multiplicative inverse of a number is the value you multiply the original number by to get a product of 1. It is often called the reciprocal of the number. If you have a number “a”, its multiplicative inverse is written as 1/a or a⁻¹.
Two numbers are considered multiplicative inverses of each other if their product is 1. For example, for the number 3, its multiplicative inverse is 1/3 because 3 × 1/3 = 1. Similarly, the multiplicative inverse of 110 is 1/110 because 110 × 1/110 = 1.
Multiplicative Inverse of Integers
The concept of the multiplicative inverse applies similarly to integers, both positive and negative.
- Natural Numbers: These are positive integers starting from 1. For a natural number “a”, its multiplicative inverse is 1/a. For instance, the multiplicative inverse of 5 is 1/5 because 5 × 1/5 = 1.
- Fractions: The reciprocal of a fraction like 2/3 is 3/2. Multiplying 2/3 by its reciprocal 3/2 results in 1.
- Negative Numbers: The multiplicative inverse of a negative number works the same way. For example, the multiplicative inverse of -4 is 1/-4 because -4 × 1/-4 = 1.
- Unit Fractions: A unit fraction is a fraction with a numerator of 1. To find the multiplicative inverse of a unit fraction 1/x, simply take x as the result. For example,
- The multiplicative inverse of 1/7 is 7 because 1/7 × 7 = 1.
- The multiplicative inverse of 1/50 is 50, since 1/50 x 50 = 1.
Multiplicative Inverse of a Mixed Fraction
To find the multiplicative inverse of a mixed fraction, follow these steps:
- Convert the Mixed Fraction to an Improper Fraction: First, transform the mixed fraction into an improper fraction. For example, for 3 1/2, convert it to 7/2.
- Find the Reciprocal: Determine the reciprocal of the improper fraction. For 7/2, the reciprocal is 2/7. So, Multiplicative Inverse of 3x 1/2, the reciprocal is 2/7 therefore, 7/2 x 2/7 = 1
Multiplicative Inverse of 0
The multiplicative inverse of a number is defined as the value that, when multiplied by the original number, gives out a product of 1. However, for the number 0, this concept does not apply.this is because multiplying 0 by any number always results in 0, not 1. Hence, there is no number that you can multiply by 0 to get a product of 1.
The concept of the multiplicative inverse involves division. The multiplicative inverse of 0 would be represented as 1/0. However, division by zero is undefined in mathematics because it does not result in a finite or meaningful number.
Multiplicative Inverse Property
The multiplicative inverse property is a fundamental concept in mathematics. It states that the product of a number and its reciprocal or multiplicative inverse is always equal to 1.
- If you have a number “n”, its multiplicative inverse is 1/n.
- According to the property, when you multiply “n” by its multiplicative inverse 1/n, the result will be 1. In other words, n x 1/n = 1.
For example: The multiplicative inverse of 4 is 1/4. Multiplying them together, we get 4 × 1/4 = 1.
How to Find the Multiplicative Inverse?
Finding the multiplicative inverse or reciprocal of a number is simple. Follow these steps:
- If the number is not already in fraction form, convert it to a fraction. For example, if the number is 5, express it as 5/1.
- Swap the numerator and the denominator of the fraction. If the fraction is a/b, its multiplicative inverse is b/a.
- Simplify the fraction if necessary.
Additive Inverse and Multiplicative Inverse
There are two properties of numbers: the additive inverse and the multiplicative inverse. These properties are related to addition and multiplication, respectively.
| Additive Inverse | Multiplicative Inverse |
| To find the additive inverse of a number, change the sign of the number. | To find the multiplicative inverse, take the reciprocal of the number. |
| The additive inverse is added to the original number to get 0. | The multiplicative inverse is multiplied by the original number to get 1. |
| For any number “n”, the equation for additive inverse is: n + (-n) = 0. | For any number “n”, the equation for multiplicative inverse is: n × 1/n = 1. |
Multiplicative Inverse: Solved Examples
Ques: Find the multiplicative inverse of 9.
Answer: The multiplicative inverse of 9 is 1/9.
Ques: Find the multiplicative inverse of 6 1/3.
Answer: The improper fraction of 6 1/3 is 19/3. Therefore, the multiplicative inverse of 6 1/3 is 3/19.
Ques: Find the multiplicative inverse of 187.
Answer: The multiplicative inverse of 187 is 1/187.
Multiplicative Inverse: Practice Questions
- Find the multiplicative inverse of 7/9.
- What is the multiplicative inverse of 12?
- Determine the multiplicative inverse of 4×1/2.
- Find the multiplicative inverse of the complex number 2 − 3i.
- Find the modular multiplicative inverse of 7 x 5/11.
Multiplicative Inverse FAQs
What is the multiplicative inverse of a number?
The multiplicative inverse of a number x is another number that, when multiplied by x, yields a product of 1. It is also known as the reciprocal of x. For example, the multiplicative inverse of 4 is 1⁄4, because 4 × 1⁄4 = 1.
How do you find the multiplicative inverse of a fraction?
To find the multiplicative inverse of a fraction a⁄b, simply flip the numerator and the denominator. The multiplicative inverse of a⁄b is b⁄a
Can the multiplicative inverse of 0 be defined?
No, the multiplicative inverse of 0 is not defined. Since multiplying 0 by any number always results in 0.
