MathsMultiplicative Inverse – Infinity Learn

Multiplicative Inverse – Infinity Learn

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.

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    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.

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    Multiplicative Inverse of a Mixed Fraction

    To find the multiplicative inverse of a mixed fraction, follow these steps:

    1. 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.
    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:

    1. 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.
    2. Swap the numerator and the denominator of the fraction. If the fraction is a/b, its multiplicative inverse is b/a.
    3. 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.


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    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 14, because 4 × 14 = 1.

    How do you find the multiplicative inverse of a fraction?

    To find the multiplicative inverse of a fraction ab, simply flip the numerator and the denominator. The multiplicative inverse of ab is ba

    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.

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