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Multiplicative Inverse of Decimal Numbers – Definition & Property (With Examples)

multiplicative inverse

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In mathematics, there is a term called ‘inverse’. In ordinary language, the word ‘inverse’ means something opposite. What could be the meaning of the word ‘multiplicative inverse’?

The term multiplicative inverse is related to the term multiplicative identity.  The multiplicative inverse of a number is a number that when multiplied by the number gives a multiplicative identity.

For example, multiplicative inverse of $2$ is $\frac {1}{2}$, since $2 \times \frac {1}{2} = 1$.

What is Multiplicative Inverse?

The multiplicative inverse of a number is its opposite number. If a number is multiplied to its multiplicative inverse, the product of both the numbers becomes $1$. 

The simple rule to get a multiplicative inverse of a number is to write its reciprocal.

For example, the multiplicative inverse of $5$ is $\frac {1}{5}$ and the multiplicative inverse of $\frac {2}{3}$ is $\frac {3}{2}$. 

Observe that $5 \times \frac {1}{5} = 1$ and $\frac {2}{3} \times \frac {3}{2} = 1$. For more on multiplication check Multiplication and Division of Fractions.

Note: $1$ is a multiplicative identity of a number.

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Multiplicative Inverse Property

The multiplicative inverse property of numbers states that when a number is multiplied by its multiplicative inverse, the product obtained is a multiplicative identity, i.e., $1$.

Mathematically it is represented as: For any number $a$, if $a \times b = 1$, then $b$ is the multiplicative inverse of $a$.

For example, $15 \times \frac {1}{15} = 1$, therefore, $\frac {1}{15}$ is multiplicative inverse of $15$.

Let’s find out which sets of numbers have a multiplicative inverse.

Multiplicative Inverse of Natural Numbers

A set of natural numbers is a set of numbers that starts with $1$ and moves on to $2$, $3$, $4$, and so on. That is $N = \{1, 2, 3, 4, …\}$.

The set of natural numbers does not contain the numbers of the form $\frac {p}{q}$, therefore, the numbers in the set of natural numbers do not have a multiplicative inverse.

For a set of natural numbers, the properties – closure property, commutative property, associative propertydistributive property, and multiplicative identity property hold, but additive identity property, additive inverse property, and multiplicative inverse property do not hold.

Difference Between Infinity & Not Defined

Multiplicative Inverse of Whole Numbers

A set of whole numbers is a set of numbers that starts with $0$ and moves on to $1$, $2$, $3$, and so on. That is $W = \{0, 1, 2, 3, …\}$.

The set of whole numbers does not contain the numbers of the form $\frac {p}{q}$, therefore, the numbers in the set of whole numbers do not have a multiplicative inverse.

For a set of whole numbers, the properties – closure property, commutative property, associative propertydistributive propertyadditive identity property, and multiplicative identity property hold but, additive inverse property, and multiplicative inverse property do not hold.

Multiplicative Inverse of Integers

A set of integers is a set of numbers that contains all the natural numbers $1$, $2$, $3$, … and their corresponding negative values $-1$, $-2$, $-3$, …, alongwith $0$, ie., $Z = \{…, -3, -2, -1, 0, 1, 2, 3, … \}$.

The set of integers does not contain the numbers of the form $\frac {p}{q}$, therefore, the numbers in the set of integers do not have a multiplicative inverse.

For a set of integers, the properties – closure property, commutative property, associative propertydistributive property, additive identity property, multiplicative identity property, and additive inverse property hold but, the multiplicative inverse property does not hold.

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Multiplicative Inverse of Rational Numbers

Rational numbers are the numbers that can be expressed in the form $\frac {p}{q}$, where both $p$ and $q$ are integers and $q \ne 0$.

As the set of rational numbers contains the numbers of the form $\frac {p}{q}$, and for every number $\frac {p}{q}$, there exists a number $\frac {p}{q}$, therefore, the set of rational numbers has a multiplicative inverse for every number.

For a set of rational numbers, the properties – closure property, commutative property, associative propertydistributive propertyadditive identity propertymultiplicative identity property,  additive inverse property, and multiplicative inverse property hold.

Multiplicative Inverse of Irrational Numbers

Irrational numbers are the numbers that when expressed in decimal form have non-terminating and non-recurring decimal places.

The multiplicative inverse of any irrational number will be a reciprocal of a number, therefore the set of an irrational number will have a multiplicative inverse for every irrational number.

For a set of irrational numbers, the properties – commutative property, associative propertydistributive property, additive inverse property, and multiplicative inverse property holds, whereas, closure property, additive identity property, and multiplicative identity property do not hold.

6 Amazing Facts About Numbers

Multiplicative Inverse of Real Number

A set of real numbers is a set consisting of all the sets – natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Therefore, the set of real numbers will have a multiplicative inverse for every real number.

For a set of real numbers, all the properties – closure property, commutative property, associative propertydistributive property, additive identity property, multiplicative identity property, additive inverse property, and multiplicative inverse property holds.

Difference Between Multiplicative Inverse and Additive Inverse

Following are the differences between multiplicative inverse and additive inverse.  

Multiplicative InverseAdditive Inverse
To find the multiplicative inverse of a number, you have to take its reciprocalTo find the additive inverse of a number, you have to just change the sign of the number
The formula for finding the multiplicative inverse of a number $a$ is $\frac {1|{a}$ The formula for finding the additive inverse of a number $a$ is $-a$
When a multiplicative inverse is multiplied by a number the result is $1$When an additive inverse is added to a number the result is $0$

Conclusion

A multiplicative inverse of a number is a number that when multiplied with the number results in a multiplicative identity. Some sets have multiplicative inverse for the numbers whereas some sets do not have multiplicative inverse for the numbers.

Practice Problems

  1. Which of the following sets have multiplicative inverse for the numbers?

Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers

  1. Which of the following sets do not have multiplicative inverse for the numbers?

Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers

  1. Find the multiplicative inverse of
    • $56$
    • $-92$
    • $13.65$
    • $-0.05$
    • $\frac {2}{7}$
    • $-\frac {56}{23}$
    • $7 \frac {2}{9}$
    • $\sqrt{5}$
    • $11 + \sqrt{5}$
    • $-5 – \sqrt{7}$

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FAQs

What is the meaning of multiplicative inverse?

The multiplicative inverse of a number is its opposite number. If a number is multiplied to its multiplicative inverse, the product of both the numbers becomes $1$.

What is the difference between reciprocal and multiplicative inverse?

The reciprocal and multiplicative inverse is the same thing. When you multiply a number by its reciprocal, the product is $1$ and when you multiply a number by its reciprocal then also the product is $1$.

How to find the multiplicative inverse?

The formula to find the multiplicative inverse of a number $a$ is $\frac {1}{a}$.

For example, the multiplicative inverse of $4$ is $\frac {1}{4}$ and the multiplicative inverse of $\frac {1}{7}$ is $\frac {1}{\frac {1}{7}} = 7$ and that of $\frac {2}{3}$ is $\frac {1}{\frac {2}{3}} = \frac {3}{2}$.

What is the multiplicative inverse of a rational number?

The multiplicative inverse of any number is its reciprocal. Therefore, the multiplicative inverse of a rational number of the form $\frac {p}{q}$, where $q \ne 0$ is $\frac {q}{p}$.

What is the multiplicative inverse property?

The multiplicative inverse property states that the product of a number and its multiplicative inverse is always $1$. For example, $3 \times \frac {1}{3} = 1$.

What is the multiplicative inverse of $0$?

Since the multiplicative inverse of any number $a$ is its reciprocal, i.e., $\frac {1}{a}$, and as $\frac {1}{0}$ is undefined, therefore, the multiplicative inverse of $0$ is undefined or does not exist.

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