Which Equation Shows the Inverse Property of Multiplication

What Is Multiplicative Inverse?

The meaning of the word “inverse” is something opposite in result. The multiplicative inverse of a number is a number that, when multiplied by the given number, gives 1 every bit the product. By multiplicative inverse definition, information technology is the reciprocal of a number.

The multiplicative inverse of a number “a” is represented as a^-1
or $\frac{1}{a}$.

Multiplicative Inverse Property

The multiplicative changed belongings states that if we multiply a number with its reciprocal, the product is always equal to ane. The image given below shows that $\frac{1}{a}$ is the reciprocal of the number “a”.

A pair of numbers, when multiplied to give production 1, are said to be multiplicative inverses of each other. Here, a and $\frac{1}{a}$ are reciprocals of each other.

How to Observe the Multiplicative Inverse?

Consider that nosotros have seven apples. To brand them into groups of i each, we demand to separate them by seven. Since division is the reverse procedure of multiplication, dividing by a number is equivalent to multiplying by the reciprocal of the number.

Thus, 7 ÷ 7 = seven × $\frac{one}{seven}$ = 1

Here, $\frac{ane}{vii}$ is chosen the multiplicative inverse of 7.

Let u.s.a. understand the multiplicative changed of unlike types of numbers like natural numbers, integers and fractions.

Natural Numbers

The numbers that are used for counting, such as ane, 2, iii, and and then on, are known as natural numbers. The reciprocal of “a” is $\frac{1}{a}$.

For example, if we multiply 7 past $\frac{i}{7}$, we go $7\frac{1}{vii}=1$. So, the multiplicative inverse of 7 is $\frac{1}{seven}$.

Integers

The reciprocal of a positive integer “a” is $\frac{one}{a}$. Negative numbers are the numbers that prevarication on the left side of zero or whose value is always less than 0. Negative numbers take minus (-) sign in front of them. The product of a negative number and its reciprocal must be 1. The reciprocal of a negative number will be a negative number. For example: If the number is -2, then information technology’due south reciprocal will exist $-\frac{1}{two}$ not $\frac{ane}{two}$, as  $-2\frac-{1}{2}=1$

Fractions

To find the reciprocal of a fraction we can only flip it over. The reciprocal of any fraction $\frac{a}{b}$ is $\frac{b}{a}$, because $\frac{a}{b}\times\frac{b}{a}=ane$.

Unit Fractions

Unit fraction is the fraction in which the numerator is 1 irrespective of the number in the denominator. The reciprocal of unit fraction $\frac{1}{10}$ is x, a whole number. For example, reciprocal or multiplicative inverse of ¼ is 4.

Mixed Fraction

A mixed fraction is a combination of a whole number and a proper fraction. For example: $2\frac{three}{7},5\frac{four}{five}$, etc.

In guild to discover the reciprocal of a mixed fraction, we convert it to an improper fraction and and then find its reciprocal.

For example, to discover the reciprocal of $ii\frac{3}{7}$,  nosotros convert $2\frac{3}{7}$ to improper fraction, that is $\frac{17}{7}$.

Since the reciprocal of $2\frac{3}{7}$ or $\frac{17}{7}$ is $\frac{7}{17}$.

Fun Facts

1. The word “reciprocal” comes from the Latin word “reciprocus”, which means dorsum and forth.
2. The multiplicative inverse of a proper fraction is an improper fraction.

Conclusion

In this article, we learned about the reciprocal of dissimilar types of numbers. To read more such informative articles on other concepts, do visit our website. We, at SplashLearn, are on a mission to make learning fun and interactive for all students.

Solved Examples

Example i: What is the multiplicative inverse of -100?

Solution: The multiplicative changed of -100 is -$\frac{ane}{100}$.

Example 2: The reciprocal of a number is
$2\frac{3}{5}$. Find the number.

Solution: A pair of numbers when multiplied to give product as 1, they are said to be reciprocals of each other.

So, the reciprocal of $2\frac{3}{5}$ or $\frac{xiii}{5}$ is the original number. Since the reciprocal of $\frac{13}{v}$ is $\frac{5}{13}$, the original number is $\frac{v}{13}$.

Example 3: What is the multiplicative inverse of
$\frac{2}{three} + \frac{3}{2}$?

Solution: To find the reciprocal, we need to simplify the expression get-go.

$\frac{2}{3} + \frac{iii}{2} = \frac{13}{6}$The reciprocal of $\frac{13}{vi}$
is $\frac{6}{thirteen}$.

Practise Problems

Ofttimes Asked Questions