The multiplicative inverse is used to simplify mathematical expressions. Words 'inverse' means somepoint opposite/contrary in impact, order, place, or direction. A number that nullifies the impact of a number to identity 1 is dubbed a multiplicative inverse.

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1.What is Multiplicative Inverse?
2.Multiplicative Inverse of a Natural Number
3.Multiplicative Inverse of a Unit Fraction
4.Multiplicative Inverse of a Fraction
5.Multiplicative Inverse of a Mixed Fraction
6.Multiplicative Inverse of Complex Numbers
7.Modular Multiplicative Inverse
8.FAQs on Multiplicative Inverse

What is Multiplicative Inverse?


The multiplicative inverse of a number is defined as a number which when multiplied by the original number offers the product as 1. The multiplicative inverse of 'a' is dedetailed by a-1 or 1/a. In various other words, when the product of 2 numbers is 1, they are said to be multiplicative inverses of each various other. The multiplicative inverse of a number is defined as the division of 1 by that number. It is also called the reciprocal of the number. The multiplicative inverse property says that the product of a number and its multiplicative inverse is 1.

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For instance, let us consider 5 apples. Now, divide the apples right into five groups of 1 each. To make them into teams of 1 each, we should divide them by 5. Dividing a number by itself is tantamount to multiplying it by its multiplicative inverse . Hence, 5 ÷ 5 = 5 × 1/5 = 1. Here, 1/5 is the multiplicative inverse of 5.


Multiplicative Inverse of a Natural Number


Natural numbers are counting numbers beginning from 1. The multiplicative inverse of a herbal number a is 1/a.

Examples

3 is a herbal number. If we multiply 3 by 1/3, the product is 1. As such, the multiplicative inverse of 3 is 1/3.Similarly, the multiplicative inverse of 110 is 1/110.

Multiplicative Inverse of a Negative Number

Just as for any positive number, the product of an adverse number and its reciprocal need to be equal to 1. Hence, the multiplicative inverse of any negative number is its reciprocal. For instance, (-6) × (-1/6) = 1, therefore, the multiplicative inverse of -6 is -1/6.

Let us think about a few more examples for a far better expertise.


Multiplicative Inverse of a Unit Fraction


A unit fractivity is a fraction with the numerator 1. If we multiply a unit fractivity 1/x by x, the product is 1. The multiplicative inverse of a unit fraction 1/x is x.

Examples:

The multiplicative inverse of the unit fraction 1/7 is 7. If we multiply 1/7 by 7, the product is 1. (1/7 × 7 = 1)The multiplicative inverse of the unit fraction 1/50 is 50. If we multiply 1/50 by 50, the product is 1. (1/50 × 50 = 1)

Multiplicative Inverse of a Fraction


The multiplicative inverse of a portion a/b is b/a because a/b × b/a = 1 as soon as (a,b ≠ 0)

Examples

The multiplicative inverse of 2/7 is 7/2. If we multiply 2/7 by 7/2, the product is 1. (2/7 × 7/2 = 1)The multiplicative inverse of 76/43 is 43/76. If we multiply 76/43 by 43/76, the product is 1. (76/43 × 43/76 = 1)

Multiplicative Inverse of a Mixed Fraction


To discover the multiplicative inverse of a mixed fraction, convert the blended fraction into an imcorrect fractivity, then determine its reciprocal. For instance, the multiplicative inverse of (3dfrac12)

Step 1: Convert (3dfrac12) to an improper fraction, that is 7/2.Step 2: Find the reciprocal of 7/2, that is 2/7. Hence, the multiplicative inverse of (3dfrac12) is 2/7.

Multiplicative Inverse of Complex Numbers


To uncover the multiplicative inverse of complicated numbers and also genuine numbers is rather challenging as you are handling rational expressions, via a radical (or) square root in the denominator part of the expression, which renders the fraction a bit complicated.

Now, the multiplicative inverse of a complicated number of the form a + (i)b, such as 3+(i)√2, wright here the 3 is the actual number and also (i)√2 is the imaginary number. In order to discover the reciprocal of this facility number, multiply and divide it by 3-(i)√2, such that: (3+(i)√2)(3-(i)√2/3-(i)√2) = 9 + (i)22/3-(i)√2 = 9 + (-1)2/3-(i)√2 = 9-2/3-(i)√2 = 7/3-(i)√2. Thus, 7/3-(i)√2 is the multiplicative inverse of 3+(i)√2

Also, the multiplicative inverse of 3/(√2-1) will certainly be (√2-1)/3. While finding the multiplicative inverse of any expression, if tright here is a radical present in the denominator, the fraction deserve to be rationalized, as presented for a portion 3/(√2-1) below,

Tip 2: Solve. (frac3 sqrt2+12 - 1)Tip 3: Simplify to the lowest form. 3(√2+1)

Modular Multiplicative Inverse


The modular multiplicative inverse of an integer p is another integer x such that the product px is congruent to 1 through respect to the modulus m. It deserve to be represented as: px (equiv ) 1 (mod m). In various other words, m divides px - 1 entirely. Also, the modular multiplicative inverse of an integer p can exist through respect to the modulus m just if gcd(p, m) = 1

In a nutshell, the multiplicative inverses are as follows:

TypeMultiplicative InverseExample

Natural Number

x

1/xMultiplicative Inverse of 4 is 1/4

Integer

x, x ≠ 0

1/xMultiplicative Inverse of -4 is -1/4

Fraction

x/y; x,y ≠ 0

y/xMultiplicative Inverse of 2/7 is 7/2

Unit Fraction

1/x, x ≠ 0

xMultiplicative Inverse of 1/20 is 20

Tips on Multiplicative Inverse

The multiplicative inverse of a fraction deserve to be obtained by flipping the numerator and also denominator.The multiplicative inverse of 1 is 1.The multiplicative inverse of 0 is not identified.The multiplicative inverse of a number x is composed as 1/x or x-1.The multiplicative inverse of a mixed fractivity have the right to be derived by converting the mixed fractivity into an imcorrect fractivity and determining its reciprocal.

Important Notes

The multiplicative inverse of a number is also dubbed its reciprocal.The product of a number and also its multiplicative inverse is equal to 1.

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Example 1: A pizza is sliced into 8 pieces. Tom keeps 3 slices of the pizza at the counter and also leaves the rest on the table for his 3 friends to share. What is the portion that each of his friends get? Do we apply multiplicative inverse here?

Solution:

Due to the fact that Tom ate 3 slices out of 8, it implies he ate 3/8th part of the pizza.

The pizza left out = 1 - 3/8 = 5/8

5/8 to be mutual among 3 friends ⇒ 5/8 ÷ 3.

We take the multiplicative inverse of the divisor to simplify the department.

5/8 ÷ 3/ 1

= 5/8 × 1/3

= 5/24

Answer: Each of Tom's friends will certainly be acquiring a 5/24 percentage of the left-over pizza.


Example 2: The total distance from Mark's house to school is 3/4 of a kilometer. He can ride his cycle 1/3 kilometer in a minute. In just how many type of minutes will he reach his institution from home?

Solution:

Total distance from house to school = ¾ km

Distance extended in a minute = 1/3 km

The time taken to cover the complete distance = complete distance/ distance covered

= 3/4 ÷ 1/3

The multiplicative inverse of 1/3 is 3.

3/4 × 3 = 9/4 = 2.25 minutes

Answer: As such, the moment taken to cover the complete distance by Mark is 2.25 minutes.


Example 3: Find the multiplicative inverse of -9/10. Also, verify your answer.

Solution:

The multiplicative inverse of -9/10 is -10/9.

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To verify the answer, we will certainly multiply -9/10 with its multiplicative inverse and inspect if the product is 1.