Multiplicative Inverse

The multiplicative inverse of a number say, N is represented by 1/N or N-1. It is also called reciprocal, derived from a Latin word ‘reciprocus‘. The meaning of inverse is something which is opposite. The reciprocal of a number obtained is such that when it is multiplied with the original number the value equals to identity 1. In other words, it is a method of dividing a number by its own to generate identity 1, such as N/N = 1.

Consider the examples, the multiplicative inverse of 3 is 1/3, of -1/3 is -3, of 8 is 1/8 and of 4/7 is -7/4. But the multiplicative inverse of 0 is infinite, because of 1/0 = infinity. So, there is no reciprocal for a number ‘0’. Whereas the multiplication inverse of 1 is 1 only

Table of contents:

Multiplicative Inverse Definition

The multiplicative inverse of a number for any n is simply 1/n. It is denoted as:

 1 / x  or  x-1 (Inverse of x)

It is also called as the reciprocal of a number and 1 is called the multiplicative identity.

Finding the multiplicative inverse of natural numbers is easy, but it is difficult for complex and real numbers.

For example, the multiplicative inverse of 3 is 1/3, of 47 is 1/47, of 13 is 1/13, of 8 is 1/8, etc, whereas the reciprocal of 0 will give an infinite value or 1/0 = . Now to check whether the inverse of a number is correct or not, we can perform the multiplication operation, such that;

3 x 1/3 = 1

47 x 1/47 = 1

13 x 1/13 = 1

8 x 1/8 = 1

Hence, you can see in all the above four cases, we get the identity number 1. So it is proved.

Multiplicative Inverse Property

The product of a number and its multiplicative inverse is 1.

x. x-1 = 1

For example, consider the number 13.

The multiplicative inverse of 13 is 1/13.

According to the property,

13. (1/13) = 1

Hence Proved.

Multiplicative Inverse Modulo and Proof

Let us see some of the methods to the proof modular multiplicative inverse.

Method 1: For the given two integers say ‘a’ and ‘m’, find the modular multiplicative inverse of ‘a’ under modulo ‘m’.

The modular multiplicative inverse of an integer ‘x’ such that.

ax ≡ 1 ( mod m )

The value of x should be in the range of {0, 1, 2, … m-1}, i.e., it should be in the ring of integer modulo m.

Note that, the modular reciprocal exists, that is “a modulo m” if and only if a and m are relatively prime.

gcd(a, m) = 1.

Method 2: If a and m are coprime, multiplicative inverse modulo can also be found using the Extended Euclidean Algorithm

From the Extended Euclidean algorithm, that takes two integers to say ‘a’ and ‘b’, finds their gcd and also find ‘x’ and ‘y’ such that

ax + by = gcd(a, b)

To find the reciprocal of ‘a’ under ‘m’, substitute b = m in the above formula. We know that if a and m are relatively prime, the value of gcd is taken as 1.

ax + my = 1

Take modulo m on both sides, we get

ax + my = 1(mod m)

We can remove the second term on the left side as ‘my (mod m)’ because for an integer y will be 0. So it becomes,

ax ≡ 1 (mod m)

So, the value of x can be found using the extended Euclidean algorithm which is the multiplicative inverse of a.

It mostly used in equations for simplifications. Mostly it is used for cancellation of the terms. Remember that if you want to find the multiplicative inverse of a number then take the reciprocal of a number.

Multiplicative Inverse of Complex Numbers

Find reciprocal is quite difficult for complex numbers and real numbers. When you consider both the numbers, there is a significant similarity. However, when you are dealing with rational expressions, there is an instance of having a radical (or) square root in the denominator part of the expression.

Consider an example,

2/√3+2

The radicals in the denominator make the fraction more complex. In order to remove the radical in the denominator, it is needed to manipulate the fraction. To simplify the fraction, multiply the entire fraction by the conjugate. It means that conjugates are like their counterparts, but the signs between the parts should be different.

Therefore, it becomes,

\(\frac{2}{\sqrt{3}+2}\times \frac{\sqrt{3}-2}{\sqrt{3}-2}\) \(\frac{2\sqrt{3}-4}{3-4}\) \(-(2\sqrt{3}-4)\)

If there is any minus sign inside the radical part, then take the minus outside and substitute with the letter i.

For example: 4+√-3  is a complex number.

It can be written as;

4+i√3

Where the 4 is the real number and i√3 is the imaginary number. Now to find the reciprocal of this complex number, we have to multiply and divide it by 4-i√3, such that:

4+i√3 x [(4-i√3)/(4-i√3)]

[42 – (i√3)2]/4-i√3

(16 – i23)/4-i√3

Since, i2 = -1

Therefore,

16 – (-3)/4-i√3

19/4-i√3 is the reciprocal of 4+i√3.

Examples

Example 1 : Find the multiplicative inverse of -5

Solution : The reciprocal of -5 is -1 / 5

Check : Number x Multiplicative inverse = 1

(-5) x (-1/5) = 1

1 = 1

So, the multiplicative inverse of -5 is -1 / 5.

Example 2 : Find the reciprocal of 7/74

Solution : Multiplicative inverse of 7/74 = (1/7) / (1/74)

= 74/7

Check : Number x Multiplicative inverse = 1

(7/74) x (74/7) = 1

Therefore, the solution is 74/7.

Example 3: Find the reciprocal of x2

Solution: The reciprocal of x2 is 1/x2 or x-2

Check: x2 × x-2 = 1

1 = 1

Example 4: What is the reciprocal of 11/33.

Solution: The reciprocal of 11/33 is 33/11.

If we further simplify. we get;

11/33 = 1/3

So, the reciprocal of 1/3 is 3.

Because, 1/3 × 3 = 1. Hence it satisfies the reciprocal property.

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