Multiplicative Inverse – How it Works

1/N or N^-1 is the multiplicative inverse of a number, such as N. It’s also known as reciprocal, which comes from the Latin word’reciprocus.’ The word inverse means the opposite of something. The reciprocal of a number obtained is one whose value equals identity 1 when multiplied by the original number. In other terms, it is a way of generating identity 1 by dividing a number by its own, such as N/N = 1.

The resultant value of a number multiplied by its own multiplicative inverse is 1.

Consider the following examples: 1/3 is the multiplicative inverse of 3, -1/3 is -3, 8 is 1/8, and 4/7 is -7/4. However, because 1/0 = infinity, the multiplicative inverse of 0 is infinite. As a result, the number ‘0’ has no reciprocal. The multiplication inverse of one is only one.

Definition:

For any n, the multiplicative inverse of a number is simply 1/n. It’s written as:

1 / x  or  x^-1 (Inverse of x)

It’s also known as a number’s reciprocal, and 1 is known as the multiplicative identity.

Finding the multiplicative inverse of natural numbers is simple, but complex and real numbers are more complicated.

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, and so on, whereas the reciprocal of 0 is 1/0 = ∞. We can now use the multiplication operation to see if the inverse of a number is valid or not, as follows:

● 3 x 1/3 = 1

● 47 x 1/47 = 1

● 13 x 1/13 = 1

● 8 x 1/8 = 1

As you can see, the identity number 1 is obtained in all four circumstances. This has been established.

Multiplicative Inverse of Natural Number:

The multiplicative inverse of x is 1/x if x is any natural integer (0,1,2,3,4,5,6,7,…) The multiplicative inverse of 5 is 1/5, for example.

Multiplicative Inverse Property:

A number’s product with its multiplicative inverse is 1.

x. x^-1 = 1

For example, consider the number 18.

The multiplicative inverse of 18 is 1/18.

According to the property,

18. (1/18) = 1

Hence Proved.

Multiplication inverse of fraction:

If p/q is a fraction, then the multiplicative inverse of p/q should equal 1, when multiplied by the fraction. As a result, q/p is the multiplicative inverse of p/q.

p/q x q/p = 1

For example: 2/7 x 7/2 = 1

Examples:

● Mul. Inverse of 2/7 is 7/2: 2/7 x 7/2 = 1

● Mul. Inverse of ½ is 2: ½ x 2 = 1

● Mul. Inverse of ¾ is 4/3: ¾ x 4/3 = 1

● Mul. Inverse of 2/9 is 9/2: 2/9 x 9/2 = 1

Multiplication inverse of unit fraction:

½, 1/3, ¼, 1/5, and so on are all termed unit fractions because the numerator is always 1. As a result, the values in the denominator will be the multiplicative inverse of these unit fractions.

● ½ x 2 = 1

● 1/3 x 3 = 1

● ¼ x 4 = 1

● 1/5 x 5 = 1

The multiplicative inverse of Mixed Fraction:

First, convert the mixed fraction to a proper fraction to determine the multiplicative inverse. Let’s look at a few examples.

● 2½ = 5/2: ⅖

● 3⅔ = 11/3: 3/11

Multiplicative inverse modulo:

Let’s look at various proof modular multiplicative inverse approaches.

Method 1:

Find the modular multiplicative inverse of ‘a’ under modulo’m’ for the two integers ‘a’ and’m’.

The modular multiplicative inverse of an integer ‘x’ in the sense that

ax ≡ 1 ( mod m )

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

The modular reciprocal, or “a modulo m,” exists if and only if a and m are relatively prime.

gcd(a, m) = 1.

Method 2:

If a and m are coprime, the Extended Euclidean Algorithm can also be used to find multiplicative inverse modulo.

The Extended Euclidean method takes two integers, say ‘a’ and ‘b,’ finds their gcd, as well as ‘x’ and ‘y,’ and finds them in such a way that

ax + by = gcd(a, b)

Substituting b = m in the following formula to determine the reciprocal of ‘a’ under’m.’ If a and m are relatively prime, we know that the value of gcd is 1.

ax + my = 1

Take modulo m on both sides, we get

ax + my = 1(mod m)

Because y is 0 for an integer, we can delete the second term on the left side as’my (mod m)’. As a result,

ax ≡ 1 (mod m)

The multiplicative inverse of a can be used to find the value of x using the extended Euclidean algorithm.

It is primarily used to simplify equations. It is most commonly used to cancel terms. Remember that the reciprocal of a number is used to obtain the multiplicative inverse of a number.

The multiplicative inverse of complex numbers:

For both complex and real numbers, finding the reciprocal is challenging. When both numbers are compared, they are strikingly comparable. When dealing with rational expressions, however, it is possible to have a radical (or) square root in the denominator section of the expression.

Solved examples:

Q1. Find the multiplicative inverse of -10

Solution: The reciprocal of -10 is -1 / 10

Check : Number x Multiplicative inverse = 1

(-10) x (-1/10) = 1

1 = 1

So, the multiplicative inverse of -10 is -1 / 10.

Q2. What is the reciprocal of 88/44.

Solution: The reciprocal of 88/44 is 44/88.

If we further simplify. We get;

88/44 = 1/4

So, the reciprocal of 1/4 is 4.

Because, 1/4 × 4 = 1. Hence it satisfies the reciprocal property.

Conclusion:

The reciprocal of a given number is the multiplicative inverse. It is used to make mathematical expressions easier to understand. In effect, order, position, or direction, the word ‘inverse’ suggests anything opposite/contrary. When you multiply an integer by its multiplicative inverse, you get 1.