Multiplicative Inverse

A number’s reciprocal is its multiplicative inverse. The multiplicative inverse of a number 'n' is written as \(\frac{1}{n} \). Here, 1 becomes the numerator, and the number becomes the denominator.

When a number is multiplied by its reciprocal, the result will always be 1. 

Multiplying the number ‘n’ with its reciprocal: \(n \times \frac{1}{n} = 1 \)
For example, let’s take the number 4

According to the multiplicative inverse property:

n =  4

\(\frac{1}{n} = \frac{1}{4} \)

Therefore, \(n \times \frac{1}{n} = 1 \;\Rightarrow\; 4 \times \frac{1}{4} = 1 \) 

Hence, multiplying 4 by its reciprocal gives 1 as the final result.

The multiplicative inverse is found by dividing 1 by the given number. Now let’s learn how to find the multiplicative inverse. Follow the steps given below:

Step 1: Write the given number as a fraction. For example, if the number is 7, write it as \(\frac{7}{1} \).

Step 2: Now switch the numerator and denominator. So 71 will become 17. Now the numerator is 1 and the denominator is 7

Step 3: Now multiply both the fractions obtained in Step 1 and Step 2 to make sure that the product is 1

\(\frac{7}{1} \times \frac{1}{7} = \frac{7}{7} = 1 \)

Hence, the multiplicative inverse of 7 is \(\frac {1}{7}\) and their product is always 1.

An integer is a number that can be positive or negative. It can never be a decimal or fraction.

For positive integers, the product of the number and its reciprocal is always 1. Similarly, the product of a negative integer with its reciprocal is also 1. 

Let the negative integer be -n.

The multiplicative inverse for -n will be \(\frac{1}{-n} \)

Multiplying \((-n) \times \frac{1}{-n} \) gives the product as 1

For example, take -9. The multiplicative inverse for \(-9 \text{ is } \frac{1}{-9} \) 

Multiplying \(-9 \text{ is } \frac{1}{-9} \) gives the product of 1. 

Let \(\frac{m}{n}\) be the fraction. The multiplicative inverse of m/n will be \(\frac{n}{m}\). 

Here, both m and n cannot be zero.

Multiplying \(\frac{m}{n} \times \frac{n}{m} \) will give the product as 1.

For example, take the fraction \(\frac{5}{10} \). The multiplicative inverse will be \(\frac{10}{5} \)

Multiplying \(\frac{5}{10} \times \frac{10}{5} \) gives \(\frac{50}{50} = 1 \). Both numerator and denominator simplify to 1.

To find the multiplicative inverse of a mixed fraction, first convert the given mixed fraction into an improper fraction. After converting, change the position of the fraction upside down to get the multiplicative inverse.

Let’s take 4\(\frac{1}{2} \) as the mixed fraction.

Converting 4 \(\frac{1}{2} \) into an improper fraction results in \(\frac{9}{2} \)

Since the improper fraction is \(\frac{9}{2} \), its multiplicative inverse is \(\frac{2}{9} \)

Multiplying the fractions \(\frac{9}{2} \) and \(\frac{2}{9} \) will give 1 as the product:

\(\frac{9}{2} \times \frac{2}{9} = 1 \)

The multiplicative inverse of a number is its reciprocal, which when multiplied by the number gives 1.

The multiplicative inverse of 0 is not possible because multiplying any number by 0 will always be 0. The multiplicative inverse of 0 is written as \(\frac{1}{0} \), but it is not defined.

Complex numbers are made of two parts, a real part (any number) and an imaginary part (i). Let Z be a complex number, where \(Z = a + ib \).

Here, ‘a’ is the real part, and ‘ib’ is the imaginary part. The multiplicative inverse of Z is \(\frac{1}{Z} \), which is 

For example, take 3 + i√2

In \(3 + i\sqrt{2} \):

3 is the real part, and \(i\sqrt{2} \) is the imaginary part.

The multiplicative inverse of \(3 + i\sqrt{2} \) is \(\frac{1}{3 + i\sqrt{2}} \)

Now, to find the multiplicative inverse of complex numbers, follow the steps given below:

Step 1:  Let the complex number \(Z = a + ib \). The reciprocal form of the given complex will be \(\frac{1}{a + ib} \)

Step 2: Take the conjugate of \((a+ib)\), which is \((a-ib)\). We take the conjugate to remove the imaginary part by multiplying and dividing the inverse with \((a-ib) \)

→ \(\frac{1}{a + ib} \times \frac{a - ib}{a - ib} = \frac{a - ib}{a^2 + b^2} \)

Using the identity \((a + ib)(a - ib) = a^2 - (ib)^2 \quad \text{and} \quad i^2 = -1 \), we solve the denominator as
\(a^2 + b^2\)
 

Step 3: Simplify to the simplest form

\(\frac {1}{z} = \frac {a}{a^2 + b^2} - \frac {b}{a^2 + b^2}\)

The modular multiplicative inverse of the number q is another number p, such that ‘q × p’  will always be 1 (remainder) when divided by ‘x’ 

It is represented as \(q \times p \equiv 1 \) (mod x)

This means that ‘x’ can completely divide \(q × p - 1\)

For example, let’s find the modular inverse of 3 mod 7

We write the expression as \(q × p ≡ 1\) (mod x)

Here, q is 3 and x is 7. We have to find p.

Therefore, the expression is written as (mod 7)

\(3 × p  ≡ 1\)

This means that the result should be when 3 is multiplied by some number ‘p’ when divided by 7.

Now we need to check for values of p. Start with number 1.

Here, p is 5 because \(3 × 5 = 15\). When 15 is divided by 7, the remainder is 1.

Therefore, the modular multiplicative inverse of 3 mod 7 is 5

To master finding the multiplicative inverse, follow the given tips and tricks.

• To find the inverse, just change the position of the numerator and denominator. For the number 3, the multiplicative inverse will be \(\frac{1}{3}\).
   
• If you are asked to find the inverse of 0, remember that it can never be written in its reciprocal form.
   
• Always convert the mixed fraction into an improper fraction before finding the multiplicative inverse.
   
• Check your final answer by multiplication: After finding the multiplicative inverse, multiply it with the original number. The result should always be 1.
   
• Use in algebraic expressions carefully: When finding the inverse of variables or expressions, treat them the same way as numbers, flip the numerator and denominator.

Multiplicative inverse is not just used in daily life but also in professional fields. Given below are some real-life applications of the multiplicative inverse:

• We use multiplicative inverse in cryptography to enable secure encryption and decryption.
   
• In finance, multiplicative inverse helps calculate the reciprocal exchange rates for currency conversions.
   
• It is also used in scaling to adjust the given quantities proportionally.
   
• Measurements for the conversion of units are done with the help of multiplicative inverse.
   
• In computer graphics and physics simulations to calculate inverse transformations, such as undoing rotations, scaling, or other linear operations we apply multiplicative inverse.