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 property says that when you multiply a number by its reciprocal, the result will always be 1. In the image, you can see that 1/n is the reciprocal of n, and their product gives 1.

For example, imagine you have five apples, and you want to split them into five groups of 1 apple each. To do that, you divide the apples by 5. Dividing a number by itself is the same as multiplying it by its reciprocal. So, 5 ÷ 5 = 5 × ⅕ = 1.

Here, ⅕ is the multiplicative inverse of 5.

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 a number p is another number x such that their product px leaves a remainder of 1 when dividend by m. This is written as,

\(px ≡ 1 (mod m)\)

This means that m perfectly divides px - 1.

A modular multiplicative inverse of p exists only when p and m are coprime, that is, gcd (p, m) = 1.
 

• A number’s multiplicative inverse is also known as its reciprocal.
   
• When you multiply a number by its reciprocal, the result is always 1.
   
• Only non-zero numbers have a multiplicative inverse.
   
• The multiplicative inverse of a fraction is found by swapping its numerator and denominator.
   
• For a unit fraction like 1/x, the multiplicative inverse is simply x.

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

• To find the inverse, switch the numerator and denominator. For the number 3, the multiplicative inverse will be 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 by 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.
   
• Parents can use everyday examples, such as dividing snacks, to help children understand reciprocals more easily.
   
• Teachers can use visual objects and classroom activities to help students practice finding multiplicative inverses.
   
• Children should think of the multiplicative inverse as the partner that, when multiplied, makes 1.

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.