Multiplicative Inverse:

The multiplicative inverse of a number is also known as the reciprocal of that number. It is a number that, when multiplied by the original number, gives the product of 1. In other words, the multiplicative inverse of a number 'a' is a number 'b' such that a x b = 1.

Example:
Let's consider the number 3. The multiplicative inverse of 3 is 1/3 because 3 x (1/3) = 1.

Properties of Multiplicative Inverse:

1. Every non-zero number has a multiplicative inverse except for zero.
2. The multiplicative inverse of a number is unique.
3. The product of a number and its multiplicative inverse is always 1.

Finding the Multiplicative Inverse:

To find the multiplicative inverse of a number, follow these steps:

1. Take the number 'a' for which you want to find the multiplicative inverse.
2. Set up an equation: a x b = 1, where 'b' is the multiplicative inverse.
3. Solve the equation for 'b' by dividing both sides of the equation by 'a'.
4. The value of 'b' obtained is the multiplicative inverse of 'a'.

Examples:

1. Find the multiplicative inverse of 5.
- Set up the equation: 5 x b = 1
- Divide both sides by 5: b = 1/5
- The multiplicative inverse of 5 is 1/5.

2. Find the multiplicative inverse of -2.
- Set up the equation: -2 x b = 1
- Divide both sides by -2: b = -1/2
- The multiplicative inverse of -2 is -1/2.

3. Find the multiplicative inverse of 1/4.
- Set up the equation: (1/4) x b = 1
- Divide both sides by 1/4: b = 4/1 = 4
- The multiplicative inverse of 1/4 is 4.

Importance of Multiplicative Inverse:

The concept of the multiplicative inverse is important in various mathematical operations and concepts, including:

1. Division: Division can be represented as multiplying by the multiplicative inverse. For example, dividing by 5 is the same as multiplying by 1/5.
2. Solving equations: The multiplicative inverse is used to solve equations involving fractions or variables. It allows us to isolate the variable on one side of the equation.
3. Rationalization: The concept of the multiplicative inverse is used to rationalize the denominators of fractions. By multiplying the numerator and denominator by the multiplicative inverse, we can eliminate radicals or complex numbers from the denominator.

Conclusion:
The multiplicative inverse, also known as the reciprocal, is a number that, when multiplied by the original number, gives the product of 1. It is an important concept in mathematics and is used in various operations and concepts such as division, solving equations, and rationalization.