Additive Inverse and Multiplicative Inverse

Additive inverse of a number is what you add to the original number to get a sum of zero. On the other hand, multiplicative inverse of a number is what you multiply the original number by to get a product of one.

Additive Inverse and Multiplicative Inverse in Math

Let’s learn about the Additive Inverse and Multiplicative Inverse with the help of solved examples.

Additive Inverse Definition

The additive inverse of a number is a value that, when added to the original number, results in a sum of zero

For a number ‘a’, it is denoted as ‘-a’. This represents the value that, when added to ‘a,’ results in a sum of zero.

For example, the additive inverse of 5 is -5.

Illustration of Additive Inverse

Additive Inverse of 0

For any number ‘a,’ the additive inverse of 0 is still 0. This is because when 0 is added to 0, the result is, unsurprisingly, 0.

While other numbers have both positive and negative additive inverses, 0 is unique in having an additive inverse that is the same number itself.

Symbolically, 0 + 0 = 0.

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• Additive Inverse

Multiplicative Inverse Definition

The multiplicative inverse, or reciprocal, of a number is a value that, when multiplied with the original number, results in a product of one. It is also called the reciprocal of a number.

The multiplicative inverse, denoted as ‘1/a’ or ‘a^-1‘ is the reciprocal of a non-zero number ‘a,’ such that their product equals 1.

For instance, the multiplicative inverse of 3 is 1/3, as 3 × (1/3) equals 1.

Multiplicative Inverse of 0

The multiplicative inverse, or reciprocal, of 0 is undefined. In other words, there is no real number ‘a’ such that 0 multiplied by ‘a’ equals 1.

Symbolically, 0×a=1 has no real solution for ‘a’.

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• Multiplicative Inverse
• Multiplicative Inverse of Zero

Difference between Additive Inverse and Multiplicative Inverse

Let’s discuss the differences between additive and multiplicative inverses.

Additive Inverse vs. Multiplicative Inverse
Aspect	Additive Inverse	Multiplicative Inverse
Definition and Notation	The additive inverse of a number ‘a’ is represented as ‘-a.’	The multiplicative inverse of a non-zero number ‘a’ is denoted as ‘1/a’ or ‘a-1.’
Similarities	Both inverses involve operations with a given number to result in a specific outcome: zero for additive inverses and one for multiplicative inverses.	Both concepts play fundamental roles in algebraic manipulations and equation-solving.
Properties	Adding a number to its additive inverse yields zero: a+(−a)=0.	Multiplying a number by its multiplicative inverse results in one: a x (1/a) =1.
Applications	Used in solving equations, balancing expressions, and understanding symmetries.	Applied in solving equations involving division, scaling, and proportionality.

Additive Inverse and Multiplicative Inverse Examples

Let’s solve some example questions on Additive Inverse and Multiplicative Inverse.

1. Find the additive inverse of -12.

The additive inverse of -12 is 12, as -12 + 12 equals 0.

2. Determine the multiplicative inverse of 1/3.

The multiplicative inverse of 1/3 is 3, as (1/3) × 3 equals 1.

3. Solve for ‘x’ in the equation 2x + 5 = 0 using additive inverses.

Subtract 5 from both sides to get 2x = -5. Then, divide by 2 to find x = -5/2.

4. Apply the multiplicative inverse to solve 4y = 8.

Divide both sides by 4 to find y = 2.

5. Use additive inverses to balance the equation 2a – 7 = 5.

Add 7 to both sides to get 2a = 12. Then, divide by 2 to find a = 6.

Additive Inverse vs Multiplicative Inverse- FAQs

1. What is additive inverse of a number?

The additive inverse of a number ‘a’ is ‘-a,’ and the sum of ‘a’ and ‘-a’ is always zero.

2. What is multiplicative inverse of a non-zero number?

The multiplicative inverse of a non-zero number ‘a’ is ‘1/a,’ and the product of ‘a’ and ‘1/a’ is always one.

3. What are the practical applications of additive inverse?

Additive inverses are used in solving equations, balancing chemical equations, and understanding symmetries in mathematical structures.

4. Can a number have both additive and multiplicative inverses?

Yes, a non-zero number can have both an additive inverse (negation) and a multiplicative inverse (reciprocal).

5. How are additive and multiplicative inverses relevant in geometry?

Additive inverses are linked to geometric transformations like reflections, while multiplicative inverses play a role in dilations and scaling.