 An additive inverse of a number is defined as the value, which on adding with the original number results in zero value. It is the value we add to a number to yield zero. Suppose, a is the original number, then its additive inverse will be minus of a i.e.,-a, such that;

a+(-a) = a – a = 0

Example:

• Additive inverse of 10 is -10, as 10 + (-10) = 0
• Additive inverse of -9 is 9, as (-9) + 9 = 0

Check: Multiplicative Inverse

Additive inverse is also called the opposite of the number, negation of number or changed sign of original number.

 Fact: Additive inverse of zero is zero only.

## How to Find the Additive inverse?

The additive inverse of any given number can be found by changing the sign of it. The additive inverse of a positive number will be a negative, whereas the additive inverse of a negative number will be positive. However, there will be no change in the numerical value except the sign. For example, the additive inverse of 8 is -8, whereas the additive inverse of -6 is 6.

## Properties

Additive inverse simply means changing the sign of the number and adding it to the original number to get an answer equal to 0.

The properties of additive inverse are given below, based on negation of the original number. For example, x is the original number, then its additive inverse is -x. So, here we will see the properties of -x.

• −(−x) = x
• (-x)2 = x2
• −(x + y) = (−x) + (−y)
• −(x – y) = y − x
• x − (−y) = x + y
• (−x) × y = x × (−y) = −(x × y)
• (−x) × (−y) = x × y

## Additive Inverse of Different Numbers

We have understood that an additive inverse is added to a value to make it zero. Now this value can be a natural number, integer, rational number, irrational number, complex number, etc. Let us find the additive inverse of different types of numbers.

### Additive inverse of Natural or Whole Numbers

As we know, natural numbers are the positive integers. Therefore, the additive inverse of positive integers will be negative.

 Whole numbers/Natural numbers Additive Inverse Result 0 0 0+0 = 0 1 -1 1+(-1) = 0 2 -2 2+(-2) = 0 3 -3 3+(-3) = 0 4 -4 4+(-4) = 0 5 -5 5+(-5) = 0 10 -10 10+(-10) = 0 20 -20 20+(-20) = 0 50 -50 50+(-50) = 0 100 -100 100+(-100) =0

### Additive Inverse of Complex Numbers

Complex numbers are the combination of real numbers and imaginary numbers. A + iB is a complex number, where A is the real number and B is the imaginary number.

Now the additive inverse of A + iB should be a value, that on adding it with a given complex number, we get a result as zero. Therefore, it will be -(A + iB)

Example: Additive inverse of 2 + 3i is -(2+3i)

2+3i + [-(2+3i)]

= 2+3i -2-3i

= 0

### Additive Inverse of Rational Numbers

Suppose a/b is a rational number such that the additive inverse of a/b is -a/b and vice versa.

 Fraction Additive Inverse Result 1/2 -1/2 (½) + (-½) = 0 1/4 -1/4 (¼) + (-¼) = 0 3/4 -3/4 (¾) + (-¾) = 0 2/5 -2/5 ⅖ + (-⅖) = 0 10/3 -10/3 10/3 + (-10/3) = 0

## Difference Between Additive Inverse and Multiplicative Inverse

Additive inverse and multiplicative inverse, both have different properties. See the below table to know the differences.

 Additive Inverse Multiplicative Inverse It is added to the original number to get 0 It is multiplied to the original number to get 1 Results in 0 Results in 1 Sign of the original number is changed and added Reciprocal of the original number is multiplied Example: 66 + (-66) = 0 Example: 66 × (1/66) = 1

### Solved Examples

Example 1: What is the additive inverse of 2/3?

Solution:

Given number is 2/3.

We know that the additive inverse of a/b is -a/b.

Hence, the additive inverse of 2/3 is -2/3.

Example 2: What is the additive inverse of -5/9?

Solution:

Given number is -5/9.

We know that the additive inverse of -a/b is a/b.

Hence, the additive inverse of -5/9 is 5/9.

### Practice Problems

1. Find the additive inverse of 4/5.
2. What is the additive inverse of -7/9?
3. Write the additive inverse of 5 and -7.

### What is an additive inverse?

An additive inverse of a number is the value, which on adding with the original number results in zero value.

No, both are different. Additive inverse is the added to get the result as zero.Additive identity is the value that is added to get the original number, which is zero.

Examples:

3 + (-3) = 0

3 + 0 = 3

-17

17+(-17) = 0

### What is the additive inverse of 23?

-23

23 + (-23) = 23 -23 = 0

### What is the additive inverse of ab?

(-ab)

ab + (-ab) = ab – ab = 0