What is Additive Inverse?

Your Complete Guide to the Inverse Property of Addition

In math, the additive inverse property refers to the fact that, for evert real number, A, there is an inverse value, -A, such that the equation A + (-A) = 0 is true. While this definition of additive inverse may seem complex at first glance, the additive inverse property is a relatively simply concept to grasp once you see a few examples. This Guide on The Additive Inverse Property will do just that, as we will take a deep dive into the definition of the inverse property of addition as well as several examples.

Before we get started, we will note that, in math, an additive inverse is simply a number that can be added to any given number where the resulting sum is zero. The additive inverse is sometimes referred to as a number’s opposite (although this is a more casual definition).

For example, consider the number 7. If you wanted to find the additive inverse of 7, you would have to ask “what number can be added to 7 where the resulting sum is zero?” In this case, if we took 7 and added -7 to it, the result would indeed be zero: 7 + -7 = 0. So, the additive inverse of 7 is -7.

Again, if you find this introduction on the inverse property of addition to be confusing, continue onto the next sections where we will take a deeper dive into the definition of additive inverse and several more examples.

What is Additive Inverse?

Let’s start by revisiting the definition of the additive inverse property, which states:

Definition: The Additive Inverse Property says that, for every real number, A, there is another number, -A, such that:

A + (-A) = 0

In other words, the additive inverse of a number is that number’s opposite. Whenever you take a number and it’s additive inverse and add them together, the sum will always equal zero (i.e. they cancel each other out).

For example, the number 4 has an additive inverse of -4 because: 4 + -4 = 0.

On the other hand, the number -9 has an additive inverse of 9 because -9 + 9 = 0.

So, we can apply this simple rule to any occasion where you have to determine the additive inverse of a number:

If the number is positive (+) → the additive inverse will be the negative (-) form of that number.
If the number is negative (-) → the additive inverse will be the positive (+) form of that number.

This simple rule to determining the additive inverse of any number is illustrated in Figure 02 below.

The Inverse Property of Addition Explained

Now that you understand the meaning of the additive inverse property and how it applies to numbers like 4, 7, and -9, we can take the next step to extending this understanding to the entire universe of real numbers.

Remember that the additive inverse property says that, for any real number A, there is an opposite number, -A, such that the sum of A and (-A) equals zero, or A + (-A) = 0. Also remember that if the number is positive, then its additive inverse will be negative and vice versa.

So, we can say that A and -A are considered to be negative inverses of each other:

-A is the additive inverse of A
A is the additive inverse of -A

The next important thing to understand about the inverse property of addition is that this rule applies to all real numbers, including integers, fractions and decimals.

For example, the fraction 4/5 has an additive inverse of (-4/5) because:

4/5 + (-4/5) = 0

In this example, (-4/5) is the additive inverse of 4/5 and vice versa, meaning that 4/5 is the additive inverse of (-4/5). The fact that the number in this example was a fraction did not change anything because the additive inverse property applies to all real numbers.

Let's now look at another example which involved a decimal.

For example, the decimal number -3.6 has an additive inverse of 3.6 because:

-3.6 + 3.6 = 0

In this example, 3.6 is the additive inverse of -3.6 and vice versa, meaning that -3.6 is the additive inverse of 3.6.

The chart in Figure 03 below shows a few more examples of a given integer, fraction, or decimal, and its corresponding additive inverse.

Again, notice how the additive inverse of each example number, whether it be an integer, a fraction, or a decimal, follows the simple rule that if the number is positive (+), then the additive inverse will be the negative (-) form of that number and vice versa.

The Additive Inverse of Zero

Now that you understand the additive inverse property, let’s consider a special case: zero.

Does zero have an additive inverse? If we are given the number zero and we’re tasked with finding its additive inverse, we would have to look for an “opposite” number that we can add to zero where the result would be zero.

Given the fact that zero is neither positive nor negative, we can not say that the additive inverse of zero is “negative zero,” since there is no such thing. However, we do know that 0 + 0 = 0, so we can conclude that the additive inverse of zero is zero.

In fact, zero is the only number whose additive inverse is itself! Note that zero is a special case and it is the only real number whose additive inverse does not have the opposite sign.

The Additive Inverse Property in the Real World

At this point, you may be wondering how the additive inverse property applies to the real world.

The act of taking a value and cancelling it by adding it to its opposite is actually quite common and applies to a variety of real life scenarios such as:

Finance and Accounting: In finances and accounting, the concepts of credits an debits is a real life application of the additive inverse property. Whenever you have to balance out a transaction, you have to consider the debit (eg. $200) and the corresponding credit (eg. -$200) so that the two values cancel each other out.
Temperature: Given that positive and negative values are associated with temperatures, it is often the case that positive and negative changes in temperature can cancel each other out. For example, if the temperature outside increases by 10 degrees and then later on decreases by 10 degrees, the +10 and -10 will cancel each other out and will not affect the average temperature value.
Sports: The scoring systems in many sports have elements similar to the additive inverse property. For example, in golf, the goal is to score as far below par as possible. If a golfer scores +2 over par on the first hole and then -2 under par on the next hole, the +2 and -2 will cancel each other out and the golfer’s score will remain at net zero over par.

These are just a few examples of how the additive inverse property applies to the real world. Understanding the inverse property of addition will give you a better grasp of the number system in general and will help strengthen your overall number sense. The key conceptual takeaway is that, for every value, there is an opposite value that can be added to it where the sum is zero (i.e. they cancel each other out and get you back to zero).