Additive Inverse Property: Definition & Examples

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  • 0:02 Definition
  • 1:23 Graphical Representation
  • 2:11 Example
  • 3:12 Lesson Summary
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Lesson Transcript
Instructor: Norair Sarkissian
In this lesson, we will examine the additive inverse of a given number. We will also learn about the additive inverse of a variable and how it can be represented graphically. Finally, we will explore the most widely employed application of the additive inverse.

Definition

The additive inverse of a number is what you add to a number to create the sum of zero. So in other words, the additive inverse of x is another number, y, as long as the sum of x + y equals zero. The additive inverse of x is equal and opposite in sign to it (so, y = -x or vice versa). For example, the additive inverse of the positive number 5 is -5. That's because their sum, or 5 + (-5) = 0.

What about the additive inverse of a negative number? Using the same approach, if x is a negative number, then its additive inverse is equal and opposite in sign to it. This means that the additive inverse of a negative number is positive. For instance, if x equals -12, then its additive inverse is y = 12. We can verify that the sum of x + y equals zero, since when x = -12 and y = 12, we have -12 + 12 = 0.

It should be noted that the additive inverse of 0 is 0. Zero is the only real number, which is equal to its own additive inverse. It is also the only number for which the equation x = -x is true.

Graphical Representation

We can also think of the additive inverse visually. Let's consider the real number line, which is usually drawn horizontally, with 0 near the middle, the negative numbers to its left, and the positive numbers on the right. Two numbers of opposite sign fall on either side of 0 on the number line at equal distance.

Once we place the point corresponding to a number x on the number line, we know that the additive inverse, or -x, will fall on the opposite side of the number line with respect to 0. In fact, the point 0 is the midpoint between x and its additive inverse -x. For example, when x = 5, its additive inverse is -5.

A number and its additive inverse are equidistant from the 0
Number line

It is clear to see that the point 0 is the midpoint of the segment between -5 and 5.

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