Propriedade inversa da adição

Combining Opposite Numbers to Make $0$

When you add a negative number to its positive counterpart, the answer will always be $0$.

Let’s take a look at an example. $2+(-2)$.

First, let’s draw a number line and start at $2$.

Next, add $-2$ to $2$. This is essentially the same as subtracting $2$ from $2$.

The answer is $0$ because $-2$ is a counterpart of $2$. As a result, we say $-2$ is the additive inverse of $2$. Also, $2$ is called the additive inverse of $-2$.

Let's look at another example: $-19+19$.

We can solve this problem without drawing a number line. Looking at the equation, we can observe that $19$ is the counterpart of $-19$. Adding these two numbers will result in $0$, so $-19+19=0$. This can be thought of solving the equation

$$x - 19 = 0$$

where $x$ is a variable. Adding $19$ to both sides is the same as finding the additive inverse.

Now, let's try a couple of examples without drawing a number line. You can still draw it yourself if you need to. After a while you probably won't need to. The picture will pop up in your head like magic!

$$\begin{align} 5+(-5)&= 5-5\\ &=0 \\ \end{align}$$

To make zero, add the opposite of $7$, which is $-7$.

Adding a negative and a positive of the same number will equal $0$

The Inverse Property of Addition states the following:

Adding a number and it's negative version of itself yields $0$.

In other words, if you add $-3 + 3$ or $152 + (-152)$, the answer will always be $0$.

Also, we can add either the positive or the negative number first. This is because addition is commutative.

Opposite quantities cancel each other out

When you add positive and negative versions of the same number together, they will always equal zero. This is because when we add numbers, we essentially connect their positions relative to zero. For example, if we add $4 + (-4)$, we would do the following:

This will also work the other way around if we add a positive number to a negative number (so in the case of this example, $(-4) + 4$).

(Video) Opposites and the Addition Property of 0

by mathman1024

This video by mathman1024 shows how adding a number to its opposite always equals zero. He then goes over the Addition Property of Zero.

$\large{\text{Summary}}$

The first example he looks at is: $8+(-8)=0$. If you increase something by $8$ and then decrease it by $8$, you're back where you started.

It doesn't matter if you start with a negative first either. For example, $-12+12=0$. As long as the signs are different the addition of them will always be $0$. Why can we do either order? Addition obeys the commutative property.

This leads us to the Addition Property of Zero. $$a+0=a$$ Any number plus $0$ doesn't change the number at all.

• $19+0=19$
• $-27+0=-27$
• $0+(-83)=-83$

Inverse Property of Addition

The inverse property of addition states:

If you add the positive and negative versions of a number, the sum will be $0$. $$x+(-x) =0$$

Unless $x = 0$, we are adding a positive number and a negative number together. The two numbers look almost the same. However, the second number has a negative sign in front.

Let's look at some problems you might encounter about the inverse property of addition.

$$-3-(-3) = -3+3=\boxed{0}$$

$$11+(-11)=11-11=\boxed{0}$$

$$17+(-17)= 17-17=\boxed{0}$$

$$-19-(-19)= -19+19=\boxed{0}$$

(Video) Inverse Property of Addition and Multiplication

by William Chermak

Here's a video by William Chermak that shows the inverse property of addition. It also discusses the inverse property of multiplication. However, in this explanation we'll focus on addition.

$\large{\text{Summary}}$

The inverse property of addition states that any number added to its opposite will always equal $0$. $$\boxed{a+(-a)=0}$$

Example

$5+(-5)=0$

We might be asked to find the additive inverse of a number. This additive inverse acts as a variable you'll need to find. In other words, it's an equation that can be solved with addition or subtraction.