乘以-1

Flip the sign

When you multiply a number by $-1$, the only thing that changes is the sign of the number. That means that if the number was positive, it becomes negative. Also, if the original number was negative, it becomes positive.

This is effective for the following reason. When you multiply a number by a negative, it always switches the sign of a number. In other words, it flips the number around $0$ on the number line. Meanwhile, when you multiply a number by $1$, it stays the same. So, when you multiply a number by $-1$, we switch the sign. Everything else about the number stays the same. In particular, the number keeps the same absolute value. Flipping the sign does not change the distance from $0$ on the number line. It only changes the direction!

Let's take a look at this on the number line. If we're multiplying $4 \times -1$, we can find $4$ first:

Since we're only adding $4$ once, that's all we have to do. But, since we're multiplying by $-1$, we also have to flip the sign (so that we're essentially adding $-4$ once):

Notice that $-4 < 4$. This is because $-4$ is to the left of $4$ on the number line. We get this same type of comparison whenever we switch a number's sign. This is called an inequality because it lets us compare two numbers. The number line simply helps us see what is going on.

And that's it! Think of this as moving to the right on the number line, then reflecting over the $0$. It's just like looking at your reflection in the mirror.

Finally, if we want to flip the sign of $0$, well, we can't. The number $0$ doesn't have a sign. So multiplying $0$ by $-1$ just gives $0$. We don't move around on the number line at all.

$$0 \times -1 = 0$$ $$-1 \times 0 = 0$$

Multiplying by $-1$ flips a number around $0$

When you multiply a number by $-1$, all you have to do is change the sign. If it was negative it becomes positive, and if it was positive it becomes negative:

$$-1(x) = -x$$ $$\text{OR}$$ $$-1(-x) = x$$

Think of it as $-1$ telling you to flip your number around $0$. The image below illustrates what is happening.

Here's another graphic about the Multiplication Property of $-1:$

If a number is written $-a$, that means we're multiplying $a$ by negative one. In this case $a$ can be positive OR negative itself.

Finally zero is a special case. We have something called the zero property of multiplication. It says that if we multiply any number by zero, we get zero. This includes $-1$, so

$$-1 \times 0 = 0$$ $$0 \times -1 = 0$$

We wrote the multiplication above in two different ways. We can flip the order of writing the numbers on the left. This is called the commutative property of multiplication.

(Video) Integers Multiplication and Division

by Math Antics

Here's a great video by mathantics that shows you what multiplying by $-1$ does to numbers.

$\large{\text{Summary}}$

Multiplying by $-1$ switches the number to the opposite side of zero on the number line. In other words, it flips the number's sign. $$\boxed{a\times-1=-a}$$ $$\boxed{-1\times a=-a}$$

Example 1

$5\times-1=-5$

Example 2

$-3\times-1=3$

Even if we multiply a number by $-1$, its absolute value stays the same.