绝对值是什么？

(Video) Absolute Value and Evaluating Numbers

by PatrickJMT

This video by PatrickJMT explains absolute values.

$\large{\text{Summary}}$

An absolute value essentially just turns negative values into positive values. In more technical terms, it expresses the distance between a number and zero on a number line. Since distance isn't negative, the absolute value makes all values nonnegative.

Example 1: $|5|$ and $|-5|$

The central idea of absolute value is that it takes in any number and makes it nonnegative. Since $5$ is positive, absolute value does not affect it. Since $-5$ is negative, absolute value makes it positive. $$|5|=5$$ $$|-5|=5$$ Note: absolute value also keeps $0$ the same, so $|0|=0$.

Example 2: $|2-8|$

When there are expressions within absolute value bars, that expression is evaluated before the absolute value. This means that solving $|2-8|$ would first require solving $2-8$ and then taking its absolute value. $$\begin{align} |2-8|&=|-6|\\ &=6 \end{align}$$

This is very similar to how parentheses work in order of operations. Evaluate the $2-8$ first as it's inside one set of absolute value signs.

Example 3: $|2| - |10|$

The absolute value bars do not affect anything outside of them, so the minus between the two absolute value expressions is preserved. $$\begin{align} |2| - |10|&=2-10\\ &=-8 \end{align}$$

More complicated expressions

We can now introduce absolute value into our existing arithmetic equations. $$\begin{align} 5|-3| +|-8|^2&=5(3)+(8)^2\\ &=15+64\\ &=79 \end{align}$$

$$\begin{align} |3(4-8)| - | 2(1+4)|^2&=|3(-4)|-|2(5)|^2\\ &=|-12| -|10|^2\\ &=12-(10)^2\\ &=12-100\\ &=-88 \end{align}$$

In these examples simplify what's inside the absolute value signs first. If there is an exponent outside, evaluate that afterward.

Visualizing Absolute Value

We can think of every number as having two components: a value and a sign. The value of a number is the positive distance between that number and zero on a number line. The sign of a number is either positive or negative—this is the direction that the number is relative to zero on the number line.

For example, the number $3$ has a value of $3$ and a sign of $+$. Similarly, $-3$ has a value of $3$ and a sign of $-$.

Absolute value comes down to caring about the value of a number regardless of its sign. Let's take a look at how this works.

What are the absolute values of $3$ and $-3$?

The absolute value of $3$ is $3$. We can see that there are $3$ units between $0$ and $3$ on the number line.

The absolute value of $-3$ is also $3$! We can see that there are $3$ units between $0$ and $-3$ on the number line.

Mathematically, we write these as: $$\lvert 3\rvert=3$$ $$\lvert -3\rvert=3$$ Absolute value cares about the value of the number, regardless of its sign.

Absolute value practice

The absolute value of a number can be thought of as the distance of the number from $0$ on the number line. Absolute value is always nonnegative, since distance is always nonnegative.

For example, the absolute value of $8$ is $8$, since $8$ is $8$ units from $0$ on the number line. The absolute value of $-8$ is also $8$, since $-8$ is also $8$ units from $0$ on the number line.

If you start with a positive number or zero, its absolute value is itself. If you start with a negative number, its absolute value is its opposite.

$10$ is a positive number, so its absolute value is itself, $10$.

$-12$ is a negative number, so its absolute value is its opposite, $12$.

The absolute value of $7$ is $7$, as is the absolute value of $-7$, since they are equidistant from $0$ on the number line.

Since absolute value denotes distance—a nonnegative value—there are no numbers with a negative absolute value.