Understanding Absolute Value as Distance from Zero

Welcome back. In the previous lesson, you learned that opposite integers lie on different sides of $0$ but are the same distance from it. Absolute value keeps the distance part of that idea and ignores the direction.

In this lesson, you will determine an integer’s absolute value by viewing it as its distance from zero on a number line. You will also learn the absolute-value symbol and apply the idea to situations such as elevations and temperatures.

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Distance, not direction

A number on a number line gives two pieces of information:

1. its direction from zero: right for positive, left for negative;
2. its distance from zero: the number of equal units between it and $0$.

For example, $5$ is five units to the right of zero. The integer $-5$ is five units to the left of zero. Their directions differ, but their distances are identical.

The absolute value of an integer is its distance from $0$. The notation uses two vertical bars around the number:

$$|5|=5$$ $$|-5|=5$$

Read the second statement as “the absolute value of negative five is five.”

Notice what changed: $-5$ tells you a location five units left of zero. But $|-5|$ tells you only the distance: five units. A distance cannot be negative.

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Finding absolute value on a number line

To find an integer’s absolute value, imagine starting at $0$ and counting the unit intervals until you reach the integer. Do not count left intervals as negative; you are counting a distance.

For $-8$, begin at $0$, move eight units left, and stop at $-8$. The distance traveled is $8$ units:

$$|-8|=8$$

For $12$, begin at $0$, move twelve units right, and stop at $12$. The distance traveled is also $12$ units:

$$|12|=12$$

This leads to a useful shortcut for integers:

Integer inside the bars	Absolute value
A positive integer, such as $14$	Keep it: $|14|=14$
A negative integer, such as $-14$	Use its distance: $|-14|=14$
Zero	$|0|=0$

The shortcut is reliable, but keep the definition in mind: absolute value is distance from zero. That definition explains the shortcut instead of making it a fact to memorize.

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The important case of zero

Zero is neither positive nor negative. It is already at the reference point, so it is zero units from itself.

$$|0|=0$$

Absolute value is therefore never negative. For a nonzero integer, its absolute value is positive; for zero, its absolute value is zero.

For example, these results are impossible:

$$|-6|=-6$$ $$|0|=1$$

The first cannot be true because $-6$ is six units, not negative six units, from zero. The second cannot be true because there is no distance to travel from $0$ to $0$.

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Absolute value and opposites: related but different

The previous lesson on opposites is helpful here, but the words opposite and absolute value ask different questions.

• “What is the opposite of $-8$?” asks for the number on the other side of zero: $8$.
• “What is the absolute value of $-8$?” asks for the distance from zero: $8$.

For a negative integer, the answers happen to look the same. That can be confusing. Use a positive integer to see the difference:

Question	Answer	Why
Opposite of $8$	$-8$	The direction switches sides of zero.
Absolute value of $8$	$8$	The distance from zero is eight units.

So absolute value does not mean “find the opposite.” It means “find the distance from zero.”

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Absolute value in real situations

Integers often describe locations relative to a reference point. Absolute value tells how far away something is from that reference point, even when the original integer tells whether it is above or below, warmer or colder, ahead or behind.

Elevation

Suppose a diver is $-18$ meters relative to sea level. The negative sign means the diver is below sea level. Their distance from sea level is:

$$|-18|=18$$

The diver is 18 meters below sea level, or simply 18 meters from sea level. The absolute value alone does not tell you “below”; it only gives the distance.

Temperature

A temperature of $-7$ degrees is seven degrees below zero. Its absolute value is:

$$|-7|=7$$

This does not mean the temperature is $7$ degrees. It means the temperature is 7 degrees away from zero.

Money

If a bank balance is $-25$ dollars, the negative sign can represent owing money. Its absolute value is:

$$|-25|=25$$

The amount owed is 25 dollars. Again, the original sign supplies the meaning of the situation; absolute value supplies the size of the amount.

A helpful sentence frame is:

The integer tells the position or direction relative to zero; its absolute value tells the distance or amount.

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A quick accuracy check

When you see an expression such as $|-270|$, use this brief check:

1. Identify the reference point: on a number line, it is $0$.
2. Ask how many units separate the integer from $0$.
3. Write that number as a nonnegative answer.

Thus,

$$|-270|=270$$

Likewise,

$$|44|=44$$

Both $270$ and $44$ are distances from zero, so neither answer should have a negative sign.

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Key takeaways

Absolute value is the distance of an integer from zero.

• The symbol for absolute value is a pair of vertical bars, such as $|-9|$.
• Positive and negative opposites have the same absolute value:

$$|9|=|-9|=9$$

• Absolute value is never negative.
• Zero is zero units from zero:

$$|0|=0$$

• In real-world contexts, absolute value gives the size of a distance or amount, while the sign gives its direction or meaning.

Next, you will use positions on the number line to compare integers using $<$, $>$, and $=$.