Absolute Value
Distance from Zero
Learn how absolute value measures the distance a number is from \(0\) on a number line: always positive, always useful.
6th Grade Math
What Is Absolute Value?
The absolute value of a number tells you how far that number is from \(0\) on a number line. It doesn't care about direction: left or right doesn't matter. Only distance matters.
Visualize It on the Number Line
The number \(-3\) sits 3 units to the left of zero. Absolute value ignores the direction and reports only the distance: \(|{-3}| = 3\).
Why Does Absolute Value Matter?
Temperature
"It's \(-8\,^\circ\text{F}\)" outside" tells direction (below zero). The magnitude is \(|{-8}| = 8\) degrees from zero.
Money
A debt of \(-\$50\) has a magnitude of \(|{-50}| = 50\) dollars: the size of what you owe.
Elevation
A submarine at \(-200\) ft is \(|{-200}| = 200\) feet from sea level.
The Formal Rule
Don't let "\(-a\)" confuse you. When \(a\) is already negative, the extra negative sign makes it positive. For example: if \(a = -7\), then \(-a = -(-7) = 7\).
Both Directions, Same Distance
A number and its opposite are the same distance from zero. That's why their absolute values are always equal: \(|{-4}| = |4| = 4\).
Negative Sign vs Absolute Value
| Expression | Meaning | Result |
|---|---|---|
| \(-6\) | Negative six (a position) | \(-6\) |
| \(|{-6}|\) | Distance of \(-6\) from \(0\) | \(6\) |
| \(-|6|\) | The negative of the absolute value of \(6\) | \(-6\) |
| \(-|{-6}|\) | The negative of the absolute value of \(-6\) | \(-6\) |
Three Key Properties
1
Always non-negative: \(|a| \geq 0\) for every number \(a\). Distance can never be negative.
2
Zero only at zero: \(|a| = 0\) if and only if \(a = 0\). The only number with zero distance from zero is zero itself.
3
Opposites are equal: \(|a| = |{-a}|\). A number and its opposite share the same distance from zero.
How to Find Absolute Value
1
Look at the number inside the bars. Is it positive, negative, or zero?
2
If positive or zero: the absolute value is the number itself. \(|7| = 7\), \(|0| = 0\).
3
If negative: drop the negative sign. \(|{-7}| = 7\).
4
Check for a negative sign outside the bars. If present, apply it after: \(-|{-7}| = -7\).
Example 1: Simple Integers
Example 2: Fractions and Decimals
\(\left|{-\frac{3}{4}}\right|\)
The fraction \(-\frac{3}{4}\) is \(\frac{3}{4}\) units from zero. Answer: \(\frac{3}{4}\)
\(|{-2.5}|\)
The decimal \(-2.5\) is \(2.5\) units from zero. Answer: \(2.5\)
Example 3: Negative Outside the Bars
1
Problem: Find \(-|{-9}|\).
2
Inside first: \(|{-9}| = 9\). The distance from zero is \(9\).
3
Apply the outside negative: \(-9\).
4
Final answer: \(-|{-9}| = -9\).
Absolute Value as Magnitude
Real-World Magnitude Examples
Bank Account
A balance of \(-\$120\) means you owe money. The magnitude of the debt is \(|{-120}| = \$120\).
Altitude
Death Valley sits at \(-282\) ft. Its distance below sea level is \(|{-282}| = 282\) ft.
Football
A loss of \(-8\) yards has a magnitude of \(|{-8}| = 8\) yards. The play moved the ball 8 yards.
Common Mistakes to Avoid
✕ "Absolute value makes things negative"
Wrong: absolute value never gives a negative result. \(|{-5}| = 5\), not \(-5\).
✕ Ignoring the outside negative
Students write \(-|3| = 3\). The correct answer is \(-|3| = -3\). The negative outside applies after.
✕ "Absolute value of 0 is undefined"
Zero is 0 units from itself: \(|0| = 0\). It's perfectly defined and equals zero.
Quick Reference Table
| Expression | Value | Why? |
|---|---|---|
| \(|12|\) | 12 | Already positive |
| \(|{-12}|\) | 12 | 12 units from 0 |
| \(|0|\) | 0 | Zero distance from 0 |
| \(-|12|\) | \(-12\) | Negative applied after |
| \(\left|{-\frac{1}{2}}\right|\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) unit from 0 |
| \(|{-3.7}|\) | 3.7 | 3.7 units from 0 |
Question 1 of 5
What is \(|{-14}|\)?
The number \(-14\) is 14 units from \(0\) on the number line. Absolute value always gives the non-negative distance, so \(|{-14}| = 14\).
Question 2 of 5
What is \(-|7|\)?
First, evaluate inside: \(|7| = 7\). Then apply the negative sign outside: \(-7\). The negative sign is not inside the bars, so it stays.
Question 3 of 5
A scuba diver is at an elevation of \(-45\) feet. What is the magnitude of the diver's depth?
Magnitude means absolute value: \(|{-45}| = 45\). The diver is 45 feet below sea level. We don't say the magnitude is negative: magnitude is always a non-negative quantity.
Question 4 of 5
Which statement is true?
\(|{-3}| = 3\) and \(|2| = 2\). Since \(3 > 2\), the correct comparison is \(|{-3}| > |2|\). Absolute value converts to distance first, then you compare the distances.
Question 5 of 5
What is \(\left|{-\frac{5}{8}}\right|\)?
Absolute value works the same for fractions. \(-\frac{5}{8}\) is \(\frac{5}{8}\) units from \(0\). Drop the negative sign: \(\left|{-\frac{5}{8}}\right| = \frac{5}{8}\).