Absolute Value: Definition, Examples and How to Find It
Part of the Math Concepts guide for Grades 3 to 8.
What Is Absolute Value?
The absolute value of a number is its distance from zero on the number line, regardless of direction. Absolute value is always zero or positive; it can never be negative. The absolute value of a number x is written as |x|.
- |5| = 5 (5 is 5 units from zero)
- |−5| = 5 (−5 is also 5 units from zero)
- |0| = 0 (zero is zero units from itself)
Key Formulas
| Rule | Formula | Example |
|---|---|---|
| Positive number | |a| = a | |8| = 8 |
| Negative number | |−a| = a | |−8| = 8 |
| Zero | |0| = 0 | Zero is always zero units from itself |
| Formal definition | |x| = x if x ≥ 0; |x| = −x if x < 0 | |−3| = −(−3) = 3 |
Absolute Value Examples
Example 1: Basic values
Problem: Evaluate |−7|, |4|, |0|, |−15|
- |−7| = 7 (7 units from zero)
- |4| = 4 (4 units from zero)
- |0| = 0
- |−15| = 15
Answer: 7, 4, 0, 15
Example 2: Expression inside absolute value
Problem: Evaluate |3 − 8|
- Evaluate inside the bars first: 3 − 8 = −5
- Apply absolute value: |−5| = 5
Answer: 5. Always simplify what is inside the absolute value bars before applying the absolute value.
Example 3: Comparing absolute values
Problem: Which is greater: |−10| or |7|?
- |−10| = 10
- |7| = 7
- Compare: 10 > 7
Answer: |−10| > |7|. A negative number can have a larger absolute value than a positive number.
Example 4: Real-world context
Problem: A submarine is 200 metres below sea level (−200 m). What is its absolute depth?
- Depth = −200 m
- Absolute depth = |−200| = 200 m
Answer: 200 metres. Absolute value removes the direction and gives us just the size of the number.
Absolute Value in Everyday Life
Here are some situations where your students will encounter absolute value without even realising it.
- Temperature change: If the temperature drops from 5°C to −3°C, the absolute change is |5 − (−3)| = |8| = 8 degrees.
- Distance: Distance is always positive. If you travel backwards 4 km, the distance travelled is |−4| = 4 km.
- Bank accounts: A balance of −$50 means you owe $50. The absolute value |−50| = $50 represents the amount owed.
- Elevation: A location 300 metres below sea level sits at −300 m. Its absolute elevation is 300 m.
Common Mistakes to Avoid
Mistake 1: Thinking absolute value always makes a number positive
This is almost true but slightly off. Absolute value makes a number non-negative, which means it can be zero or positive. |0| = 0, not a positive number. A more precise statement is: absolute value is always greater than or equal to zero.
Mistake 2: Applying absolute value before simplifying
When you have an expression inside the bars, simplify it first. |3 − 8| is not |3| − |8| = 3 − 8 = −5. You must evaluate 3 − 8 = −5 first, then take the absolute value: |−5| = 5.
Mistake 3: Confusing absolute value with a negative sign
|−x| does not equal −x. It equals x (assuming x is positive). For example, if x = 5, then |−5| = 5, not −5. The absolute value bars always give a non-negative result.
Common Questions About Absolute Value
What is absolute value?
Absolute value is a number's distance from zero on the number line. It is always zero or positive. The absolute value of −5 is 5 because −5 is 5 units away from zero. The absolute value of 5 is also 5.
Why is absolute value always positive?
Absolute value represents distance, and distance is always zero or positive. You cannot travel a negative distance. Both 5 and −5 are the same distance from zero, so both have an absolute value of 5.
What does |x| mean?
The vertical bars | | are the notation for absolute value. |x| means the absolute value of x. For example, |−8| = 8 and |8| = 8.
What is the difference between absolute value and negative numbers?
A negative number has a value less than zero. Absolute value is always zero or positive; it converts negatives to positives. The absolute value of a negative number is the same number without the minus sign: |−6| = 6.
Can absolute value be negative?
No. Absolute value is always zero or positive. The expression |x| ≥ 0 for all real numbers x.
What grade do students learn absolute value?
Absolute value is introduced in 6th grade in the United States (Common Core 6.NS.C.7), as part of the unit on positive and negative rational numbers and number line understanding.
