Absolute Value

Learn what absolute value means and how to find the distance from zero.

Intermediate20 minLesson
Printable worksheet

Definition

The absolute value of a number is its distance from zero on the number line. We write absolute value using vertical bars: ∣n∣|n|.
Since distance is always positive (or zero), absolute value is always non-negative:
∣5∣=5and∣−5∣=5|5| = 5 \quad \text{and} \quad |-5| = 5
Both 55 and −5-5 are exactly 5 units away from zero, so they have the same absolute value.
Key insight: Absolute value tells us "how far" without caring about direction (left or right).

Try it now

What is ∣−6∣|-6|?

Worked Examples

Find: ∣8∣|8| and ∣−8∣|-8|
1

For ∣8∣|8|: How far is 8 from zero?

Count from 0 to 8 on the number line → ∣8∣=8|8| = 8
2

For ∣−8∣|-8|: How far is −8-8 from zero?

Count from −8-8 to 0 on the number line → ∣−8∣=8|-8| = 8
3

Compare the results

Both 8 and −8-8 are 8 units from zero → ∣8∣=∣−8∣=8|8| = |-8| = 8

Common Mistakes

Thinking ∣−5∣=−5|-5| = -5

Why it's wrong: Students may not apply the "distance" concept and just keep the negative sign.

Correct: Absolute value is always non-negative. ∣−5∣=5|-5| = 5 because −5-5 is 5 units from zero.

Confusing −∣5∣-|5| with ∣−5∣|-5|

Why it's wrong: These look similar but mean different things!

Correct: ∣−5∣=5|-5| = 5 (find absolute value of −5-5), but −∣5∣=−5-|5| = -5 (take the negative of ∣5∣|5|).

Thinking absolute value changes the number inside

Why it's wrong: Students might think ∣−3∣|-3| means "remove the negative" for any expression.

Correct: Absolute value gives the distance from zero. It doesn't change the number; it measures how far it is from zero.

Interactive Visual

to

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy
What is ∣−6∣|-6|?

Why It Matters

Absolute value appears everywhere when we care about magnitude rather than direction:
  • Error measurement: If a measurement is off by −3-3 or +3+3, the error size is the same: ∣−3∣=∣3∣=3|{-3}| = |3| = 3
  • Distance: The distance between two cities is always positive, regardless of which way you travel
  • Temperature difference: Whether it's 55 degrees warmer or colder, the change is still 55 degrees
  • Sports: In golf, a score of −2-2 (under par) and +2+2 (over par) are both "2 away from par"
Absolute value strips away the sign and tells us only the size!

Real World Applications

Measurement Error

Scientists and engineers use absolute value to measure how accurate their measurements are.

Example:

If the actual temperature is 20 degrees Celsius and your thermometer reads 18 degrees Celsius, the error is ∣18−20∣=∣−2∣=2|18 - 20| = |-2| = 2 degrees.

1Try It Yourself

A recipe calls for exactly 250 grams of flour. You measure 243 grams.

What is the absolute error in your measurement?

Step 1: Write the mathematical expression

Calculate ∣243−250∣|243 - 250|

Temperature Changes

When comparing temperature changes, we often care about the size of the change, not the direction.

Example:

If the temperature dropped from 15 degrees Celsius to 8 degrees Celsius, the change is ∣8−15∣=∣−7∣=7|8 - 15| = |-7| = 7 degrees.

2Try It Yourself

Monday's high was 22 degrees Celsius. Tuesday's high was 17 degrees Celsius.

What was the temperature difference?

Step 1: Write the mathematical expression

Calculate ∣17−22∣|17 - 22|

Key Takeaways

  • 1Absolute value ∣n∣|n| represents the distance from zero on the number line
  • 2Absolute value is always non-negative: ∣n∣≥0|n| \geq 0
  • 3Opposites have the same absolute value: ∣n∣=∣−n∣|n| = |-n|
  • 4For any number nn: if n≥0n \geq 0, then ∣n∣=n|n| = n; if n<0n < 0, then ∣n∣=−n|n| = -n
  • 5Distance between two numbers: ∣a−b∣|a - b| or ∣b−a∣|b - a|

Frequently Asked Questions

Absolute value represents distance, and distance cannot be negative. You can't walk a negative number of steps! Whether you go left or right, you still cover a positive distance.
Absolute value represents distance, and distance cannot be negative. You can't walk a negative number of steps! Whether you go left or right, you still cover a positive distance.
∣0∣=0|0| = 0. Zero is zero units away from itself. It's the only number whose absolute value equals zero.
No! ∣−x∣=∣x∣|-x| = |x| (same distance from zero), but −∣x∣-|x| is the negative of the absolute value. For example: ∣−5∣=5|-5| = 5, but −∣5∣=−5-|5| = -5.

Glossary

Absolute value
The distance of a number from zero on the number line, written as ∣n∣|n|
Distance
The amount of space between two points, always non-negative
Non-negative
A number that is greater than or equal to zero
Magnitude
The size or amount of something, without regard to direction or sign
Opposites
Two numbers with the same absolute value but different signs, like 55 and −5-5

Formula Card

Definition

∣n∣=n|n| = n if n≥0n \geq 0; ∣n∣=−n|n| = -n if n<0n < 0

Absolute value equals the number if positive, or its opposite if negative

Opposites

∣n∣=∣−n∣|n| = |-n|

A number and its opposite have the same absolute value

Distance

d=∣a−b∣d = |a - b|

Distance between two numbers on the number line

Non-negativity

∣n∣≥0|n| \geq 0

Absolute value is always zero or positive

Today's challengeOne problem, three tries

More in This Topic

Introduction to Negative Numbers

Elementary20 min

Adding and Subtracting Integers

Elementary25 min

Comparing Integers

Elementary20 min

Ordering Integers

Elementary20 min