Skip to main content
Daily Math Minute

The Number System

Absolute Value

Interpreting absolute value as distance from zero.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Negative Numbers & the Number Line

Distance Doesn't Care Which Direction

A submarine is 300 feet below sea level; a weather balloon is 300 feet above it. Before reading on, think about this: are these two vehicles the same distance from sea level? What would you say if asked 'how far' each one is, without mentioning up or down?

Definition — Absolute Value

The distance a number is from zero on the number line, always expressed as a non-negative value. Written with vertical bars, like |−5|, which equals 5.

Both the submarine and the balloon are exactly 300 feet from sea level — the direction is different, but the distance is identical. That's exactly what absolute value measures: |−300| = 300 and |300| = 300, because distance itself never comes with a direction attached. A number and its opposite always share the same absolute value.

Worked Example — Evaluating Absolute Value

Find |−9| and |9|. Both represent a distance of 9 units from zero, regardless of sign: |−9| = 9 and |9| = 9.

Worked Example — Comparing Values Using Absolute Value

Which is farther from sea level: an elevation of −45 feet, or an elevation of 30 feet? Compare their absolute values: |−45| = 45, and |30| = 30. Since 45 is greater than 30, the −45 foot elevation is farther from sea level, even though −45 is a smaller number than 30.

Tip

Don't confuse 'which number is greater' with 'which number has the greater absolute value' — they're different questions. 30 is greater than −45 as a number, but −45 is farther from zero.

Common Mistakes

  • Treating absolute value bars as if they simply remove a negative sign wherever one appears in a longer expression, without first evaluating what's inside.

    Evaluate whatever is inside the absolute value bars completely first, then take the distance from zero of that one final result.

  • Confusing 'greater value' with 'greater absolute value,' such as assuming the number farther from zero is always the larger number.

    A number's value and its absolute value answer different questions — a very negative number can have a large absolute value while still being a small (very negative) number overall.

Key Takeaways

  • Absolute value measures a number's distance from zero, always as a non-negative value.
  • A number and its opposite always have the same absolute value.
  • Comparing absolute values answers 'which is farther from zero,' a different question from 'which number is greater.'

Summary

Absolute value strips away direction to measure pure distance from zero. With negative numbers and their distances understood, the next lesson extends the coordinate plane from Grade 5 to include all four directions at once.

Sign in to track your progress and mark this lesson complete.

Track your progress
← Negative Numbers & the Number LineBack to The Number SystemThe Coordinate Plane with Negative Numbers→