Absolute value
The idea
Absolute value answers one question: how far is this number from zero? Distance has no direction, so |−48| and |48| are both 48. You already know how the number line stretches in both directions, and absolute value simply measures the length of the walk back to zero, ignoring which side you started on. It matters whenever size counts more than sign — how big an error is, how extreme a temperature is, how far below sea level something sits.
The trap is treating absolute value as a button that 'makes things positive' inside any expression. It is really a measurement taken after you know the number: |3 − 8| means find 3 − 8 = −5 first, then take its distance from zero, 5. That same idea gives a beautiful tool: the distance between any two numbers a and b is |a − b|, no matter their signs. Comparing |−48| with |35| compares sizes while a plain comparison of −48 with 35 compares positions.
Worked example
A delivery drone hovers 35 m above sea level while a submarine sits at −48 m. Which one is farther from sea level, and how far apart are they vertically?
- Distance from sea level is an absolute value: the drone is |35| = 35 m away and the submarine is |−48| = 48 m away.
- Compare the distances, not the raw numbers: 48 > 35, so the submarine is farther from sea level even though −48 < 35 as positions.
- The vertical gap between them is the distance between their positions: |35 − (−48)| = |35 + 48| = 83 m.
- Interpret why the distances add: the two are on opposite sides of sea level, so the gap is the drone's 35 m plus the submarine's 48 m.
Answer. The submarine is farther from sea level (48 m versus 35 m), and the two are 83 m apart vertically.
Check your understanding
- Why can an absolute value never be negative, and what would a negative distance even mean?
- How is comparing |−48| and |35| a different question from comparing −48 and 35?
- Why does |a − b| give the distance between a and b even when both numbers are negative?
- Where could blindly making every number positive lead you to a wrong answer in a multi-step problem?
Build the foundations first
Absolute value builds on these concepts. If any feel shaky, start there.