Key takeaway: Absolute value measures how many steps a number is from zero, ignoring direction, so the answer is never negative; bars show it, as in |−35| = 35 and |48| = 48.
Grade 66.NS.C.7Numbers and Operations
Definition of Absolute Value
Absolute value tells how far a number lies from 0 when you disregard which way you walked. A diver at −35 feet and a hiker at +35 feet stand in different places, yet each is 35 feet away from sea level, so both share an absolute value of 35.
A length cannot be negative, so the outcome is positive unless the number itself is 0. In everyday settings it often stands for a size without a direction: the amount of a debt, the depth of a dive, or the size of a temperature drop.
Formula and notation
Two vertical bars around a number or expression ask for its distance from zero:
\[|x| = x \text{ if } x \ge 0 \qquad |x| = -x \text{ if } x < 0\]
The bars work like parentheses: do the arithmetic inside first, then take the distance. A minus sign outside the bars is applied last.
Key facts
- A number and its opposite share the same value for this distance.
- The result is never negative, and only 0 gives 0.
- Comparing distances from zero is different from comparing the numbers themselves.
- A minus sign written outside the bars still makes the final answer negative.
Absolute Value example
Relative to the water’s surface, a scuba diver floats at \(-35\) feet and a camera drone hovers at \(48\) feet. Find how far each one is from the surface, then how far apart they are.
-
Diver: \(|-35| = 35\) feet.
-
Drone: \(|48| = 48\) feet.
-
One is below and one above the surface, so the two lengths add: \(35 + 48 = 83\), which matches \(|48 – (-35)| = 83\).
The distances are 35 ft and 48 ft, and the two are 83 ft apart.
Common mistake
Watch out
Kenji simplifies \(-|-9|\) and writes \(9\).
Why it is wrong
He removed both signs. Work from the inside: \(|-9| = 9\), then the minus sign outside the bars makes it \(-9\). The bars only act on what is inside them.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations