Absolute Value
In short: Absolute value is the distance between a number and zero on the number line, so |-9| = 9 and |9| = 9, and it is never negative.
6th GradeNumbers and Operations6.NS.C.7
What is Absolute Value?
The absolute value of a number answers “how far from zero?” without caring about direction. A hiker 85 feet below a trailhead and a climber 85 feet above it are the same distance from the start, even though one height is negative and the other is positive. On a number line, it is simply the length of the segment that joins the number to the origin, measured in whole or partial units, which is why it can never come out negative.
Precisely, \(|a| = a\) when \(a \ge 0\) and \(|a| = -a\) when \(a < 0\). The bars are read "the absolute value of", and \(|0| = 0\). Because it measures a distance, \(|a| \ge 0\) for every number \(a\), and two numbers that are opposites share the same value.
How to spot it
- A story asks for a distance, a depth or a size where the direction of travel does not matter.
- Vertical bars surround a number or expression, as in \(|-14|\).
- You are asked which of two values is farther from zero, even though one is negative.
- A balance or elevation is given with a sign, and the question asks for its magnitude.
The method
- Place the number on a horizontal number line, with 0 in the middle as the starting point.
- Count the steps from 0 to the number, ignoring whether you walk left or right.
- Write that count as a nonnegative number: \(|-9| = 9\).
- For an expression, simplify inside the bars first, then take the distance.
Absolute Value: solved example
Problem
A cave explorer in Kentucky is at an elevation of \(-124\) feet relative to the entrance, while a bat flies at \(+87\) feet. Which one is farther from the entrance, and by how much?
Solution
Distances: \(|-124| = 124\) and \(|87| = 87\).
The explorer is farther, by \(124 – 87 = 37\). So the explorer is 37 feet farther from the entrance level.
Try it
Question 1
A savings account balance is \(-\$46\) after a fee. What is the absolute value of the balance, and what does it tell you?
Show the answer
\(|-46| = 46\). The account is 46 dollars below zero, so the owner is overdrawn by \$46.
Question 2
Which is greater, \(|-15|\) or \(|12|\)? Explain.
Show the answer
\(|-15| = 15\) and \(|12| = 12\), so \(|-15|\) is greater: it is farther from zero.
Absolute Value vs. Opposite Numbers
An opposite changes the sign: the opposite of \(-9\) is 9, and the opposite of 9 is \(-9\). The absolute value of both is 9, and it always comes out nonnegative. Taking absolute value erases the sign; taking the opposite flips it.
Keep going
Lessons
Worksheets and quizzes
- Ordering Rational Numbers and Absolute Value – Worksheet
- Ordering Rational Numbers and Absolute Value – Quiz
Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations