Opposites and absolute value
Distinguish a signed position from its distance from zero.
Before you begin
- Ordering negative numbers
- Distance on a number line
Reflect a number across zero
Opposite numbers lie equally far from zero on opposite sides. The opposite of 4 is −4, and the opposite of −4 is 4. Applying the opposite operation twice returns to the starting number, so −(−a) = a. Zero is its own opposite. This is a reflection of position, not an instruction to make every number negative.
Compare magnitude separately from value
Absolute value measures distance from zero. Both |−4| and |4| equal 4 because distances are nonnegative. A balance of −$60 is smaller than −$25, but its debt has the greater magnitude. State whether you are comparing signed values or the size of an amount; those questions can give different orderings for negative numbers.
Worked example
Two divers are at −8 m and −3 m relative to the surface. Which has the smaller elevation and which is farther from the surface?
Show the worked solution
- On the number line, −8 is below −3, so −8 m is the smaller elevation.
- Compute the distances from zero: |−8| = 8 and |−3| = 3.
- The diver at −8 m is 8 m from the surface, which is farther.
The diver at −8 m has the smaller elevation and is farther from the surface.
Try it yourself
What is −(−2.7)?
Common mistakes
- Absolute value does not preserve the ordering of negative values.
- The opposite of a negative number is positive.
What can you explain now?
Compare account balances of −$42 and −$18. Which balance is greater, and which debt is larger?
Compare with the explanation
−$18 is the greater balance; the $42 debt is larger.
−18 is to the right of −42. Debt size uses absolute value: 42 is greater than 18.
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Common Core mathematics standards ↗