Integers & Absolute Value
Negative numbers on the number line, and what the bars around a number really mean.
The explanation
Integers are whole numbers including the negatives: … −3, −2, −1, 0, 1, 2, 3 … They sit evenly spaced on a number line with zero in the middle.
Absolute value, written |x|, asks one question: how far is this number from zero? Distance has no direction, so the answer is never negative. |−7| = 7 and |7| = 7.
Careful, though: |x| is not "delete the minus sign." It is distance. −|−7| is still −7, because the absolute value happens first and the minus sign outside stays.
The integers ℤ extend the naturals so that subtraction is always defined. On the number line, −a is the reflection of a across 0, which makes ℤ closed under addition, subtraction and multiplication (but not division).
Absolute value is formally a piecewise function:
|x| = x if x ≥ 0
|x| = −x if x < 0
Note that the second branch produces a positive output precisely because negating a negative gives a positive. Geometrically |x| is the distance from x to 0, and more generally |a − b| is the distance between a and b on the line — a reading that makes absolute value equations and inequalities far easier to solve later, since |x − 3| < 5 simply says "x is within 5 units of 3."
Worked example
Evaluate |−12| + (−|4|) − |3 − 10|.
- |−12| = 12.
- −|4| = −4, because the bars are applied before the outside sign.
- |3 − 10| = |−7| = 7.
- 12 + (−4) − 7.
Answer: 1
Common mistakes
- Writing |−5| = −5. Distance cannot be negative.
- Simplifying |3 − 10| as |3| − |10|. Do the inside first; absolute value does not distribute over subtraction.