Absolute value: distance, never negative
The absolute value of a number, written |x|, is its distance from 0 on the number line — and distance is never negative. So |3| = 3 and |−3| = 3, because both 3 and −3 are three steps from zero. The operation simply throws away the sign: |x| = x when x ≥ 0, and |x| = −x when x < 0 (and yes, −x is positive when x is negative).
Opposite vs. reciprocal: two different undos
Two numbers can “cancel” each other in two completely different senses, and beginners mix them up constantly. The opposite of x is its additive inverse −x: it cancels under addition, because x + (−x) = 0. The reciprocal of x is its multiplicative inverse 1/x: it cancels under multiplication, because x · (1/x) = 1. The opposite of 5 is −5; the reciprocal of 5 is 1/5. They are not the same thing.
number x opposite −x reciprocal 1/x
──────── ─────────── ──────────────
5 −5 1/5
−4 4 −1/4
2/3 −2/3 3/2 (flip it!)
0 0 undefined (no 1/0)
Checks:
5 + (−5) = 0 ← opposites add to 0
5 · (1/5) = 1 ← reciprocals multiply to 1
(2/3)·(3/2) = 6/6 = 1 ✓For a fraction the reciprocal is gorgeously simple: just swap top and bottom, so the reciprocal of 2/3 is 3/2. The one outlaw is zero: 0 has an opposite (itself), but no reciprocal, because 1/0 is undefined — you cannot divide by zero. Keep that exception in your pocket; it will explain a great many “undefined” answers later in algebra.