The sign gives the direction. The number gives the distance. Start at the first number and jump.
1Add on a number line
−3 + 7 = 4
Owing $3, then earning $7: you cross zero and finish with 4. Adding a negative would jump left instead.
2Add on a vertical number line
3 + (−7) = −4
A bird at 3 m dives 7 m: it passes the surface and ends 4 m under.
Same rule, vertical: + jumps up, − jumps down. Elevation and depth use this line.
3Rewrite subtraction as addition
p − q = p + (−q)
3 − 8→3 + (−8)→−5
Every subtraction becomes an addition you can draw.
Subtracting 8 and adding −8 land on the same spot. Every time, for any p and q.
4Distance has no sign
Distance is |p − q|. |−3 − 4| = |−7| = 7. The change from −3 to 4 is +7; the change back is −7; the distance both ways is 7. Distance is never negative.
