Negative Numbers — Walking the Other Direction
Last lesson, every step you took was to the right. This one just adds a second direction — and that's all a negative number has ever been.
Intuition First
Go back to that sidewalk from last lesson. So far you've only ever walked to the right of home (0) — 3 steps right, 8 steps right. But nothing stops you from turning around and walking left instead. A negative number is exactly that: the same number of steps, just walked in the opposite direction. −3 isn't a different kind of number from 3 — it's the identical 3 steps, facing the other way.
This shows up everywhere once you look for it. A bank balance of −$30 means you're 30 dollars short of zero, the same distance from "even" as +$30 is, just on the debt side. An elevation of −20 meters means 20 meters below sea level — same distance from the shoreline, opposite side. In every case, the number tells you the size of the step, and the sign tells you which way you're facing.
A Balance You Can Check by Hand
Your bank account is overdrawn by $30 — a balance of −$30. You deposit $50. What's your new balance?
You started 30 steps left of home, then walked 50 steps right — 30 of those steps just get you back to zero, and the remaining 20 land you on the positive side. Adding a positive number always means walking right, no matter where you started.
Try It — Walk in Either Direction
Set a starting position, then drag the steps slider — positive steps walk right, negative steps walk left. Watch where you land.
Formal Definition
Every number a on the line has an opposite, −a — the same distance from zero, but on the other side:
Adding two numbers means walking one, then walking the other from wherever you land. Adding a negative number is just walking backward, which is the same as subtracting its positive twin: