The Number Line — Numbers as Positions You Can Walk To

Picture a long sidewalk with evenly spaced marks painted on it, like a giant tape measure stretched out on the ground. Stand at the mark labeled 0 — call that "home." Every other mark is just a distance from home: walk 3 marks to the right and you're standing at 3. Walk 7 marks to the right and you're standing at 7. The number isn't a mysterious symbol — it's simply how many steps you took, and which way you're facing.

This is exactly what a thermometer or a treadmill display is doing. The thermometer's liquid doesn't "mean" 72 degrees in some abstract sense — it has physically risen to the mark labeled 72, the same way you'd walk to that mark on the sidewalk. Once numbers are positions on a line, two of them can be compared just by looking at which one is farther right, and the distance between them is just the length of sidewalk you'd have to cover to get from one to the other.

You're standing at the mark for 3. A friend is standing at the mark for 8. How many steps do you have to walk to reach them?

Subtracting the smaller position from the larger one gives you exactly the number of steps between them — the distance. It doesn't matter which direction you're facing when you start; the sidewalk between 3 and 8 is always 5 marks long.

Drag the slider to walk your marker along the line. Watch your position update, and watch the distance to the fixed flag change as you get closer or farther from it.

The number line is an ordered arrangement of numbers along a straight line, where every number has exactly one position, and position increases as you move right. For any two positions a and b, the distance between them is:

The vertical bars mean "strip off any negative sign" — they guarantee a distance is never negative, no matter which point you subtract from which: