Lesson 01 · From a single choice
A line is just a line. Nothing on it is special — until you choose one point to call 0, and a second to call 1. That one choice fixes every whole number. Build it yourself below.
Definition · H. Wu
“…a line each of whose points is uniquely identified with a (real) number.” — Understanding Numbers in Elementary School Mathematics, §1.4 (quoted with permission)
“Fix a horizontal straight line and designate a point on it as 0.”— Hung-Hsi Wu, where every number line begins
An empty line. Choose where 0 goes to begin.
Watch what just happened. You didn't write the numbers 2, 3, 4, 5 down one by one — you laid the unit down, end to end, and they appeared where it landed. That's the whole idea: counting is the unit, stamped again and again. The number line turns counting into a picture you can see and measure.
Beginners often think the number line is handed down by nature. Most of it isn't a law — it's a convention we agree on; Wu calls drawing it rightward “entirely a matter of convention.” But once the conventions are set, the placement of every whole number is forced. Keeping these straight is the difference between memorizing a picture and understanding one.
The line is drawn horizontal, and the numbers grow to the right. Nothing mathematical depends on this — flip it or stand it up and the math is identical. We just agree on it.
ConventionYou may put 0 anywhere, and you may pick the length of [0, 1] — the unit — to be anything you like.
ForcedOnce 0 and the unit are chosen, every whole number has exactly one home: equally spaced points marching off to the right. You don't get to choose where 2 goes.
“Once a unit segment has been chosen on a given straight line, all the whole numbers are fixed on the line.”— Hung-Hsi Wu, the one choice that decides everything
Because the unit's length is yours to pick, a different choice gives a line that looks different — longer or shorter — yet behaves exactly the same. Drag the unit and watch: the picture stretches, but every whole number still lands once, equally spaced.
Stretched or squeezed — the spacing stays uniform and 2 is always twice as far as 1.
It helps to call the finished line a ruler — but that's an analogy, and it's worth saying so out loud. A ruler is a physical object; the number line is the idea of equally spaced points that the ruler happens to illustrate. Good visuals earn their keep by being honest about where the picture ends and the mathematics begins. That principle — visual intuition first, but always grounded in correct mathematics — is what every lesson here is built on.
Once the line is a ruler, addition stops being a fact to memorize and becomes something you can see: lay one length after another and measure where you land. That's Lesson 02.
“…addition is simple: concatenate, then measure the length.”— Hung-Hsi Wu, turning arithmetic into geometry