Lesson 05 · The two-sided number line
So far every number has lived to the right of 0. Now we use the whole line. One geometric move — reflection across 0 — builds the rational numbers. You will not see a minus sign anywhere on this page. That is deliberate, and Wu will explain why.
“The fractions together with their reflected images will be seen to form a number system in the sense that we can perform the four arithmetic operations on them all, in a way that is consistent with the operations already defined on the fractions.”— Hung-Hsi Wu, Ch. 25, p. 375
Recall from Lesson 01 that a number is, by definition, a point on the number line. The rational numbers begin with a single new idea:
Definition · H. Wu
“Take any point p on the number line—p could be on either side of 0 and, in particular, it does not have to be a fraction—and we will define its mirror reflection p* as follows. If p = 0, we define 0* to be 0. If p ≠ 0, then p* is the point of the number line on the opposite side of 0 so that p and p* are equidistant from 0.” — Understanding Numbers in Elementary School Mathematics, Ch. 25, p. 375
Drag the teal point p — to either side of 0. Its mirror reflection p* follows. The two distances from 0 stay equal because equidistance is the definition, not a coincidence. And watch the labels when you drag p across 0: reflecting an already-starred point stacks a second star, and the two stars cancel. Keyboard: focus p and use ← →.
Equidistance is marked under the line. Drag p across 0 to see p* pick up a second star — and shed it by (25.1).
0* = 0 and p** = p. In Wu’s words: “The first is just a definition, and the second is nothing but a succinct way of expressing the fact that reflecting a point across 0 twice in succession brings the point back to itself.” (p. 375)
With reflection in hand, the definition takes one sentence. Fractions live to the right of 0; their reflections — 1*, 2*, (⅔)* — live to the left:
Wu’s own example points (p. 376). Remember that the fractions include the whole numbers.
Definition · H. Wu
“The set of all the fractions and their mirror reflections with respect to 0, i.e., the numbers m/n and (k/l)* for all whole numbers k, l, m, n (l ≠ 0, n ≠ 0), is called the rational numbers.” — Ch. 25, p. 376
Definition · H. Wu
“The set of whole numbers and their mirror reflections,”
…, 3*, 2*, 1*, 0, 1, 2, 3, …
“is called the integers.” — Ch. 25, p. 376
Using “⊂” to denote “is contained in”, the three number systems built so far nest inside one another:
whole numbers ⊂ integers ⊂ rational numbers
On the two-sided line, comparing numbers stays what it has always been on this site: a statement about location.
Definition · H. Wu
“We extend the concept of order among numbers from fractions to all numbers: for any x, y on the number line, x < y means that x is to the left of y. An equivalent notation is y > x.” — Ch. 25, p. 376
Drag both points. The sentence below rewrites itself from their positions alone — no rules about signs are needed, because no signs exist yet.
“To the left of” is the entire definition of <. Each point also reports whether it is positive, negative, or neither.
Definition · H. Wu
“Numbers which are to the right of 0 (thus those x satisfying x > 0) are called positive, and those which are to the left of 0 (thus those that satisfy x < 0) are negative. So 2* and (⅓)* are negative, while all nonzero fractions are positive. The number 0 is, by definition, neither positive nor negative.” — Ch. 25, p. 376
You have surely noticed what this page is not doing: writing 2* as “negative two.” That is deliberate, and Wu’s reasoning is the pedagogical heart of this whole approach:
“The negative sign, having to do with the operation of subtraction, simply will not figure in our considerations until we begin to subtract rational numbers. Moreover, the terminology of ‘negative sign’ carries certain psychological baggage that may interfere with the learning of the basic facts about rational numbers the proper way. For example, if a = −3, then there is nothing ‘negative’ about −a, which is 3. It is therefore best to withhold introducing the negative sign until its natural appearance in the context of subtraction in Chapter 27.” — Hung-Hsi Wu, Ch. 25, pp. 376–377
In other words: “−” is an operation waiting to be defined, not a name for the left half of the line. The star lets students learn where rational numbers are before any subtraction enters the picture. When subtraction is defined — next lesson — the negative sign arrives with an honest job to do.
Wu’s answer: “To answer this question, we take up the problem of solving equations” (p. 379). Each expansion of the number system earns its keep by solving equations the old system could not. Compare the same two numbers, 7 and 5, under multiplication and under addition:
Which whole number, multiplied by 7, equals 5? None. But allow x to be a fraction and, in Wu’s words, “the situation changes radically: the problem now has an answer and it is 5⁄7.” (p. 379)
✓ solved by the fractions: x = 5/7
Which whole number, added to 7, equals 5? Again none — and this time, Wu notes, “allowing x to be a fraction does not improve the situation.” (p. 380)
✗ unsolvable by the fractions — a genuine gap
The gap is visible on the line: starting at 7, you must land on 5 — a move to the left. No fraction can take you there. Press solve and watch which number does.
The equation demands a leftward move of length 2 — and the rational number 2* is exactly the number that provides it.
“Now if we add the negative fractions—the mirror reflections of the fractions—to fractions to obtain the rational numbers, and if the concept of addition is correctly extended from fractions to the rational numbers, then there will be solutions to every equation a + x = b for any fractions a and b. For example, we shall see that x = 2* is the solution to 7 + x = 5.” — Hung-Hsi Wu, Ch. 26, p. 380
“Correctly extended” is a promise, and the next lesson keeps it. Extending addition to the whole two-sided line requires exactly one new geometric idea — a segment with a direction — and it is the subject of Wu’s Chapter 27. Definitions first, then the machinery; the star notation means that when the negative sign finally appears, it will mean precisely one thing.