Lesson 06 · Wu, Chapter 27
In Lesson 02, addition was concatenation: lay two lengths end to end and measure. That worked because every number sat to the right of 0. Now numbers live on both sides — so a segment is no longer enough. It needs a direction. One upgrade, and the whole two-sided line opens up.
“The surprising twist here is that the subtraction of rational numbers will end up being defined in terms of addition.”— Hung-Hsi Wu, Ch. 27, p. 381
Wu names the plan in one sentence: the way to bring the left half of the line into addition and subtraction is “to make every rational number into a ‘vector’, and then define the addition of vectors” (p. 381). So first: what is a vector?
Definition · H. Wu
“A vector is a segment on the number line together with a designation of one of its two endpoints as a starting point and the other as an endpoint. We shall refer to the length of the segment as the length of the vector, and say the vector is left-pointing if the endpoint is to the left of the starting point, and right-pointing if the endpoint is to the right of the starting point. The direction of a vector refers to whether it is left-pointing or right-pointing.” — Understanding Numbers in Elementary School Mathematics, §27.1, p. 382
Drag either end. The circle is the starting point; the arrowhead is the endpoint. Notice what the readout tracks: just two facts, length and direction. Wu’s own examples are one button away — K is left-pointing with length 1, L is right-pointing with length 2.
A vector is a segment plus a choice: which end is the start, which is the end.
For adding rational numbers we don’t need arbitrary vectors — only the ones tied to a number:
Definition · H. Wu
“Let x be a number (not necessarily rational), then we define the vector x to be the one with its starting point at 0 and endpoint at x.” … “A special vector is a vector whose starting point is 0.” — §§27.1–27.2, pp. 382–383
“A special vector is completely determined by its length and its direction.” Wu’s examples: if x has length 3.7 and is left-pointing, then x = (3.7)*. If y has length 23/7 and is right-pointing, then y = 23/7. (p. 383)
Here is the whole mechanism. To add x and y, pick up y and slide it — without turning it — until its tail rests at x:
Definition · H. Wu
“Let x and y be numbers. Then the sum x + y is the special vector whose endpoint is the new endpoint of y after we slide y along the number line until its original starting point (which is 0) rests at x.”
“In intuitive language, the sum x + y is the special vector whose endpoint is obtained by ‘going from the point x along a vector with the same length and same direction as y.’” — §27.2, p. 383
Set x and y, then press slide. The teal vector x stays put; the brass vector y travels until its tail lands on x; the vertical arrow marks where its tip ends up. Wu’s two worked examples are preset — and try y = x* to see two opposite vectors cancel to the zero vector.
Set two numbers, slide, and read the endpoint. The Key Observation narrates each result below.
Because x and x* have the same length but opposite directions, x + x* = 0 — the slid vector lands its tip right back on 0. (p. 384)
Every result the machine just showed you is governed by one two-part fact, which Wu proves by picture-drawing across all the cases:
Key Observation · H. Wu
“Let x and y be rational numbers, then
(i) the direction of the sum x + y is the direction of the vector whose length is greater, and
(ii) the length of the sum x + y is the sum of the lengths of the vectors if they have the same direction, and is the difference of the lengths of the two vectors if they have different directions.” — §27.2, p. 384. Wu’s footnote: “We understand ‘difference’ to be the longer length minus the shorter length. In particular, it is always positive.”
Two consequences fall out immediately, because a special vector is determined by nothing but its length and direction — and the Key Observation computes both without caring about order:
“The addition of special vectors is commutative, i.e., x + y = y + x for all rational numbers x and y.” … “The addition of special vectors is associative.” And the closure fact that makes the next section legal: “If x and y are rational numbers, then there is a rational number z so that z = x + y.” (pp. 385–386) Try the Swap x ↔ y button above: different journey, same endpoint. That is commutativity, seen.
Everything is now in place. Addition of rational numbers is one line:
Definition · H. Wu
“Formally, we define the sum x + y for any two numbers x and y to be the endpoint of the vector x + y. In other words,”
x + y = the endpoint of x + y
— §27.3, p. 386
And because of consequence (iii) of the Key Observation, when x and y are both fractions this new addition is exactly the concatenation-of-lengths addition from Lesson 02 — the new definition extends the old one instead of replacing it. Wu closes the section with the picture to keep:
“If x and y are rational numbers, then x + y is a rational number whose distance from x is equal to the length of y, and is to the left of x if y is negative and to the right of x if y is positive.”— Hung-Hsi Wu, §27.3, p. 386, consequence (vi)
Notice what never appeared on this page: a rule like “same signs add, different signs subtract.” Students don’t memorize that sentence here — they watch it happen, and the Key Observation states it precisely, as a provable fact about lengths and directions. And Wu’s twist from the top of the chapter is still waiting: subtraction will not be a new idea at all, but addition in disguise. That is where this series goes next.