Section 1-8: One-Dimensional Figures (Day 5)

Section 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").

As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:

"A number is between two others if it is greater than one of them and less than the other."

Then the text can define betweenness for points:

"A point is between two other points on the same line if its coordinate is between their coordinates."

But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:

"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."

Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.

I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:

"If B is between A and C, then AB + BC = AC."

Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.

Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.

We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC = c - a. And so we calculate:

AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
               = c - a (simplification -- cancelling terms b and -b)
               = AC

The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED

Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.

The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:

"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."

The definitions of ray and opposite ray are similarly defined in terms of betweenness.

The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough AB, but not overline. For the purpose of the rest of this post, let's pretend that the strikethrough AB is really the overline for segment AB.

Now if AB and CD are both of, say, unit length, then AB = CD, but we can't write AB = CD. After all, AB and CD are real numbers -- both are 1 -- and those numbers are equal. But the segments AB and CD can't be equal unless they have the same endpoints (that is, A and C are the same point, as are B and D, or vice versa). The numbers (the lengths) are equal, while the segments are congruent. But students and teachers alike confuse a segment with its length, and confuse equality with congruence.

To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that AB literally equals BA (but ray AB is not the same ray as BA). Notice that I tried to draw all of the one-dimensional figures in red.