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Betweenness of Points on a Line Idea and Proof

Date: 06/26/2008 at 10:14:55
From: Kyle
Subject: Betweenness of Points Theorem

Could someone explain the betweenness of points theorem and why it's
important?  I don't understand the way my book proves it:

It starts with the Ruler Postulate 3, which says: 

The points on a line can be numbered so that positive number 
differences measure distances.

Definition: (Betweenness of Points)
A point is between two other points on the same line iff its 
coordinate is between their coordinates. (More briefly, A-B-C iff 
a<b<c or a>b>c.)

Example of Definition: the coordinates of U,S,and A are 9,4,and 1.
Computing distances we get US=9-4=5, SA=4-1=3,and UA=9-1=8:
US + SA = UA because 5+3=8.
From this it defines the theorem.

Theorem 1. The Betweenness of Points Theorem
If A-B-C, then AB + BC = AC.
Proof for case in which a<b<c

Statements                                         Reasons    
1. A-B-C.                                           Hypothesis.
2. a<b<c.                                           Definition of 
                                                   betweenness of
3. AB = (b-a) and BC= (c-b).                       Ruler Postulate.
4. AB+BC =(b-a) +(c-b) = c-a.                      Addition and 
5. AC= c-a.                                        Ruler Postulate.
6. AB+BC = AC.                                     Substitution 
                                                  (steps 4 and 5)

I understand the idea of a point between two other points, its just
the proof I'm confused about, and why the whole idea is important. 

Thank you.

Date: 06/27/2008 at 12:24:01
From: Doctor Peterson
Subject: Re: Betweenness of Points Theorem

Hi, Kyle.

First, let's deal with your original question: Why is this important, 
and what is it all about?

The concept of betweenness was not explicitly mentioned until 
relatively recently (the 20th century, perhaps); writers like Euclid 
would just draw a picture and do things like adding angles or 
distances without saying how they knew whether to add or subtract 
distances.  That makes some of the early proofs technically invalid; 
and it is the basis for some classic fallacies--supposed "proofs" 
that, say, all triangles are isosceles--in which the picture LOOKS as 
if you would add this angle and that angle to get another, but an 
accurate picture would show it is otherwise.  (I can show you one of 
those if you're interested.)  When mathematicians tried to make a more 
complete set of axioms (postulates) than Euclid's, in order to make 
sure that all proofs could be shown to be valid, they had to include 
some sort of "betweenness" axioms and definitions.

Modern school versions of geometry often include these ideas, but try 
to keep it simple by using ideas like the Ruler Postulate relating all 
this to the number line.  Your version of some of these ideas seems a 
little awkward to me, but we can work with it.

The notation A-B-C means that A, B, and C are points on a line, and B 
is between A and C. Your book's way of defining this is that if you 
(using the Ruler Postulate) assign numbers (coordinates) to the points 
on the line, either a<b<c or c<b<a.  In other words, it looks like

                A     B             C


                C     B             A

Now, your Betweenness Theorem says that whenever B is between A and C,
you can add the distances AB and BC to get the distance AC:

                |  AB        BC     |
                A     B             C

If B were NOT between A and C, you couldn't do that:

                A     C             B

Here you'd have to SUBTRACT BC from AB to get AC.  So betweenness is 
important in proving theorems later on.

Now let's look at the proof:

>Theorem 1. The Betweenness of Points Theorem
>If A-B-C, then AB + BC = AC.
>Proof for case in which a<b<c
>Statements                                         Reasons    
>1. A-B-C.                                           Hypothesis.
>2. a<b<c.                                           Definition of 
>                                                   betweenness of
>                                                   points.
>3. AB = (b-a) and BC= (c-b).                       Ruler Postulate.
>4. AB+BC =(b-a) +(c-b) = c-a.                      Addition and 
>                                                  Simplification.
>5. AC= c-a.                                        Ruler Postulate.
>6. AB+BC = AC.                                     Substitution 
>                                                  (steps 4 and 5)

They've assumed that a<b<c; you'd have to repeat the whole thing if 
c<b<a, but it would work exactly the same.

Statement 2 uses the definition to translate from betweenness of 
points to an algebraic statement about their coordinates (again, 
assuming this order).  Statements 3 and 5 use your Ruler Postulate to 
find the distances AB and BC, and AC, in terms of their coordinates. 
(The postulate says that the distance is given by the difference of 
the coordinates, so that AB = b-a; they don't explicitly mention it, 
but you have to subtract the smaller coordinate from the larger one-- 
that's what they mean by "positive number difference".)

The core of the proof is to add the expressions for AB and BC and 
simplify the resulting expression.  The result is AC, which is what 
they wanted to prove.

Now, if the order had been A-C-B as in my picture above, we would 
instead have

  AB + BC = (b-a) + (b-c) = b - a + b - c = -a + 2b - c

which is not equal to AC (as we expected).  In fact,

  AB - BC = (b-a) - (b-c) = b - a - b + c = c-a = AC

As with many of the first proofs in a text, this one can be hard to 
follow simply because its result seems so obvious and the foundational 
facts are stated in a complicated way.  I hope this helps a bit.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Euclidean/Plane Geometry

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