Can we explain to undergraduates how points make a line?

Question

Many of my students arrive in college believing that lines are (in some way) made out of points.

They also believe that points have no length.

They want to know how a bunch of zero length points make a line of length 1?

Edit I did ask this question in math.stackexchange, but because it is about teaching it was declared off topic and closed. I changed to question there in hopes that if I understood it better, then I might be able to explain it.

I have asked my original question here in mathematics education in hopes of hearing teaching advice. I see it as a question about geometry and teaching geometry although it comes up in a class on ideas of infinity.

Answer 1

This is what I came up with thinking about your question.

I would start by exploring what it means to say that a line is 'made up of' points, because I think that is a really important thing that doesn't get taught in schools. Students know 'the equation of a line', but usually have no real concept of what that means.

Admittedly, that will be somewhat tricky if they have not done an introduction to sets. I would perhaps try using plotting points:

Imagine using a computer to plot '$y=3x+2$' on the screen by looking at each pixel at a time. Pick one pixel. It has an $x$ coordinate and a $y$ coordinate. We check whether these points satisfy the equation. That is, if we put the two numbers into the equation '$y=3x+2$', do we end up with a statement that is true, or a statement that is false? If it's true, that means the pixel is on the line, and we colour it red. It it's false, that means it is not on the line, so we colour it blue. Once we've done this for all the pixels, we'll have our picture of the line.'

I'd then look at the Paradoxes of motion. The ideas go a bit beyond what most of the students will need, but the starting point is within reach, and it's reasonably easy for students to find out more if they are interested. If nothing else, it might make some of them think.

I'd end by saying that really understanding the answer to the original question means studying measure theory. There are two reasons I see for pointing this out. Firstly, it assures the students that it's ok that they do not understand, because what they are asking is more advanced than the maths they have done. But it also points out that is isn't magic, or made up. There is a real answer out there, that they could reach if they really put their minds to it.

Answer 2

They don't. You need additional information to organize a set of points into anything resembling geometry; a bare set of points is simply lacking the context to allow anything more interesting than counting them (i.e. the cardinality of the set).

A typical form of context boils down to "remembering" how the points are embedded in a Euclidean plane.

Below are two answers I've given to similar questions at https://math.stackexchange.com. Both questions have a variety of other answers to browse through as wel.

• https://math.stackexchange.com/a/1084093/14972
• https://math.stackexchange.com/a/1805439/14972

Answer 3

I agree with the premise, lines are (in some way) made out of points, and points have no length.

If, restricting ourselves to a straight line, we can consider these as subsets of the real numbers $\Bbb R$.

A line is a subset of this with the property, if $a, b \in \Bbb L$ then any real number between $a, b$ is in $\Bbb L$.

Calling elements of a line points, we can define the length of the line between the two points (on a straight line) say $a, b$ as the distance between them, and this distance is $\left|b - a\right|$. The length here being a property of two points, on a line, rather than the sum of a property of all individual points.

We could mention subsets of the real numbers, even with uncountably many elements, where ways to measure a length gives zero, such as the Cantor set, and these do not contain line segments.

They want to know how a bunch of zero length points make a line of length 1?

I find this tough. We could consider a line $[0,1]$ of length $1$ as divided into $n$ segments of length $\frac{1}{n}$.

The total length of these segments is $\frac{n}{n} = 1$ while as $n \rightarrow \infty$ the length of the segments $\rightarrow 0$

Answer 4

I suppose one of the problems with discussing the issue is coming to some agreement with your students about what a point, a line, and length are, that is, the axioms or definitions of the terms. As Abraham Lincoln wrote, "One would start with great confidence that he could convince any sane child that the simpler propositions of Euclid are true; but, nevertheless, he would fail, utterly, with one who should deny the definitions and axioms."

For instance, one might consider that Aristotle proves the opposite of the OP's proposition in a way that should appeal to students who ask, "How [can] a bunch of zero length points make a line of length 1?":

A line cannot be made of points, if the line is continuous and the point indivisible. (Aristotle, Physics 231a)

Or that in Hilbert's Foundations of Geometry, points and (infinite straight) lines are distinct systems of "things," and lines are not made of anything. However there are "mutual relations" between them, such as given a line and a point, the point is either "on" the line or not. And with the aid of other axioms, there are infinitely many points on a line. A "line segment" consists of two points lying on a line, and Hilbert defines the "points within the segment"; however, the segment does not consist of the points within the segment. (OTOH, a circle is defined to be a set of points, via a definition in common use today.) One could prove, I suppose, that a line is completely determined by the points on it (since any two distinct points completely determine the line); and there is nothing else "on" the line by defintion, or the lack of definition: The only things defined to be "on" a line are points. They are not "on" in the same sense that there are birds sitting "on" the telephone line outside.

In coordinate plane geometry, one may posit the existence of a plane, but one could also simply begin with the set of ordered pairs of real numbers. One can then define a line to be a set of all ordered pairs satisfying a certain algebraic relation on it coordinates. Then a line is by definition made up of points. There is no prior definition of a line from which we have to start to prove there is nothing "on" or "in" a line that makes up its composition.

An entirely different approach: Start with a line segment of length 1. Any two distinct points on it have a length between them greater than zero. (1) You cannot form a line by adding points one at a time, without adding length at each step. What about adding infinitely many points at a time? "I can't imagine how that would work — can you? Do you mean adding a bit of a line segment?" (2) Paradox: The length between two points is the same whether we include the points or not, right? So let's throw out the end points. Now pick a point within the segment and throw it out. It divides the segment into two pieces whose lengths add to 1. Now repeat. At each step, we have thrown out some points of the segment but still have a total length of 1. Continue ad infinitum until you've thrown out all the points. Now you have no points left, but you still have a total length of 1. [An appropriate conclusion might be: Infinity is hard to think about. To an insistent student, I might ask, "Do you think you could really exhaust all the points? If not, then there's no paradox. But you'll need to study more analysis to understand what can and cannot be done when you carry out a process ad infinitum." It's still a matter of definitions and axioms.]

I am dodging the problem that students might be thinking that points and lines exist prior to mathematics, and it is the job of mathematics to investigate them and bring into clear definition what they really are and how they relate to each other. This tries to make mathematics related to geometry in the same way natural science is related to nature, as if geometry actually existed and was not merely a way of thinking about space that we construct. Well, I don't accept these hypotheses, which I attributed to the students. Referring back to Lincoln, we would fail to convince each other if we cannot agree to common starting points.

Answer 5

This question alludes to a deep developmental issue. Your students are starting to think hard about the continuum, even if they do not realize it. A countable collection of points (for example rational numbers) in [0,1] has Lebesgue measure zero. Yet the collection of all points in [0,1] has Lebesgue measure 1. But of course the notion of Lebesgue measure was not fully developed until the early 20th century. Perhaps an undergraduate would get to this understanding in an upper division analysis course, but most do not.

Another perspective is from projective plane geometry. Here points and lines are both undefined terms, and they are on the same footing. By projective duality, ranges of points (points incident with a given line) are logically the same as pencil of lines (lines incident with a given point). In other words, instead of thinking as points as fundamental and deriving lines as ranges of points which is what we tend to do, we could take lines as the fundamental objects and ``derive'' points by considering pencils of lines.

I point out these two notions not because they will be of immediate help to younger undergraduate mathematics students, but to indicate to the OP just how deep the question is.

Answer 6

Often the fact that words have connotations in common parlance which are at odds with the more technical way these words are used within mathematics causes tensions in getting ideas across to students. But the notion of point and line are especially subtle.

Euclid gives definitions for the words point and line but by the time Hilbert and others came along, point and line were left as undefined in axiom systems that were developed to understand the Euclidean plane and other kinds of geometry. So one can construct "models" that satisfy a collection of axioms that involve point and line as undefined terms and give an interpretation to these words in the model. If one picks any of Hilbert's axioms, say A, one can construct a model where the interpretations of point and line satisfy each of the axioms other than A and where A fails to hold.

To help show what can go on I like to show students finite geometries. Some students get the "point" here but most just don't accept the fact that the "lines" of these finite geometries are "really" lines! You might find this brief introduction to finite geometries of interest:

http://www.ams.org/samplings/feature-column/fcarc-finitegeometries

And then there is the model where points are pairs (x,y) where x and y are rational numbers and lines are linear equations with rational coefficients - a geometry which is not the Euclidean plane but has lines with lots of points between any pair of points, but lots of "holes," too.

Answer 7

They want to know how a bunch of zero length points make a line of length 1?

It may help to think of the Euclidean plane as a set, each element of it being a point in the plane. Then any geometric figure (e.g. a line, or a circle) would be a subset of the Euclidean plane, each element of that subset being a point in the plane.

Example 1: A subset of the plane P is a circle iff there exists a point C in P and a distance r such that every point in that subset is a distance r (the radius) from the point C (the center).

$$\{ X \in P : |CX| = r\}$$

Example 2: A subset of the plane P is a straight line iff there exists a pair of distinct points A and B in P such that every point in that subset is equidistant from both A and B.

$$\{ X\in P : |AX| = |BX|\}$$