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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.

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    $\begingroup$ Whether lines are made up of indivisible points was a big discussion topic among Medieval philosophers. $\endgroup$ – Dag Oskar Madsen Sep 6 '17 at 0:27
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    $\begingroup$ Carl Boyer: The History of the Calculus and Its Conceptual Development, Chapter III is one reference. $\endgroup$ – Dag Oskar Madsen Sep 6 '17 at 0:32
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    $\begingroup$ Just want to chime in that this wasn't just a question among medievals, but for instance was a big bone of contention at the time of development of calculus. Cavalieri, Torricelli, and Hobbes are all good (if divergent) primary sources on this. $\endgroup$ – kcrisman Sep 6 '17 at 0:40
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    $\begingroup$ By the way, to be helpful to the actual question, since you have identified the course as such, I think you should "teach the question". The answer is "yes", there is a paradox and a lot of people have thought about this a lot without coming to a good conclusion. As much as I dislike its stridency and tail-wagging-the-dog, Amir Alexander's book Infinitesimal takes on a lot of these concepts well; Doug Jesseph's book on the Hobbes/Wallis controversy is better but definitely at a higher level. $\endgroup$ – kcrisman Sep 6 '17 at 1:08
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    $\begingroup$ You have made the same question in other community! math.stackexchange.com/questions/2417029/… $\endgroup$ – Cragfelt Sep 6 '17 at 13:40
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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.

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  • $\begingroup$ Thank you. I phrased the question as I did in hopes that others might find any answers interesting and useful. I teach freshman mathematics at a liberal arts college. This question arises in a writing seminar I teach centered on "Ideas of Infinity". Many of the students are liberal arts majors. I have the students read some of Aristotle's Physics. We do discuss Zeno's Paradoxes and Galileo's paradox and supertasks. I use some intuitions about sets. Late in the course I present the idea of cardinality including "countable" and "uncoutable". $\endgroup$ – Jim H Sep 7 '17 at 12:18
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    $\begingroup$ Re, pixels: A computer screen only has a finite number of discrete pixels, and it is easy to define a line that crosses the screen but does not contain any of its pixels. Actual line-drawing algorithms color pixels based on how far each one lies from the line. $\endgroup$ – Solomon Slow Sep 11 '17 at 20:51
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    $\begingroup$ @jameslarge The point isn't to actually discuss drawing a line on a computer, it's to make a picture the students can understand to help them grasp a new idea. $\endgroup$ – Jessica B Sep 12 '17 at 6:26
  • $\begingroup$ @jameslarge: Last time I took such a course, actual line-drawing algorithms were doing integral delta-steps from one endpoint to the other (no distances involved). en.wikipedia.org/wiki/Bresenham%27s_line_algorithm $\endgroup$ – Daniel R. Collins Sep 14 '17 at 3:46
  • $\begingroup$ @DanielR.Collins, The Bresenham algorithm selects the connected set of pixels that lies closest to the actual line. I emphasise "select" because it doesn't do anti-aliasing: That is, it does not attempt to truly render a sampled image of a finite-width line. en.wikipedia.org/wiki/Bresenham%27s_line_algorithm $\endgroup$ – Solomon Slow Sep 14 '17 at 15:24
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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.

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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 $\rightarrow 0$

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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.

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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.

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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.

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

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  • $\begingroup$ I do not like Example 2 (understood as a definition), as it uses the euclidean structure while "line" is an affine concept. $\endgroup$ – Benoît Kloeckner Dec 20 '17 at 13:05
  • $\begingroup$ @BenoîtKloeckner I don't understand your objection, but both examples use only the concept of the distance between a pair of points in the plane. $\endgroup$ – Dan Christensen Dec 20 '17 at 13:28
  • $\begingroup$ that is my point, circle need the notion of distance to be defined (well, even this is debatable), but clearly the notion of line does not need the notion of distance. For example, very many transformations of the plane send lines to lines, but do not (in general) preserve distances: affine transformations. It would thus be wrong to define lines using distance while "line" is an affine notion, not a metric one. $\endgroup$ – Benoît Kloeckner Dec 21 '17 at 14:04
  • $\begingroup$ @BenoîtKloeckner I tried to think of the simplest selection criterion for points in the plane that would form a line without using axioms for betweeness, etc. L = {X in P: |AX| = |BX|}. My point being that a line is just a subset of the Euclidean plane, each element of which is a point. $\endgroup$ – Dan Christensen Dec 21 '17 at 17:52
  • $\begingroup$ @BenoîtKloeckner The students in question don't seem to have a problem with the notion of finite distances, so I built on that. $\endgroup$ – Dan Christensen Dec 21 '17 at 18:17

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