## Wednesday 27 June 2012

### What is a Line?

On Monday I was tutoring a girl in year nine, and teaching her about linear graphs, and the whole y = mx + c thing. She asked what linear meant, and I told her that it comes from the Latin word linearis, which means like a line. She was momentarily puzzled: "Aren't all graphs lines?", she asked. I drew a vaguely-quadratic looking graph on the mini whiteboard and said, "This one's not". To which she replied, "Er, yeah it is".

This made me question my concept image of a line. Can a line be curved? If no, then why do we bother saying "straight line" when this would apparently be tautologous. If yes, then how can we use the word "linear" to describe relationships that correspond to strictly straight line graphs?

I decided to do some research. I asked a few members of my department how they would define a line. Here are some of the answers I got, along with an evaluation of each.

Something that joins together two points
The person who gave me this definition said that a curve would count as a line by this definition, as a curve can join two points. However, this would mean that the line joining two points would not be unique, and that makes me feel uneasy. Also, this definition seems to be more for a line segment, as opposed to a line: I have always been taught that lines are infinitely long in two directions.

The shortest path between two points
A little bit nicer, as it now has uniqueness (if by "shortest" we're referring to the usual metric). However again this is just a line segment, not a line.

The locus of points where...
This definition was never finished, because he couldn't think what the locus would be. I decided to still include this definition because I think he's on to something.

A sequence of points
Take any sequence of numbers and plot them on cartesian axes. Then join these up, and what you have is a line. I would refine this by saying "linear sequence", but that would be a bit of a circular definition, because I would define a linear sequence as one that would make a straight line when drawn on a graph. This definition isn't very satisfying because it only makes sense on a set of axes, whereas obviously lines occur elsewhere. In geometry, for example, lines can exist with absolutely no plane of reference.

I suppose the definition I would go with would be the second one, except I'd alter it to make it infinitely long:
"A line is the infinitely long path that goes between two points, such that the path from one point to the other is the shortest possible" Arggh this doesn't work! The line could be squiggly outside of the two points and this could still hold! How do I say "it's straight" when I'm trying to define straight in the first place?

I'm afraid I'll have to do what I always end up doing when it comes to maths debates:

Wikipedia to the rescue!
"Line (geometry), an infinitely-extending one-dimensional figure that has no curvature:
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects.

Thus, until seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points"
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

Is it just me or is that a really hand-wavey definition? And it's still not clear whether a line has to be straight: the pre-seventeenth century definition allows for curvature, but the analytic geometry definition does not. And what does "equally extended" mean anyway?

You might wonder why on earth I care about these things. I do too sometimes. But when I asked people in my department, we got a good discussion going, and that for me is an amazing thing: people arguing about maths. I love it! I remember year 11 RE lessons when I was at school: we used to argue all the time (and every comment would start with "Surely..." - Do any of my ex-classmates remember this?) but we never argued in maths. I love having a good debate (pro tip: never use the expression "maths debate" in a lesson. Trust me) and it is something we should be doing more of in maths. Maths is not about blindly accepting rules and definitions. We should question them, challenge them! Ask a child this and watch their mind explode: "How did the first ruler making factory make the ruler perfectly straight?"

As you can probably tell from the length of this post, I have a lot of work to be doing which I am trying to avoid. There is something quite stressful happening next week in my department, and my stress reduction technique is to immerse myself in mathematical pedantries.

Bye for now,

Emma x x x

I thought you might be interested to read the hilarious text message conversation that took place between me and my dad as I was researching this post:

Me: how would you define a line?
Dad: a set of points determined by a linear function.
Me: define "linear".
Dad: an equation that forms a line.
Me: that's a circular definition.
Me: a shape that contains no lines.
Me: our conversation has come full circle!