I Midpoint(s) of the unbounded number line

[tomwilliam]

TL;DR Summary

Midpoint(s) of the unbounded number line

Just chatting with my son about Maths and he casually mentioned that 0 would be the midpoint of the number line from -inf to +inf. I wondered whether it wouldn’t be more accurate to say there is no single midpoint. Couldn’t you make an argument that any real number is exactly halfway between -inf and +inf?

 

[PeroK]

TL;DR Summary: Midpoint(s) of the unbounded number line

Just chatting with my son about Maths and he casually mentioned that 0 would be the midpoint of the number line from -inf to +inf. I wondered whether it wouldn’t be more accurate to say there is no single midpoint. Couldn’t you make an argument that any real number is exactly halfway between -inf and +inf?

There is no geometric midpoint. Whether zero is an algebraic midpoint depends entirely on how you define midpoint.

More generally, things in mathematics are what they are defined to be. If you don't have a usable definition, then all bets are off.

 

[fresh_42]

It is hard to determine the center of a line that cannot be measured. On the other hand, zero is the midpoint of how we describe the real numbers, positive in one direction, negative in the other. In the end, it all depends on what you mean by "midpoint". It is the origin, or center, or midpoint of the real vector space that the real numbers are.

 

[PeroK]

A quick search suggests that midpoint normally applies to a (finite) line segment.

 

[FactChecker]

TL;DR Summary: Midpoint(s) of the unbounded number line

Just chatting with my son about Maths and he casually mentioned that 0 would be the midpoint of the number line from -inf to +inf. I wondered whether it wouldn’t be more accurate to say there is no single midpoint.

Yes. Any finite number is as good as any other.

Couldn’t you make an argument that any real number is exactly halfway between -inf and +inf?

That goes too far. It's impossible to say that any number is "exactly halfway between -inf and +inf".

All that being said, I would not recommend that you confuse your son. Let him accept what the class is doing.

 

[DaveC426913]

I would not recommend that you confuse your son. Let him accept what the class is doing.

Although, for all we know, this is a fourth-year university Set Theory course.

 

[WWGD]

I think there's no such thing as the middle/midpoint of an open interval in the Standard Real line. Maybe in the 1- or 2- point compactification. But a closed interval does have a midpoint. Maybe using Hausdorff distance, though. Interesting question.

 

[fresh_42]

Although, for all we know, this is a fourth-year university Set Theory course.

Yes, it all depends on the environment where you ask this question: geometry, (linear or abstract) algebra, measure theory, arithmetic, or whatever. The term midpoint is normally used in geometry and on finite lines. In the case of an infinite line, you can specify any particular point on the number line and call it the midpoint. But whenever you decide to switch from Greek geometry to analytical geometry, we end up calling zero the origin. Whether a precisely defined origin, zero, can be called a midpoint is a linguistic question.

 

[BWV]

does midpoint = median? a standard normal distribution has a median of zero and its range is (−∞,+∞)

but does that mean you have to define a convergent function on the line to define a midpoint?

 

[WWGD]

I think there's no such thing as the middle/midpoint of an open interval in the Standard Real line. Maybe in the 1- or 2- point compactification. But a closed interval does have a midpoint. Maybe using Hausdorff distance, though. Interesting question.

Just what is it you're skeptical about @PeroK ?

 

[PeroK]

Just what is it you're skeptical about @PeroK ?

Everything. A finite open interval can be defined to have a midpoint in the obvious way. Compactification and Hausdorff distance have nothing to do with a basic question of mathematical terminology.

 

[martinbn]

I think there's no such thing as the middle/midpoint of an open interval in the Standard Real line. Maybe in the 1- or 2- point compactification. But a closed interval does have a midpoint. Maybe using Hausdorff distance, though. Interesting question.

Why is it a problem to say that the midpoint if $(0, 2)$ is $1$?

 

[fresh_42]

does midpoint = median? a standard normal distribution has a median of zero and its range is (−∞,+∞)

but does that mean you have to define a convergent function on the line to define a midpoint?

I think this is a good example of how mathematics works. You have to define the terms you use precisely. If we assume "midpoint" requires the setup of Greek geometry, in which it is normally used, then the term midpoint requires a real line of finite Euclidean length. It is automatically a closed interval since the Greeks didn't have topology and their lines $\overline{AB}$ always included $A$ and $B$.

The mentioned chat begins when one of the participants generalizes the concept to fields where it wasn't meant to be applicable. This cannot be done without telling everybody in which way such a generalization should be understood. Missing this step creates arbitrariness and room for a discussion, usually with an open end. It is the same as in that sad social media example $a\div b \times c$ which makes use of such an ambiguity caused by unspoken rules of how to read $\div$.

Hence, I think it is a good example. If you use technical terms in mathematics, make sure everybody means the same thing when using them. That's why mathematics always begins with definitions.

 

[WWGD]

Why is it a problem to say that the midpoint if $(0, 2)$ is $1$?

Ok, valid point, I missspoke. I was thinking about infinite intervals, but didn't make it clear.

 

[WWGD]

Everything. A finite open interval can be defined to have a midpoint in the obvious way. Compactification and Hausdorff distance have nothing to do with a basic question of mathematical terminology.

Yes, except I was referring to infinite intervals. I thought of expanding on the idea into compactification would help illustrate the point.

 

[WWGD]

Why is it a problem to say that the midpoint if $(0, 2)$ is $1$?

I confusingly mixed up finite with infinite- and half-infinite intervals. My bad.