On a real number line, is there a number in the exact middle? I claim that there is not, but I am unaware of a proof. Is my contention correct and if so, is there a proof? Is there a proof of the opposite?

Thanks,

Rob

On a real number line, is there a number in the exact middle? I claim that there is not, but I am unaware of a proof. Is my contention correct and if so, is there a proof? Is there a proof of the opposite?

Thanks,

Rob

How do you define “middle”? I can think of definitions by which the answer is 0, and I can think of definitions by which there is no well-defined answer.

The question really makes no sense, because there is no “middle” of a line that extends infinitely in both directions. You can define zero as the middle for a given purpose, but that is just as arbitrary as defining Greenwich as zero longitude.

For every point on the left side of zero, there is a point on the right side, and vice versa. Therefore, zero is in the middle.

On the other hand…

For every point on the left side of 37.569, there is a point on the right side, and vice versa. Therefore, 37.569 is in the middle.

Infinity is weird.

Eleventy-gazillion, five hundred and thirty-two, i.e. infinity divided by two.

I think you meant negative eleventy-gazillion, five hundred and thirty-two.

And this is it right there. As a general rule, you can’t even begin to worry about proof until you have a definition for what you’re talking about.

But there’s also **negative** eleventy-gazillion, five hundred and thirty-two, which is also infinity divided by two. So you have 2 x (infinity/2) . . . or infinity.

I know exactly what the middle number is, but it’s too large to fit into the margin.

I had a fifty-fifty chance.

What definition would give 0 as the answer?

A line is a geometrical object that is straight, infinitely long and infinitely thin. It has no beginning, and no end. There is no middle of a line. Not just no middle point, but no middle at all, because the middle is also defined as everything between two extreme points, and there are no extreme points on a line. If there is no middle of the line, there is no middle point of the line because the middle point must be in the middle.

Infinity is So Cool, and so maddening.

A Guy came up with this Paradox (his name was Hilbert):

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.

Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.

It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.

Another Guy who is Worth reading About, if you’re into this sort of thing, is George Cantor.

The number which has all negative numbers to the left of it and all positive numbers to the right of it. Not a very useful definition, though.

42

Although, probably more useful than any other definition.

The absolute value of the quantity infinity/2?

Was it the definition you had in mind?

It’s actually under consideration, but mounting evidence is showing that wisserteen is perhaps the middle, central number of all numbers in the universe. Even the center of that bastard number 103.

The real number line is commonly drawn/displayed with 0 in the middle (of the portion shown). So I suppose you might say that **aesthetically** or **psychologically** 0 is the middle number.