Mike's question....

StarValleyWy asked an interesting question.....how many real numbers are >1 and <2? Is that the same number as between 8 and 9?

I will answer tomorrow unless someone else does first.

What makes a number 'real'?

**I think** (but correct me if I'm wrong...)

Real numbers are those between +infinity and -infinity. Imaginary numbers are derived from the square root of -1. Complex numbers contain a real and imaginary element.

So I guess there are infinitely many real numbers between 1 & 2 (and between 8 & 9) since you can go on adding decimal places or use larger and larger fractions forever.

Come to think of it, surely whatever number you have between 1 & 2 can just have 7 added to it to make it between 8 & 9, so there must be the same amount.

But I expect that wasn't the answer you were looking for. Any more clues before you give the game away?

Actually, you sum it up nicely. Between 1 and 2 there is a non-denumerably infinite set of real numbers (uncountable). Each can be added to seven in a unique way, as you say, so the set of numbers between 1 and 2 is in 1-1 correspondence with those between 8 and 9.

Originally posted by royalchicken
Actually, you sum it up nicely. Between 1 and 2 there is a non-denumerably infinite set of real numbers (uncountable). Each can be added to seven in a unique way, as you say, so the set of numbers between 1 and 2 is in 1-1 correspondence with those between 8 and 9.

There are also the same number of real numbers betwen 0 and 1 as there are between 0 and 10000000. And, I think, there are the same number of real numbers between 0 and 1 as there are of rational numbers between -infinity and +infinity... is that right? 😕

The first is right, the second wrong. Rationals can be put in 1-1 correspondence with natural numbers. Naturals cannot be put in 1-1 correspondence with reals. Therefore, there are 'more' reals than rationals.

EDIT Find an old thread on Cantor for more on this.

Originally posted by royalchicken
The first is right, the second wrong. Rationals can be put in 1-1 correspondence with natural numbers. Naturals cannot be put in 1-1 correspondence with reals. Therefore, there are 'more' reals than rationals.

EDIT Find an old thread on Cantor for more on this.

I'm not saying you're wrong, but where is the error in this?

Any rational number is the ratio of two naturals. Two naturals can be encoded into a real between 0 and 1 by using alternate digits of the decimal expansion of the real to represent each integer. Therefore the reals between 0 and 1 map to the rationals.

I would agree without argument that the whole set of reals is bigger than the rationals...

Originally posted by iamatiger
I'm not saying you're wrong, but where is the error in this?

Any rational number is the ratio of two naturals. Two naturals can be encoded into a real between 0 and 1 by using alternate digits of the decimal expansion of the real to represent each integer. Therefore the reals between 0 and 1 map to the rationals.

I would agree without argument that the whole set of reals is bigger than the rationals...

The reals, by virtue of being an infinite set, can be put into 1-1 correspondence with any subinterval of reals. Therefore, the reals are EXACTLY as large as any of their subintervals. Since you accept that ''all of the reals'' is larger than any set of rationals, you must also accept that any infinite set (subinterval) of reals is larger than any set of rationals.

But that does not exactly answer your question. The error in oyur argument lies in the fact that there is no 1-1 correspondence there. For example, using your method would associate 14/37 with 0.1347. However, 0.1347 = 0.13470, so this same number also associates with 140/37 and so on. Indeed, each real associates with an infinitude of rationals.

Cantor's formal proof works in a slightly similar way.

Yeah - I take it back. I was wrong😀
This guy was quite convincing: http://www.aleph.gr/en/info/aleph.html

Not a bad site there. Interesting discussion too....we'll see if it's been answered to Mike's satisfaction.

Originally posted by royalchicken
The reals, by virtue of being an infinite set, can be put into 1-1 correspondence with any subinterval of reals. Therefore, the reals are EXACTLY as large as any of their subintervals. Since you accept that ''all of the reals'' is larger than any set of rationals, you must also accept that any infinite set (subinterval) of reals is larger than any ...[text shortened]... ith an infinitude of rationals.

Cantor's formal proof works in a slightly similar way.

But are the reals perhaps in correspondance with the un-normalised rationals? I mean the set which considers 10/20 to be distinct from 1/2. I'm not sure that the un-normalised rationals can be counted...

No. The 'unnormalized rationals' is simply the set of all points in a cartesian plane with positive integer coordinates. Put your pen at (1,1). Draw a line segment to (1,2). Go over to (2,2), then to (2,1) then (3,1) then (3,2) then (3,3) and keep zigzagging in this way, hitting all lattice points. Count each one off as you hit it. you can put the 'unnormalized rationals' in 1-1 correspondence with the integers, and thus not with the reals. The more technical explanation is that the rationals are the members of the set of points with natural coordinates (x,y) such that gcd(x,y) = 1. The asymptotic density of these is positive within the whole set {(x,y): x,y are in N}, so the correspondence can be made with the unnormalized ones and thus with the integers.

Originally posted by royalchicken
No. The 'unnormalized rationals' is simply the set of all points in a cartesian plane with positive integer coordinates. Put your pen at (1,1). Draw a line segment to (1,2). Go over to (2,2), then to (2,1) then (3,1) then (3,2) then (3,3) and keep zigzagging in this way, hitting all lattice points. Count each one off as you hit it. you can put t ...[text shortened]... in N}, so the correspondence can be made with the unnormalized ones and thus with the integers.

Very good - I concede! 🙂

Was a pleasure 🙂.

Very good. I think I understand that there are an infinite number of reals between 1-2, 8-9 etc. I just boggle at the idea of ever arriving at a "terminus" as we approach "the speed of light" for example... Seems weird that in real life we can "always get there", yet in Plancks small world, and Einsteins big world, we can never get there. I have often wondered if the "energy" levels we so blithely speak of in chemistry aren't a way for the real world to "cheat" and get to integer values, or some good imitation of that... Also... your above conversion makes me wonder... what would you have left if you took any infinite set of reals between any two arbitrary limits and subtracted the "set of rationals that exactly match" the reals in the same set?... there would remain the irrationals? what about the imaginary components (if that even applies... not sure)? If we continue to map and subtract "all known types" from the remaining Reals, what is left at the end of all map-and-removals?