The question is straight:
Which range contains more rational numbers?
- 0 to 1
- 1 to 10
The question being put in such a straight manner, there was not much to think about.
First of all, we have to provide some liberty to the quizzer and lets not pounce on him saying that infinity cannot be compared with infinity, rather we would say that let the number of rational numbers between 0 and 1 is x (which is very very large).
Intuitively, we can say that if we add 1 to each of these x numbers, we will get x numbers in the range 1 to 2, and adding 1 further we get another set of x numbers in range 2 to 3 and so on until we get x numbers in range 9 to 10.
By this logic, it is pretty clear that if we have x numbers in range 0 to 1 we have 9x numbers in range 1 to 10.
Now the simplicity of this problem is seriously challenged by the following counter argument.
Assume that there are y distinct rational numbers in range 1 to 10. Consider the reciprocals of all these y numbers in the range 1 to 10. There are three things that we know about these new y numbers.
- All y are rational numbers.
- All y are distinct (Knowing that all distinct rational numbers have distinct reciprocals).
- All y lie between 0 and 1 (Reciprocals of numbers>1 lie between 0 and 1)
This means that the range 0 to 1 contains at least y rational numbers. (and more, eg the reciprocals of numbers greater that 10 as well)
By this logic, it is pretty clear that rational numbers lying in 0 to 1 range are far more than rational numbers lying in range 1 to 10.
Paradox or Anomaly?
How does one explain that?
4 comments:
I think you missed some things in your writing.
you are saying:
Assume that there are y distinct rational numbers in range 1 to 10
And again at 3rd point you are saying:
All y lie between 0 and 1 (Reciprocals of numbers>1 lie between 0 and 1)
But rational numbers can be grater than one.
I think you are missing here to define one new group based on y.
You should say that let Z is set that is reciprocal of Y.
And then say your three points on Z.
Thanks
Gyan
It's neither paradox nor anamoly...
The only thing you've missed out here is the fact that x is countably infinite and hence your calculus beats Common Sense.
Anyway, I think the answer would be that there are equal number of rational numbers in both the ranges. However, that number would be countably infinite.
Proof
=====
Consider the linear Transformation F(x) : x -> 1+9x
Now we have the following :
F(0) = 1
F(1) = 10 , and
for each x(rational) belonging to (0,1), we have F(x) (surely rational) belonging to (1,10) and vice-versa (since it's a linear transformation).
Hence, Proved.
to Gyan:
You are right that the 3 points are valid for the reciprocals and not on the original numbers themselves. But that is what it is supposed to mean when it is written that, "There are three things that we know about these NEW y numbers". When i wrote 'y' I meant to communicate the size of the set(number of elements in the set), not the name of the set.
Let me know if its still unclear.
To Vijay:
Ya, the fancy language can prove it. But its equally easy to fancify the other two explainations to prove that one of the sets is larger or smaller.
We can stay rest assured that our common sense does not allow us to think about things as large as infinity.
The topic discussed here is best handled by considering the entities as sets. The statement "assume there are y distinct ratonal numbers in the range 1 to 10" is not appropriate. For one thing, to what set does y belong? y is not even a real number! the set "all rational numbers between 1 to 10" and the set "all rational numbers between 0 to 1" cannot be compared for size equality using the notions you have used in your post... for the simple reason that the quatity 'y' is not a part of our natural numbers and hence arithmatic rules and notions of inequality defined on the natural numbers do not extend in a similar fashion to cardinalities of these sets. To overcome this, there is a special name given to the cardinality of natural numbers - aleph null (http://en.wikipedia.org/wiki/Aleph_number). In set theory we say that two sets are of same cardinality iff there exists a bijection between them, which is what you have shown by your arguments, and hence the two sets considered here are of the same cardinality. It takes some more work (not much though) to show tha this cardinality is same as aleph null!
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