A paradox about infinity?

Question

Let a and b be any two mutually prime positive integers where a < b. Then a/b is a rational number less than 1, while b/a is a rational number greater than 1.

Clearly, there are the same number of a/b numbers as there are b/a numbers, since the integer I put in the numerator is arbitrary. Also, I know that every rational number between 0 and 1 can be expressed as a/b, while every rational number greater than 1 can be expressed as b/a.

Does this mean the number of rational numbers between 0 and 1 is the same as the number of rational numbers between 1 and infinity?

Answer 1

In a very well-defined way, yes.

This is yet another example of why "infinity" is not a number. For example, by adding 1 to your rational numbers in the interval [0,1], you get infinitely many rational numbers in [1,2]. Continuing this, you get that there are as many rational numbers larger than 1 as there are in [a,a+1] for any integer a. If "infinity" was a number, this would mean that "infinity" + "infinity" + ......... = "infinity" or "infinity" * "infinity" = "infinity". That is, "infinity" is a solution of x^2=x. This is bad. We want out number system to have (at most) only two solutions for degree two polynomials.

What I meant by well-defined is the following. Call a non-finite set X "countably infinite" (or just "countable") if there is a map from the natural numbers *onto* X. Why call it this? Well, such a map just describes a way to "count" the elements of the set X. Ok, is this definition useful? Well, yes. The "size" of the real numbers *ought* to be bigger than the "size" of the integers, right? Well, the integers are countable (map 1 to 0, 2 to 1, 3 to -1, 4 to 2, etc.). But the real numbers are not. There is a great argument due to Cantor that shows this. It is enough to show that the real numbers in (0,1) are not countable. Every real number has a decimal expansion. To make them unique, don't use ones with repeating 9's at the end. Now, if this set of real numbers was countable, then we could list them (this is a consequence of being countable; the number that 1 maps to, list it first; the number that 2 maps to, list it next; etc):

0.qwerty....
0.asdfgh...
0.zxcvbn...
...

where the letters here stand for the digits of the numbers. Now consider the number formed in the following way. The tenths digit is anything except 9, 0, or q (the tenths digit of the first number above. The hundredths digit is anything except 9, 0, or s (the hundredths digit of the second number listed above). The thousandths digit is anything except 9, 0, or c (the thousandths digit of the third number listed above). Etc. If you do this forever, that is, if you define a decimal expansion of a number this way, then the number you create cannot be in the list above since we chose every number to differ from our number in at least one digit. Also, our number does not end in repeating 9's and is not 0. But it is still between 0 an 1. That means we couldn't have listed all the real numbers between 0 and 1 in our list above. So we see that the real numbers are not countable.

This is one of the things that makes countable a useful notion. Another is that we can see that the size of the rational numbers is the same as the size of the natural numbers! Or, for your problem, we can see that the size of the rational numbers between 0 and 1 is the same as the size of the rational numbers larger than 1 -- in a very well-defined way -- because they are both countably infinite.

So while "infinity" isn't a number, we can still make some sense of some infinities being larger than others and some infinities being equal to others. Fun stuff!

Answer 2

Yes, you demonstrated a one-to-one correspondence between the rational numbers between 0 and 1 and those greater than one.

It's also true that there are as many integers as there are rational numbers. There are more real numbers than rational numbers, however. See link below.

Answer 3

yes, and not every rational number greater than 1 can be expressed as a/b where a and b are primes. Consider 4/9.

Answer 4

if a and b can be any number than there are an infinite number of choices for those variables. That means that it would be infinity over infinity, giving an infinite number of solutions between 0 and 1.

Answer 5

The real paradox comes from misconceptions of "space", "expansion" and "void". Space is the distance between objects, and when there were no objects, there was no space, or "void" to "expand" into. The moment a chance fluxuation in space-time created the first particle in "our" space-time, its energy density exceeded allowable bounds and set forth a chain reaction that populated our known universe within the first billionth's of a second. The "expansion" was actually an "inflation", meaning that although the distance between newly created objects proceeded at the speed of light, making the total "expansion" of the universe seem to violate the laws of relativity, the actual inflation created not only the distance between objects, but the net size of the universe. Since there is no "out there" or "void" beyond the existing universe, "Infinity" will only exist "if" the inflation does not stop, which might be the case since it actually appears to be accelerating.

Answer 6

Yes.
If a and b are not zero, then you can construct a one to one correspondence between the numbers a/b and b/a

Answer 7

yes it does

Answer 8

yes, same cardinality