rprosperi wrote: ↑
Tue Feb 19, 2019 7:49 pm
ijabbott wrote: ↑
Tue Feb 19, 2019 6:00 pm
It seems intuitive that there should be more rational numbers than integers, but this is where intuition is wrong. The set of integers and the set of rational numbers have the same size in the same way that the set of even integers has the same size as the set of integers. However, the set of real numbers is larger (and is the same size as the set of complex numbers).
I could not begin to disagree, but could you please explain that, as neither of these things you note as having the same size feel intuitive to me, and that Reals and Complex are larger (but equal) is just as unclear. When you have time is fine, this is only a curiosity for me, and if I knew how to just google this, I'd do that.
I see Thomas Okken has already done a marvellous job, but I think the main thing that equates the cardinalities of the set of integers and the set of rationals is that both types of numbers are countable by some progressive sequence that tags each element of the set with successive natural numbers. They are infinite, but both "countably infinite".
There may be more than one way to count them, but the important point is that you can indeed progressively count them (although you'll never reach the end of course because they are endless).
For example, here is one possible way to progressively order the integers: 0, 1, -1, 2, -2, 3, -3, ....
And here is one way to progressively order the rationals (but ignore the ones in parentheses as they are not in simplest form and serve merely to show the pattern being followed): 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, 1/4, -1/4, 4/1, -4/1, (2/4, -2/4, 4/2, -4/2, ) 3/4, -3/4, 4/3, -4/3, 1/5, -1/5, ....
(In both cases, I haven't shown the natural numbers used to tag the elements, but you can just use the position of the element in the sequence as the tag.)
I haven't touched on the cardinality of the set of complex numbers versus reals as it's not quite as easy to illustrate, but as an extremely non-rigourous explanation, a complex number can be considered to be an ordered pair of real numbers (along with a set of arithmetic rules for combining with other such pairs) in the same way that a rational number can be considered as an ordered pair of integers (with common factors removed, and obeying their own rules of arithmetic), and you get the idea that the sets may be the same size (although I have by no means proved it here).