I have a 32-bit EeePC with 4GB RAM. Can I install Ubuntu 13.10 64-bit on it? I want to know whether I should install Ubuntu 32-bit or 64-bit before I download the iso.
The above answer doesn't quite apply to Ubuntu. Ubuntu can use physical address extensions: http://howtoubuntu.org/how-to-decide-if-you-should-use-32bit-or-64bit-ubuntu#.UnFNYfmnqQk. It really boils down to what software you use. eeePC is pretty standard stuff on the inside , so no driver issues on 32 or 64 bit. Also the oldest eeePCs do not support 64bit (I'm assuming yours does). Either is good, 32 with pae or 64bit. I'd go with 32bit since I'm assuming that working memory sets will be slightly more compact and memory use will be more efficient. However, Ubuntu recommends 64 bit in this article, so unless you are planning on being tight on ram, 64bit execution will be faster: https://help.ubuntu.com/community/32bit_and_64bit
To get all the bang for your buck you should install the 64 bit system. While the 32 bit can make use of 4 gigs, this refers to TOTAL addressable memory. This can include graphics cards, system memory etc etc. This doesn't mean you can't use 32 bit, it'll run; it just won't make use of all your resources. Go with 64.
For an explanation of all this 32-64 bit stuff, check this article out: http://www.zdnet.com/blog/hardware/clearing-up-the-3264-bit-memory-limit-confusion/3124