Adendum,
Actually, 32k ,48k, and 64k RAM modules were available for the TS1000/ZX81. However, only 16k was available for Basic programs; the rest could store data or machine code.
A third party also offered a 16k ram board with non-volutile memory, meaning that its memory was retained even when the computer was turned off. Small programs could be stored in this memory, and run later by transferring them to normal ram. No need to load from tape! The board could also be wired to replace the system ROM, so you could actually rewrite the computer's operating system, if you were really, really bored.
Cheapskate