The problem with
base sqrt(2) is that numbers would have
multiple possible
representations.
For example, I could represent decimal 12 as 1010000sqrt(2), because 4+8=12. But a base conversion algorithm would give the result: 100000000.01001000000001...sqrt(2) because sqrt(2)^7, or 11.313712, is closer to 12 than sqrt(2)^6, or 8. Thus, in base sqrt(2), the number 12 is rational and irrational at the same time.
A writeup in base pi points out that this problem occurs in any non-integral base.