Could anyone explain to me how every number in decimal has a unique representation in binary?
Its been a long time since i've done some conversions and they were mainly integers but I would suggest the place value for binary could be represented by $\displaystyle \dots ,2^3,2^2,2^1,2^0,2^{-1},2^{-2},2^{-3},\dots$ etc.. The negative indicies representing decimals.
Someone please correct me if I am giving the poster a bum steer.
Every natural number has a unique representation in any positive integral base, there is nothing special about decimal (or binary for that matter). If you are happy that every number (integer, natural?) has a unique representation in decimal then exactly the same reasoning is applicable to any other base.
(note: not all decimal and binary fractions have unique representations in their corresponding bases)
CB