# Polynomially related encodings

CLRS states that:

For some set $$I$$ of problem instances, we say that two encodings $$e_1$$ and $$e_2$$ are polynomially related if there exist two polynomial-time computable functions $$f_{12}$$ and $$f_{21}$$ such that for any $$i \in I$$, we have $$f_{12}(e_1(i)) = e_2(i)$$ and $$f_{21}(e_2(i)) = e_1(i)$$.

My understanding of the above statement says that, for example, if we have base 2 encoding of a problem, we can convert it to base 3 encoding of the problem in polynomial time and vice versa.

I wanted to confirm from the respected community if my understanding is correct, or am I having a flaw in my understanding?

Also, If I am correct, CLRS states one more thing: "unary encodings are expensive". I want to know, what do the authors mean by that? Do they mean to say that representing 8 by 11111111 is costlier than representing 8 by 1000?