Why notations of numbers in standard positional numeral systems when evaluated, evaluates to decimal notation?

Question

For a positional numeral system of radix b, evaluating a notation $a_3 a_2 a_1 a_0$ as shown below results in decimal notation of number,

$a_3 a_2 a_1 a_0 = a_3 \times b^3 + a_2 \times b^2 + a_1 \times b^1 + a_0 \times b^0$

Why?

PS: Please keep your answers simplified

Answer 1

You may define a numeral system in base $2b+1$ such that $$ (a_n\cdots a_2a_1a_0)=\sum a_k (2b+1)^k $$ with $-b\le a_i \le b$ (reference: Mathematics Made Difficult). This system is simpler than our usual system. You can even define Negative base.