A number $ N $ in base $ b $ can be written with an addition of powers in this base $ b $.

Example: The number $ N = 123_{(10)} $ (base 10) verifies the equality $$ N = 789 = 7 \times 100 + 8 \times 10 + 9 \times 1 = 7 \times 10^2 + 8 \times 10^1 + 9 \times 10^0 $$

$ N= $ $ c2 $ $ c1 $ $ c0 $ $ 789 $ $ 7 $ $ 8 $ $ 9 $

Take a number $ N $ made of $ n $ digits $ { c_{n-1}, c_{n-2}, ..., c_2, c_1, c_0 } $ in base $ b $, it can be written it as a polynomial:

$$ N_{(b)} = \{ c_{n-1}, ..., c_1, c_0 \}_{(b)} = c_{n-1} \times b^{n-1} + ... + c_1 \times b^1 + c_0 \times b^0 $$

To compute a base change, base $ 10 $ is the reference, or an intermediate step.