I was reading Josephus problem from book concrete maths (Donald E knuth) and at page 11 he gives the theory below after solving the problem itself (I couldn't understand the below theory and it's following parts...):
Now that we’ve done the hard stuff (solved the problem) we seek the soft: Every solution to a problem can be generalized so that it applies to a wider class of problems. Once we’ve learned a technique, it’s instructive to look at it closely and see how far we can go with it. Hence, for the rest of this section, we will examine the solution (1.9) and explore some generalizations of the recurrence (1.8). These explorations will uncover the structure that underlies all such problems. Powers of 2 played an important role in our finding the solution, so it’s natural to look at the radix 2 representations of n and J(n). Suppose n's binary expansion is
n = (b, b,-l . . bl bo)z ; // what is this?
n = bm2^m + bm-1*2^m-1 + ... + b1*2 + b0, //bm == b sub m
where each bi is either 0 or 1 and where the leading bit b, is 1. Recalling that
n = 2”+l , we have, successively,
n = (l*bm-1*bm-2...b1*b0)2, l = (0*bm-1*bm-2...b1*b0)2 , 2l = (bm-1*bm-2...b1*b0*0)2, 2l+ 1 = (bm-1*bm-2...b1*b0*1 )2 , J(n) = (bm-1*bm-2...b1*b0*bm)2
My question is what are above equation and how did author deduced them?
(I understand binomial this form of binomial I never encountered, also I'm little idiot in maths , so please easy on me and good at explanation)