Conversion between numeral systems.

Question

I would like to know if there is a "standard" for converting a number of base N to a number of base N. For example, 117 decimal to 728 of "213 base". Any random base to any random base, not only decimal to binary, etc. I would like to know if there is some obscure law/rule/trick to convert a number in a base N to a number in a base N, whatever the base is.

I thank you very much for your attention!

Answer 1

It's not very clear what OP is asking, since he is converting from base N to base N. The example does not make it clear either.

The usual process is that a number in a base, that any number is $a_0 = b a_1 + r_0$, and that one repeats this batching of $a_n$ until it becomes exhausted, eg

    120 ) 1728                  120)  16777216
    120 )  14.  r  48           120)   139810. r16
           0.   r  14           120)    1165.  r10
                                120)      9.   r85
        1728 d = 14.48 lh.     

In essence, a number like $2^{24}$, is batched into divisions of the long hundred (120), leaving a remainder here of 16. The number of batches is then rebatched into 120's to get 1165 second-batches, and 10 first-batches. One repeats until exhaustion: until there is just a series of remainders. The outcome is then 9 third-order, 85 second-order, 10 first-order and 16 units.

For the fractions, one can do much the same, but by multiplication. For example 0.125 * 120 gives 15. So decimal 0.125 = 0:15, base 120.

Some times, you may want to convert between formats of the same base. An example of this is to switch between the canonical form of base 3 (digits 0,1,2), and the balanced form (-1=M,0,1). The trick here is to add and subtract the same number, the steps done in different formats. Here is the sum done for 20

      202     dec 20 = 202
    11111     add a string of 1's
   ------ 
    12020     
    11111     subtract the strings of 1's digit-wise (no carry)
    -----
    01M1M     20 in the symmetric threes.

The process reverses completely.