How we can convert 396 into binary number in mathematical form?

Answer 1

To convert any number from one base to another base, iteratively divide by the second base, using the rules of arithmetic for the first base, and write down the remainders in reverse order, until the quotient is zero. In this example, converting 39610 into 1100011002, the process is as follows...

396 / 2 = 198 remainder 0
198 / 2 = 099 remainder 0 (Keeping the leading zeros to align the columns.)
099 / 2 = 049 remainder 1
049 / 2 = 024 remainder 1
024 / 2 = 012 remainder 0
012 / 2 = 006 remainder 0
006 / 2 = 003 remainder 0
003 / 2 = 001 remainder 1
001 / 2 = 000 remainder 1 (Stopping because the quotient is now zero.)

Look at the remainders above in reverse order, and you will see 110001100.

In mathematical form, it is far more complex. You need to solve for the coefficients of a polynomial equation of variable degree, in this case...

A28 + B27 + C26 + D25 +E24 + F23 +G22 + H21 +I20 = 396

... limiting the values of the coefficients to the valid integers in the base, 0 and 1, and then writing the coefficients in a string, ABCDEFGHI, giving 110001100.

The diophantine solution of this degree of polynomial equation seems very hard, by any other method than iteration. If someone has a better method, other than the first method stated above, please feel free to enhance this answer.