Number base conversion and arithmetic

Perhaps most of us came across base conversions when we were in high school. It's pretty easy to convert numbers from and to decimal if we remember 2 simple rules. If we desire to convert from/to base b to/from decimal we can use following rules:

Conversion from decimal to base b

Divide the given decimal number by b and store the remainder in a string. Keep on dividing until we the number can be divided no more.

num be the given decimal number

digit[i] = (num /b ^ (i+1)) % b.

resulting number in base b is digit[0] digit[1] ...digit[n-1].

Conversion from base b to decimal

num be the given number in base b and n be the number of digits

decimal = digit[n - 1] * b ^ (n -1)+...+digit[i] * b ^ (i) + digit[1] * b^1 + digit[0] * b^0


These 2 techniques are sufficient to convert between numbers in any bases.
Lets say you want to convert a number from base x into base y. To solve this you can convert base x into decimal and then convert the decimal into base y. These 2 techniques are easily transformed into programs. But it might not be a good idea to use these techniques when bases are power of 2. If bases are power of 2, bit manipulation techniques can be applied to convert. For example for converting from decimal to hex, it's always lot faster to take 4 bits at a time and convert them to hexadecimal. For example if given number is 35 (00100011, take 4 rightmost bits (0011) which is 3 and again take next 4 bits(0010) which is 2. Thus the Hex number for this is 23. This is likely to be faster than division.

Arithmetic

How would you add/subtract or do some other arithmetic operations on 2 numbers in base b? The simple idea is to convert the numbers into decimal and do the operation and then convert the resulting number back to b. The operation can be implemented for the given base itself but requires significant effort than the conversion technique.