Convert from base 10 to an arbitrary radix with proper sign extend

Question

I was wondering if there's a formal way to properly sign extend base-10 numbers in an arbitrary base when converting. For example, if I had -256 in base 10, how would I properly sign extend the result in base 7 (or base n) without assuming a fixed length for the result.

Answer 1

From wikipedia:

The radix complement of an n digit number y in radix b is, by definition, bⁿ − y

https://en.wikipedia.org/wiki/Method_of_complements#Numeric_complements

When we sign-extend the number to n+1 digits, the new representation for the value is

bⁿ⁺¹ − y = b*bⁿ − y = bⁿ − y + (b-1)bⁿ

As b-1 is the largest digit in base b and bⁿ contains all 0s in the n least significant digits (i.e. 100..0 with n zeros in base b), (b-1)bⁿ is simply a number with b-1 followed by n zeros. The remaining part (bⁿ − y) is the old n-digit radix complement in base n

So basically a 1-digit sign-extension in base b is just about prepending the digit b-1 to the left of the old value. With mathematical induction this will applies to any values of n

For example:

• -256 in base 10:

  • 3 digits: 744
  • 4 digits: 9744
  • 5 digits: 99744

• -256 in base 7:

  • 3 digits: 153
  • 4 digits: 6153
  • 5 digits: 66153