Pablo Renteria

I define the position of each digit before converting, because a positional system is defined by where each digit sits. In decimal those powers are ones, tens, hundreds. In binary they are ones, twos, fours. In hexadecimal they run to sixteens, which is why the digit for sixteen needs a letter and why hexadecimal feels odd to anyone starting from binary. Converting between bases is repeated division by the target base, collecting remainders and reading them back in reverse. That is correct for integers. For fractions the method reverses, multiplying by the target base and collecting the integer parts, and that is where hand calculation stops being easy because the loop may not terminate. The reason it may not terminate is worth stating: a fraction whose denominator has a factor other than two terminates in base two, and the analogous condition applies to other bases. A tenth in hexadecimal repeats forever, so a tool has to decide how many digits to produce and how to indicate that it stopped short. Roman numerals have the opposite problem. They are not positional, so a smaller numeral before a larger one means subtraction, the valid forms are constrained, and there is no number above a few thousand that is conventionally written at all. I cover the four rules and the forms that break them. Everything here is positional notation, and the base only decides how the positions are counted. Roman numerals are the exception to all of it, and they are the reason this page has two halves.