zlacker

I'm belatedly warming up to the idea that the digit length of the number zero is actually zero in various senses. (A footnote in the article points out that we could define it this way and then some students' simpler solutions would actually be considered correct.)

Yes, we obviously normally use a single digit to write zero, but we could logically write it as the empty string "" if we had a way to indicate that a specific number was meant. (Relative to that, writing 0 is just as unnecessary as writing 00 or 000, because we can have a rule that all leading zeroes need not be written.)

As another example of this intuition, suppose that we wrote all integers with a trailing decimal point. In that case the numbers from 1 to 10 would be "1.", "2.", "3.", "4.", "5.", "6.", "7.", "8.", "9.", and "10.", while zero could quite logically just be "." with no digits (as it has only leading zeroes, or we could say "nothing in the ones' place").

Quite a lot of arithmetic algorithms would probably work just fine with these conventions! In fact, they might have fewer exceptions or special cases than they do when they expect an explicit 0.

For instance, using the "zero is written as empty string" convention, Python int() (here simplified to assume input strings of zero or more decimal digits) and str() could look like

  def int(s):
      n = 0
      for c in s:
          n *= 10
          n += "0123456789".find(c)
      return n

  def str(n):
      s = ""
      while n:
          n, d = divmod(n, 10)
          s = "0123456789"[d] + s
      return s

Seems pretty general! Yes, it's annoying or confusing if the output is going to be viewed by a human user in a larger string context without delimiters, as we probably don't want to see things like "I ate tomatoes" instead of "I ate 0 tomatoes".

>>schoen+(OP)
Aren’t you mixing up 0 the digit and 0 the number here?

Certainly we can use the empty string to denote the number zero, but it doesn’t mean that we have a straight forward convention for how we denote any number which have some null value in some power lower than the most significant one.

Not that it’s impossible to come with some convention that ditches 0 as intermediary digit. For example 302009==3e5+2e3+9 will evaluate to true in many languages out there. In Ruby we even have `302009 === 3e5+2e3+9+''.to_i` that is evaluated to true.

But this notation loses the conveniences afforded by a positional fixed base numeral system provides.

>>psycho+M5
I mean that the number 0 specifically can be written by empty string, so that we would count like

"", "1", "2", "3", "4", ..., "9", "10", "11", "12", ..., "99", "100", "101", ...

The only change is the empty string being a valid name for a number (the number zero).

I don't mean to suggest getting rid of place value or the digit 0! This is more like making the place value system more consistent with respect to a corner case.

And the practical reason that we can't make this change is that we don't have a way to distinguish in writing between the absence of any string and the presence of the empty string.