One aspect neither of you have explicitly addressed is the speaking of numbers; speaking, after all, predates writing. We say “one billion, four hundred twenty-eight million, [...]”.
Given that that’s what we say, the first two pieces of information we need are “one” and “billion”. More generally, we need to get the first 1-3 digits (the leftmost comma-separated group), then we need the magnitude, then we can proceed reading off all remaining digits.
Given that the magnitude is not explicitly written down, we get it by counting the digits. If the digits are comma-separated into groups of 3 (and “right-justified”, so that if there are 3n+1 or 3n+2 digits, then the extra 1-2 are the leftmost group), then it’s generally possible to get the magnitude from your “peripheral vision” (as opposed to counting them one by one) for numbers less than, say, 1 billion, which are what you’d most often encounter; like, “52” vs “52,193″ vs “52,193,034”, you don’t need to count carefully to distinguish those. (It gets harder around 52,193,034,892 vs 52,193,034,892,110, but manually handling those numbers is rare.) So if getting the magnitude is a mostly free operation, then you might as well just present the digits left-to-right for people who read left-to-right.
Now, is it sensible that we speak “one billion, four hundred twenty-eight million, [...]”? Seems fine to me. It presents the magnitude and the most significant digits first (and essentially reminds you of the magnitude every 3 digits), and either the speaker or the listener can cut it off at any point and have an estimate accurate to as many digits as they care for. (That is essentially the use case of “partially running the algorithm” you describe.) I think I’d hate listening to “six hundred sixty three, six hundred twenty-seven thousand, four hundred twenty-eight million, and one billion”, or even suffixes of it like “four hundred twenty eight million and one billion”. Tell me the important part first!
One aspect neither of you have explicitly addressed is the speaking of numbers; speaking, after all, predates writing. We say “one billion, four hundred twenty-eight million, [...]”.
Given that that’s what we say, the first two pieces of information we need are “one” and “billion”. More generally, we need to get the first 1-3 digits (the leftmost comma-separated group), then we need the magnitude, then we can proceed reading off all remaining digits.
Given that the magnitude is not explicitly written down, we get it by counting the digits. If the digits are comma-separated into groups of 3 (and “right-justified”, so that if there are 3n+1 or 3n+2 digits, then the extra 1-2 are the leftmost group), then it’s generally possible to get the magnitude from your “peripheral vision” (as opposed to counting them one by one) for numbers less than, say, 1 billion, which are what you’d most often encounter; like, “52” vs “52,193″ vs “52,193,034”, you don’t need to count carefully to distinguish those. (It gets harder around 52,193,034,892 vs 52,193,034,892,110, but manually handling those numbers is rare.) So if getting the magnitude is a mostly free operation, then you might as well just present the digits left-to-right for people who read left-to-right.
Now, is it sensible that we speak “one billion, four hundred twenty-eight million, [...]”? Seems fine to me. It presents the magnitude and the most significant digits first (and essentially reminds you of the magnitude every 3 digits), and either the speaker or the listener can cut it off at any point and have an estimate accurate to as many digits as they care for. (That is essentially the use case of “partially running the algorithm” you describe.) I think I’d hate listening to “six hundred sixty three, six hundred twenty-seven thousand, four hundred twenty-eight million, and one billion”, or even suffixes of it like “four hundred twenty eight million and one billion”. Tell me the important part first!