For a typical rounding algorithm, I'm wondering how long the rounding chain goes for when you round up a number. For example, if you have a decimal like 0.4445, you round the last 5 up, which would round it to 0.4450, which would round it to 0.4500, which would round it to 0.5000, which would round to 1.0000? Or does the rounding stop after the initial rounding?
[Math] How many times can you round a number
recreational-mathematics
Related Solutions
It's certainly true that this is an instance of "chunking", but I think that writing numerals that way follows the way we name the numbers in the first place. Consider $123,456,789$. Each $3$-digit block is read as a stand-alone three digit number, followed by an appropriate big-number word: "One hundred twenty three.... million," then "four hundred fifty six... thousand," and finally "seven hundred eighty nine."
Thus, the question is really, why did we stop making new words for each place value after "thousand"? Rather than sticking with "myriad", a somewhat disused word for $10^4$, we call it "ten thousand", and then $10^5$ is "one hundred thousand", with no new word being introduced until "a thousand thousand", which we call a "million".
I suspect - and this is entirely speculative - that this happened because, in the time when this aspect of language was being developed, there wasn't much use for numbers as big as $10,000$, so they were described in terms of smaller numbers, rather than being named independently. Looking at the etymology of the word "million", it originally would have meant "a great thousand", which sounds a little less silly than "a thousand thousand". Note that, after that, the words for additional multiples of $1000$ use prefixes for $2$ (bi-llion), $3$ (tri-llion), etc.
A precise answer will likely require computational assistance, with knowledge of the specific dictionaries used in Wordle for valid guesses and valid final answers. This answer will aim to answer the question for a very permissive dictionary, where any combination of 26 letters A-Z may be used as a guess or secret word.
First, an upper bound. A full game with $n$ guesses results in $5n$ emoji shown. There are $3^{5n}$ combinations of three characters in a string of length $5n$, so $\sum_{n=1}^6 3^{5n}$, or 206,741,921,896,692, is a very rough upper bound.
We could be more careful. A game ends with GGGGG, or after six incorrect guesses. Likewise, any of the five permutations of GGGGY is impossible, since the single yellow cannot be both in the word but in the incorrect position (an extra letter that appears elsewhere is marked as Black, not Yellow). So a more careful upper bound would be $\sum_{n=0}^6(3^{5}-6)^n$, or 177,961,648,104,787, where $n$ counts the number of incorrect guesses in a game (any correct guess GGGGG is fixed), and $3^{5}-6$ counts the number of possible ways to make each incorrect guess.
Finally, we can show this upper bound may be realized (with our permissive dictionary). Given a secret word $l_0l_1l_2l_3l_4$ we should show that any emoji output besides a permutation of GGGGY is possible; we may assume the $l_n$ are distinct. Permutations of Gs/Bs are easy: use correct letters for G and distinct $x\not=l_n$ for each B otherwise.
Given emoji output with multiple Y (and any number of G,B), we may use correct letters for each G, distinct incorrect letters for each B, and rotate the correct letters among the Y. For example, GBYBY could be realized using $l_1xl_5yl_3$, where $x,y,l_n$ are all distinct.
Finally we only have the case where we have at least one B and exactly one Y. In this case, fix one B and the Y. Let $l_i$ be the correct letter for the B position; we may guess a word with $l_i$ in the Y position, and a letter $x$ distinct from every $l_n$ in the B position. We then complete the word using correct letters for each G and new incorrect (and distinct) letters for every other B position.
Thus for a permissive enough dictionary, there are 177,961,648,104,787 possible Wordle tweets. (We leave open the cases where players must use information gained from previous guesses to make future guesses.)
Best Answer
The purpose of rounding is typically to give a number to a certain number of significant digits. This is never done in such a chain as in your example. Let me try to elucidate why:
So we are given a number $0.4445$. Now if we were to round this to three decimal places we would indeed get $0.445$. Rounding the resulting number to two places gives $0.45$. However this does not represent the rounding of the original number, $0.4445$ to two places, for that would be $0.44$.
So rounding successively has no real meaning. An even more obvious example than the one above is that successive rounding of $0.4445$ yields $1$, whereas rounding the number to the ones place initially gives $0$.
Basically, don't ever round successively.