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The setup to this question is very simple: Take the numbers $\frac{1}{1}, \frac{21}{12}, \frac{321}{123},...,\frac{987654321}{123456789}$ and plot them versus the natural numbers, as seen here: https://i.stack.imgur.com/psgiz.png.

Now, seeing that this follows a linear path, and that adding more numerals seems to increase the value of the fraction by about $0.9$, the obvious next step is to ask whether this line continues for new numeral symbols. Imagine the letter $A$ was a transdecimal character for $10$, so that we've extended the common decimal number system to an undecimal system. Then we might postulate that $\frac{A987654321}{123456789A}$ is approximately $8.9$, if we follow the line of best fit. But is there a way to actually find this exact value?

Gabriel Turner
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Apparently you're taking $$f(n) = \dfrac{\sum_{j=1}^n j\cdot 10^{j-1}}{\sum_{j=1}^n j\cdot 10^{n-j}}$$ Then $$ f(n) = \dfrac{(9n-1) \cdot 10^n + 1}{10^{n+1} - 9 n - 10} $$ In particular $$ f(10) = \frac{10987654321}{1234567900} = 8.90000000891000000891\ldots$$

Robert Israel
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    The question gives us 10 fractions for the decimal system. Extending as the OP tells us, perhaps we should also incorporate for other bases, maybe change the constant 10 in your formula to an argument for the function? In that way A987654321 will become an actual base 11 number – Soham Saha Mar 03 '24 at 18:04
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    I think base is a red herring here. The first $9$ numbers are all expressed using base $10$. It would be different if they were in bases $2$, $3$, $4$, .... – Robert Israel Mar 03 '24 at 18:14