Wikipedia:Reference desk/Archives/Mathematics/2021 September 5
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September 5
[edit]Does somone ever tried to research rounding economy related to different number bases?
[edit]Does somone ever tried to research rounding economy related to different number bases?
As some example, at base 10, if you want to round to nearest number with last X digits being 0, you will need to add or remove at average a value of (10^X)/4, where X is the amount of digits you want to be 0.
Does someone ever tried to find what is the number base, where you would need at average add/remove the minimum amount of numbers to round the value you want?191.250.232.243 (talk) 17:55, 5 September 2021 (UTC)
- I suspect that there are no published articles on this concept of rounding economy, determined by the average value of |i − roundn(i)|, where rounding to n digits 0 takes place in some base b ≥ 2. We can take the arithmetic mean over the range 0 ≤ i < b n. The above formula for base 10 generalizes for even bases to:
- 1⁄4 b n.
- For odd bases, this is slightly more complicated:
- 1⁄4 (b n − b−n ).
- As n → ∞, this is asymptotically equivalent to the formula for even bases. But how to compare two bases, say 2 and 10? As shown by 210 ≈ 103, 10 binary digits have about the same information as 3 decimal digits. For a fair comparison, we must replace n by something that depends on b. The information in a base-b digit is proportional to log b, so we should replace n by an expression that is inversely proportional to log b, such as λ(log b)−1. But observe that
- b λ(log b)−1 = e λ,
- regardless of the value of b. This means the two expressions above become
- 1⁄4 e λ and
- 1⁄4 (e λ − e−λ).
- So it really does not make a difference, although odd bases have a slight edge for small values of λ. Consider 28 ≈ 35. To level the field, we compare 27.9624 = 249.41... with 35.0237 = 249.40... Then
- 1⁄4 27.9624 = 62.354..., while
- 1⁄4 (35.0237 − 3−5.0237) = 62.352...
- Somewhat unimpressive. --Lambiam 21:15, 5 September 2021 (UTC)