Wikipedia:Reference desk/Archives/Mathematics/2021 September 5

= September 5 =

Does somone ever tried to research rounding economy related to different number bases?
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｜, 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&thinsp;n. The above formula for base 10 generalizes for even bases to:
 * $1/4$&thinsp;b&thinsp;n.
 * For odd bases, this is slightly more complicated:
 * $1/4$&thinsp;(b&thinsp;n − b−n&thinsp;).
 * 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&thinsp;λ(log b) −1 = e&thinsp;λ,
 * regardless of the value of b. This means the two expressions above become
 * $1/4$&thinsp;e&thinsp;λ and
 * $1/4$&thinsp;(e&thinsp;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
 * $1/4$&thinsp;27.9624 = 62.354..., while
 * $1/4$&thinsp;(35.0237 − 3−5.0237) = 62.352...
 * Somewhat unimpressive. --Lambiam 21:15, 5 September 2021 (UTC)