docs: add comprehensive section on numerical patterns and formulas

Author: f@bi.an

docs/tech/art/numbers/index.md

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+# Numbers
+
+| name/number                      | formula                                                                                      | explaintion                                                                                                                                     |
+|----------------------------------|----------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------|
+| “Swap-tail” product (base 10)    | `123456789 × 8 = 987654312`                                                                  | Multiplying the ascending digits by 8 yields the descending digits with the last two swapped. Similar patterns hold in other bases (e.g., hex). |
+| Base-n reverse trick (exact)     | `sequence × (n − 2) + (n − 1) = reverse(sequence)`                                           | For base `n`, taking `123…(n−1)` and applying the formula gives `…321`. Works for n=6…16 (and generally).                                       |
+| Ratio close to integer (general) | `N_desc/N_asc = (b − 2) + (b − 1)^3 / (b^b − b^2 + b − 1)`                                   | Exact expression for the “almost integer” ratio of descending to ascending digits in base `b`. The fractional term shrinks ~like `1/b^{b−3}`.   |
+| Handy approximation              | `N_desc/N_asc ≈ (b − 2) + (b − 1)^3 / b^b`                                                   | Replace the denominator with `b^b` for a mental-math approximation; it slightly underestimates the exact value.                                 |
+| Decimal special case             | `987654321 / 123456789 = 8 + 729 / 9,999,999,909`                                            | From the general formula with `b=10`; the error term is `9^3/(10^{10}−91) ≈ 7.29×10⁻⁸`.                                                         |
+| Hex special case                 | `0xFEDCBA987654321 / 0x123456789ABCDEF = 14 + 1/5465701947765793`                            | Exact fractional “tail” in base 16; it’s tiny (needs ~52 bits), hence floats round to 14.0.                                                     |
+| Exact integer “fix” (base 10)    | `(987654321 − 1) / (123456789 + 1) = 8`                                                      | Nudge numerator down 1 and denominator up 1 to make the ratio exactly 8. Works analogously in other bases.                                      |
+| Missing-8 series value           | `0.12345679… = 10/81`, `0.98765432… = 80/81`                                                 | The repeating decimals with the “8” omitted equal tidy fractions; follows from geometric-series sums.                                           |
+| Geometric/derivative identity    | `∑_{k≥1} kx^k = x/(1−x)^2`                                                                   | Used to show `0.123456… = 10/81` by plugging `x=1/10`.                                                                                          |
+| Classic calculator trick         | `12345679 × (9×d) = ddddddddd` (for `d=1…9`)                                                 | Because `111,111,111/9 = 12,345,679`; multiplying by `9d` yields 9 identical digits `d`.                                                        |
+| Palindromic square               | `111…1 × 111…1 = 123…(n)…321`                                                                | E.g., `1111×1111 = 1234321`; the peak digit equals the number of 1’s.                                                                           |
+| “+9 away” observation            | `123456789×8 = 987654312` is 9 off from perfect order; `12345678×8 = 98765424` is also 9 off | A small, repeated “+9” deviation shows up when the last two digits flip.                                                                        |