| “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. |