This article explains the history and current state of fast multiplication algorithms, grounded in a concrete technical claim: the grade-school O(n²) method was proven beatable in 1960 by Anatoly Karatsuba, who achieved O(n^1.585) by trading expensive multiplications for cheap additions. Harvey and van der Hoeven advanced this to O(n log n) in 2019, which is theoretically near-optimal but only practical for astronomically large numbers. The source is clear that practitioners already use these methods—Python switches to Karatsuba at ~630 decimal digits—but the field's open question remains: is O(n log n) actually the absolute floor, or can it be beaten? The article is well-sourced and avoids overstatement, correctly noting that previous "consensus" limits were overturned and that proving the current suspected limit is now the frontier problem.