The grade‑school method multiplies each digit of one number by each digit of another, requiring n² single‑digit multiplications for two n‑digit numbers. Computer scientists measure algorithmic speed in computational steps rather than elapsed time. In 1960, a 23‑year‑old student named Anatoly Karatsuba showed that the process could be reduced to three multiplications by reusing intermediate sums, achieving O(n^1.585) complexity. Subsequent work in 2019 by David Harvey and Joris van der Hoeven introduced an algorithm with O(n log n) complexity, marking the fastest known method in theory. Both Karatsuba’s and Harvey‑van der Hoeven’s techniques outperform the grade‑school approach only when the operands are sufficiently large; for typical sizes, the simpler method remains faster due to lower overhead. Programming languages such as Python switch to Karatsuba’s method for numbers exceeding roughly 630 decimal digits, reflecting the practical threshold where the asymptotic advantage becomes measurable. Researchers continue to investigate whether O(n log n) represents the theoretical limit of multiplication, a question that remains unproven and central to the field. The open problem of establishing the optimal multiplication complexity persists, with potential impact on computer hardware, cryptography, and large‑scale data processing.