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01042857 Karatsuba, A.A. The complexity of computations. Proc. Steklov Inst. Math. 211, 169-183 (1995); translation from Tr. Mat. Inst. Steklova 211, 186-202 (1995). 1995 Moscow: Maik Nauka/Interperiodica Publishing EN fast multiplication of multiplace numbers Summary: This paper is devoted to one of the problems in the history of mathematics, namely, the history of the appearance of the basic arithmetic operations and, more specifically, the appearance of multiplication. I was repeatedly asked how the method of fast multiplication of multiplace numbers was found. In turn, I became interested in how the mankind arrived at the multiplication method that was the only known one before 1960 and was called the ``ordinary'', ``well-known'', ``school'', etc., multiplication. The aim of the present paper is to answer these questions.