The evolution of binary system from predynastic Egypt to Leibniz era

Egyptians of the predynastic era had a good decimal number system for counting and addition. Although, up to some times, they had problem in counting beyond a million, by the dawn of their history, Narmer, the founder of the first Egyptian dynasty had accountants that could record 400,000 cows and 1,422,000 goats of a war booty. Except for some ambiguities in the case of Mayan number system, specialists in the history of mathematics can guess that  how the counting system of the various civilizations evolved into one of the number systems in base 10, 20, 60, etc. There is a puzzle in the mixture of the  Egyptian decimal and binary number systems which we are going to discuss and present a justification for it. The novelty of the present paper is the study of the evolution of the binary number system from the predynastic Egypt down to the Leibniz era who, by the benefit of  Khwarazmi’s   Indian Arithmetics,  completed this evolution by representing integers in $0-1$ forms and performing the hybrid decimal/binary Egyptian arithmetic operations purely inside the $0-1$ system. The second author is pleased to dedicate his share of this paper to  Esfandiar Eslami  showing  his love and appreciation for decades of his friendship and collaboration (since 1967) and, of course, the young coauthor joins the joy of this dedication to her former professor.