Multiplication and exponentiation can be defined by equations in which one of the operands is written as the sum of powers of two. When these powers are non-negative integers, the operand is integer; without this restriction it is a fraction. The defining equation can be used in evaluation mode or in solving mode. In the former case we obtain "Egyptian" multiplication, dating from the 17th century BC. In solving mode we obtain an efficient algorithm for division by repeated subtraction, dating from the 20th century AD. In the exponentiation case we also distinguish between evaluation mode and solving mode. Evaluation mode yields a possibly new algorithm for raising to a fractional power. Solving mode yields the algorithm for logarithms invented by Briggs in the 17th century AD.

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埃及乘法及其一些后果

乘法和求幂可以通过等式来定义，其中操作数之一被写为二的幂之和。当这些幂为非负整数时，操作数为整数；如果没有这个限制，它就是一个分数。定义方程可用于评估模式或求解模式。在前一种情况下，我们获得了公元前 17 世纪的“埃及”乘法。在求解模式中，我们通过重复减法获得了一种有效的除法算法，其历史可以追溯到公元 20 世纪。在求幂的情况下，我们也区分评估模式和求解模式。评估模式可能会产生一种用于提高分数幂的新算法。求解模式产生了布里格斯在公元 17 世纪发明的对数算法。