Occasionally, a practical problem arises in which it is necessary to invert a sequence
of arithmetic operations. That is, we know the result of the operations but not
the data. The best examples of this kind of problem come from geometry, and a
typical specimen can be seen in the sangaku plaque shown in Color Plate 2. This
type of problem is the seed of the area we call algebra, whose development can be
conveniently divided into three stages. In the first stage, knowing the procedure
followed and the result, one is forced to think in the terms that Pappus referred to as
analysis, that is, deducing consequences of the formula until one arrives at the data.
The main tool in this analysis is the equation, but equations occur explicitly only
after a stock of examples has been accumulated. At the second stage, equations are
identified as an object of independent interest, and techniques for solving them are
developed. In the third stage, a higher-level analysis of the algorithms for solution
leads to the subject we now know as modern algebra. We shall devote one chapter
to each of these stages.
coco
(coco)
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