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Cracovians: The Twisted Twins of Matrices
Linear algebra is typically explained using matrices. But matrix theory is just one possible perspective. Below, I describe an alternative approach to linear algebra. Tadeusz Banachiewicz (1882–195…
In the preface to Banachiewicz’s book, Cracovian Algebra( Rachunek krakowianowy, PWN 1959), it is noted that “some calculations, generally known to be very tedious, became a sheer entertainment for the executor.” Continued fraction reduction Polynomial division Horner’s scheme Interpolation of functions of two arguments directly from function values, without creating differences Iterative solution of systems of linear equations Determination of the root of an algebraic equation with the smallest absolute value Determination of the greatest common divisor of numbers in a sequence whose terms are given by a polynomial with integer coefficients In Banachiewicz’s book, one can find charming advice, such as this: “In practice, when multiplying cracovians, we use movable objects (pen nibs, pencils, matches, etc.)
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