Approaches to Algebra - Perspectives for Research and Teaching

ISBN: 0792341686

Category: Study

Tag: Mathematics


Posted on 2019-10-15, updated at 2019-11-02, by phaelx.

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Date: 1996
Pages: 348
ISBN-10: 0792341686
ISBN-13: 978-0792341680
Author: Carolyn Kieran, Lesley Lee, Nadine Bednarz, N. Bednarz, C. Kieran, L. Lee

Size: 14 MB
Format: pdf
Download     >>    https://earn4files.com/bf6yfpu67k1o
In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an 'arithmetic' of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano.
Algebra & Trigonometry,Math Teaching Materials,Mathematics History

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