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Approaches to Algebra - Perspectives for Research and Teaching (Hardcover, 1996 ed.)
Loot Price: R7,560
Discovery Miles 75 600
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Approaches to Algebra - Perspectives for Research and Teaching (Hardcover, 1996 ed.)
Series: Mathematics Education Library, 18
Expected to ship within 10 - 15 working days
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Total price: R7,570
Discovery Miles: 75 700
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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.
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