Approaches to Algebra
Perspectives for Research and Teaching
Editors: Bednarz, N., Kieran, C., Lee, L. (Eds.)
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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 AlKhwarizmi, the other two by Cardano.
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Bibliographic Information
 Bibliographic Information

 Book Title
 Approaches to Algebra
 Book Subtitle
 Perspectives for Research and Teaching
 Editors

 N. Bednarz
 C. Kieran
 L. Lee
 Series Title
 Mathematics Education Library
 Series Volume
 18
 Copyright
 1996
 Publisher
 Springer Netherlands
 Copyright Holder
 Kluwer Academic Publishers
 eBook ISBN
 9789400917323
 DOI
 10.1007/9789400917323
 Hardcover ISBN
 9780792341451
 Softcover ISBN
 9780792341680
 Series ISSN
 09244921
 Edition Number
 1
 Number of Pages
 XVI, 348
 Topics