By Carolyn Kieran, Lesley Lee, Nadine Bednarz, N. Bednarz, C. Kieran, L. Lee
In Greek geometry, there's an mathematics of magnitudes within which, when it comes to numbers, merely integers are concerned. This concept of degree is restricted to distinctive degree. Operations on magnitudes can't be truly numerically calculated, other than if these magnitudes are precisely measured via a undeniable unit. the idea of proportions doesn't have entry to such operations. It can't be noticeable as an "arithmetic" of ratios. whether Euclidean geometry is finished in a hugely theoretical context, its axioms are basically semantic. this can be opposite to Mahoney's moment attribute. this can't be acknowledged of the speculation of proportions, that is much less semantic. basically artificial proofs are thought of rigorous in Greek geometry. mathematics reasoning is usually artificial, going from the recognized to the unknown. eventually, research is an method of geometrical difficulties that has a few algebraic features and contains a style for fixing difficulties that's diverse from the arithmetical method. three. GEOMETRIC PROOFS OF ALGEBRAIC principles until eventually the second one 1/2 the nineteenth century, Euclid's components used to be thought of a version of a mathematical conception. this can be one it is because geometry used to be utilized by algebraists as a device to illustrate the accuracy of ideas differently given as numerical algorithms. it could even be that geometry used to be a method to symbolize basic reasoning with out regarding particular magnitudes. to move a piece deeper into this, listed here are 3 geometric proofs of algebraic principles, the frrst by means of Al-Khwarizmi, the opposite through Cardano.
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Extra resources for Approaches to Algebra: Perspectives for Research and Teaching
Example text
For example, since a, b and c, d are not necessarily the same kind of magnitudes, one cannot simply use something like ad = be, the products ad and be being not necessarily defined. 3. Apollonlus and the Conics 18 A special use of equality of magnitudes can be found in Apollonius' Conics. It is in some way similar to the use of equations in analytical geometry, but it is also basically different. For example, to characterize a parabola, Apollonius uses a relation that corresponds in modem notation 19 tox 2 =key where k is the fourth proportional of A, B and C, and where A, B are two rectangles and C is a segment.
4. Conclusion In the three examples that have just been briefly discussed, the proofs and the content of the rules proven are at different levels, heterogeneous one to the other. The fonner has a geometrical support and is non-numerical. The latter is seen as purely numerical. The relation between the proof and the rule is based on the assumption that there is a way to measure magnitudes. This theory of measure is intuitive and not clearly defined. " No numbers participate in this generality. The geometric reasoning is general.
Geometrical proofs of algebraic rules seem to be different in their nature from purely Euclidean proofs. Euclidean proofs are synthetic, while the others are usually analytical. Those analytical proofs are successions of equivalent propositions. They correspond to a process of exploration. Synthetic proofs need only implications to go from one step to the next. They are unidirectional. Restrictions due to dimensional considerations limit the generality of geometrical proof. A figure cannot easily represent a situation of dimension higher than 3.