Euler.docx - Euleru2019s quote about the use of...

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Euler’s quote about the use of observation in mathematics is agreat commentary on mathematical reasoning and the inductive processitself. As Euler mentions above even in fields as abstract as puremathematics and the theory of numbers, observation and induction areimportant tools in helping identify the various intriguing behaviors andpatterns of numbers. However, observation alone is not sufficient and thetrue strength of mathematical discovery lies in the use of mathematicaltools to prove without exception what is being observed. As students ofmathematics we are no strangers to this process. Identifying a pattern in agiven example and applying inductive reasoning to generalize said patternis the basis of modern mathematics; however, as Euler notes, we should bevary of proofs by induction alone. The strongest conjectures are thosewhich can not only be supported by inductive reasoning, but proved andreproved using other methods as well. The Law of Quadratic Reciprocity isone such example in the theory of numbers.

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