Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis

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Cosimo, Inc., Jun 1, 2007 - Mathematics - 288 pages
Discusses calculating with natural numbers, the first extension of the notion of number, special properties of integers, and complex numbers; algebra-related subjects such as real equations with real unknowns and equations in the field of complex quantities. Also explores elements of analysis, with discussions of logarithmic and exponential functions, the goniometric functions, and infinitesimal calculus. 1932 edition. 125 figures.
 

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Contents

I
1
II
6
III
22
IV
37
V
55
VI
86
VII
144
VIII
162
IX
207
X
237
XI
250
XII
269
XIII
271
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Page 15 - ... Euclid, certainly does not correspond to the historical development of mathematics. In fact, mathematics has grown like a tree, which does not start at its tiniest rootlets and grow merely upward, but rather sends its roots deeper and deeper at the same time and rate that its branches and leaves are spreading upward. Just so — if we may drop the figure of speech — , mathematics began its development from a certain standpoint corresponding to normal human understanding, and has progressed,...
Page 11 - Ungern entdeck ich höheres Geheimnis. — Göttinnen thronen hehr in Einsamkeit, Um sie kein Ort, noch weniger eine Zeit; Von ihnen sprechen ist Verlegenheit. Die Mütter sind es!
Page 4 - ... role wherever mathematical thought is used. We would introduce it into instruction as early as possible with constant use of the graphical method, the representation of functional relations in the xy system, which is used today as a matter of course in every practical application of mathematics.
Page 4 - The child cannot possibly understand if numbers are explained axiomatically as abstract things devoid of content, with which one can operate according to formal rules. On the contrary, he associates numbers with concrete images. They are numbers of nuts, apples, and other good things, and in the beginning they can be and should be put before him only in such tangible form.
Page 15 - ... roots deeper and deeper at the same time and rate that its branches and leaves are spreading upward . . We see, then, that as regards the fundamental investigations in mathematics, there is no final ending, and therefore on the other hand, no first beginning, which could offer an absolute basis for instruction.

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