Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, AnalysisDiscusses 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. |
Contents
| 6 | |
| 22 | |
Concerning Special Properties of Integers | 37 |
Complex Numbers | 49 |
Real Equations with Real Unknowns | 87 |
Equations in the field of complex quantities | 101 |
The pure equation | 113 |
The tetrahedral the octahedral and the icosahedral equations | 120 |
Solution in Terms of Radicals | 138 |
Logarithmic and Exponential Functions | 144 |
The Goniometric Functions | 162 |
Concerning Infinitesimal Calculus Proper | 207 |
Transcendence of the Numbers e and л | 237 |
The Theory of Assemblages | 250 |
| 269 | |
Setting up the Normal Equation | 129 |
Other editions - View all
Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra ... Felix Klein Limited preview - 2004 |
Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra ... Felix Klein Limited preview - 2009 |
Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra ... Felix Klein No preview available - 2004 |
Common terms and phrases
a₁ according algebraic numbers angles appear arbitrary arithmetic assemblages axis biquadratic equation branch points C₁ calculation called circle coefficients commutative law complex numbers connection consider continuous functions continuum convergence coordinates corresponding course cubic equation decimal definite denumerable determined differential dihedron division elementary example expression fact factors finite number follows formula fractions function fundamental Gauss geometric give goniometric hyperbola infinitesimal calculus integers integral intuition irrational number Leibniz Leipzig limit linear logarithm mathematicians mathematics Mathematische Annalen means method multiplication negative numbers normal curve notion obtain octahedron operations parabola parameter plane polynomial positive possible power series precisely prime numbers problem proof quaternion rational functions rational numbers real numbers real roots relation Riemann surface rotation and expansion schools solution sphere spherical triangle tangent Taylor's theorem theorem theory of numbers transformation trigonometric series values variable vertices w₂ zero
Popular passages
Page 16 - ... 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 12 - 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 5 - ... 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 5 - 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 16 - ... 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.