Abstract
The introduction ended with recalling concepts in discrete mathematics as used in this book. This second chapter adds further basic concepts in continuous mathematics that are also relevant for this book, especially in the context of approximate algorithms.
I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
Carl Friedrich Gauss (1777–1855)
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Notes
- 1.
Heron of Alexandria (ca. 10–70) described this approximation method, which is also known as Babylonian method.
- 2.
Named after Baron Augustin-Louis Cauchy (1789–1857), who was central for establishing the infinitesimal calculus (e.g., of convergence of real numbers).
- 3.
Named after Bernard Placidus Johann Nepomuk Bolzano (1781–1848) and Karl Theodor Wilhelm Weierstrass (1815–1897).
- 4.
Named after Isaac Newton (1642–1727 in the Julian calendar, which was then used in England).
- 5.
It is named after Isaac Newton (see footnote on page 44) and Joseph Raphson (about 1648–about 1715).
- 6.
This paragraph was provided by Garry Tee, who also pointed out that a clear account of such convergence conditions (with illustrations) is given in Sim Borisovich Norkin’s textbook The Elements of Computational Mathematics, Pergamon Press, Oxford, 1965.
- 7.
Vatasseri Paramesvara (ca. 1380–1460) studied already mean-value formulas for the sine function. The theorem is due to A.-L. Cauchy (see footnote on page 37).
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Li, F., Klette, R. (2011). Deltas and Epsilons. In: Euclidean Shortest Paths. Springer, London. https://doi.org/10.1007/978-1-4471-2256-2_2
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