Greedy Randomized Adaptive Search Procedures
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damental flaw in earlier proofs, namely the fact that they were assuming the existence of roots and then trying to deduce properties of them. His proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our presentday standards required for a rigorous proof. He published the book 'Disquisitiones Arithmeticae' in the summer of 1801. There were seven sections, all but the last section, referred to above, being devoted to number theory. In 1814, the Swiss accountant J.R. Argand published a proof of the FTA which may be the simplest of all the proofs. His proof is based on d'Alembert's idea in 1746. Argand simplifies d'Alembert's idea using a general theorem on the existence of a minimum of a continuous function. Two years after Argand's proof appeared Gauss published in 1816 a second proof of the FTA. Gauss uses Euler's approach but instead of operating with roots which may not exist, Gauss operates with indeterminates. This proof is complete and correct. A third proof by Gauss also in 1816 is, like the first, topological in nature. Gauss introduced in 1831 the term 'complex number'. In 1849 Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots. The proof is similar to the first proof given by Gauss. However it adds little since it is straightforward to deduce the result for complex coefficients from the result about polynomials with real coefficients. It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a
Gauss, Carl Friedrich whole hierarchy of imaginary quantities of which complex numbers were the simplest. Gauss called them a shadow of shadows. The different proofs of the FTA are Gauss's most important contributions as a rigorist, that is to say, as a representative of logical strictness in method of proof [1]. Since this theorem has great significance in both algebra and function theory, it influenced many other related areas, including mathematical optimization. Gauss used infinite sequences and series in his daily work, not only in mathematics but in astronomy, geodesy, and physics. As an eleven-yearold, Gauss was already studying Newton's binomial theorem, which includes the infinite geometric series as a special case. He investigated the conditions under which an infinite binomial series has a logical meaning. He also thought about the theoretical formulation of the notion of limiting value [3]. In an unfinished article written around 1800, 'Fundamental concepts in the principles of series', he formulated the notion of the limit of a sequence in a fashion far ahead of the times. Gauss introduced there the notions of upper bound and least upper bound G; he also introduced the notions of lower bound and greatest lower bound g. Furthermore he introduced the 'final upper bound' H and the 'final lower bound' h. If H = h, then their common value was called the absolute limit
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