we do not have anything directly comparable to continued-fraction expansions for a complex quadratic irrationality. In fact, the simple, but true, answer to the problem of how to find an infinite number of rational numbers that converge to such an irrationality is that you cannot! Correspondingly, the analogue of the Pell equation has only finitely many solutions.
"Irrationality is the square root of all evil"
The system, in its irrationality, has been driven by profit to build steel skyscrapers for insurance companies while the cities decay, to spend billions for weapons of destruction and virtually nothing for children’s playgrounds , to give huge incomes to men who make dangerous or useless things, and very little to artists, musicians, writers, actors. Capitalism has always been a failure for the lower classes. It is now beginning to fail for the middle classes.
"Irrationality is the square root of all evil" -- Douglas Hofstadter
_The History of Mathematics._--The history of mathematics is in the main the history of its various branches. A short account of the history of each branch will be found in connexion with the article which deals with it. Viewing the subject as a whole, and apart from remote developments which have not in fact seriously influenced the great structure of the mathematics of the European races, it may be said to have had its origin with the Greeks, working on pre-existing fragmentary lines of thought derived from the Egyptians and Phoenicians. The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities. The great abstract ideas (considered directly and not merely in tacit use) which have dominated the science were due to them--namely, ratio, irrationality, continuity, the point, the straight line, the plane. This period lasted[11] from the time of Thales, c. 600 B.C., to the capture of Alexandria by the Mahommedans, A.D. 641. The medieval Arabians invented our system of numeration and developed algebra. The next period of advance stretches from the Renaissance to Newton and Leibnitz at the end of the 17th century. During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation. The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and Laplace. This period may be conceived as continuing throughout the first quarter of the 19th century. It was remarkable both for the brilliance of its achievements and for the large number of French mathematicians of the first rank who flourished during it. The next period was inaugurated in analysis by K. F. Gauss, N. H. Abel and A. L. Cauchy. Between them the general theory of the complex variable, and of the various "infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces. The applied mathematical sciences of light, electricity and electromagnetism, and of heat, were now largely developed. This school of mathematical thought lasted beyond the middle of the century, after which a change and further development can be traced. In the next and last period the progress of pure mathematics has been dominated by the critical spirit introduced by the German mathematicians under the guidance of Weierstrass, though foreshadowed by earlier analysts, such as Abel. Also such ideas as those of invariants, groups and of form, have modified the entire science. But the progress in all directions has been too rapid to admit of any one adequate characterization. During the same period a brilliant group of mathematical physicists, notably Lord Kelvin (W. Thomson), H. V. Helmholtz, J. C. Maxwell, H. Hertz, have transformed applied mathematics by systematically basing their deductions upon the Law of the conservation of energy, and the hypothesis of an ether pervading space. Entry: MATHEMATICS
The chief application of the theory of groups of finite order is to the theory of algebraic equations. The analogy of equations of the second, third and fourth degrees would give rise to the expectation that a root of an equation of any finite degree could be expressed in terms of the coefficients by a finite number of the operations of addition, subtraction, multiplication, division, and the extraction of roots; in other words, that the equation could be solved by radicals. This, however, as proved by Abel and Galois, is not the case: an equation of a higher degree than the fourth in general defines an algebraic irrationality which cannot be expressed by means of radicals, and the cases in which such an equation can be solved by radicals must be regarded as exceptional. The theory of groups gives the means of determining whether an equation comes under this exceptional case, and of solving the equation when it does. When it does not, the theory provides the means of reducing the problem presented by the equation to a normal form. From this point of view the theory of equations of the fifth degree has been exhaustively treated, and the problems presented by certain equations of the sixth and seventh degrees have actually been reduced to normal form. Entry: A
But the arithmetic of the ancients was inadequate as a science of number. Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number. The idea of possible incommensurability of lengths was introduced into Europe by Pythagoras; and the corresponding idea of irrationality of number was absent from a crude arithmetic, while there were great practical difficulties in the way of its introduction. Hence perhaps it arose that, till comparatively modern times, appeal to arithmetical aid in geometrical reasoning was in all possible ways restrained. Geometry figured rather as the helper of the more difficult science of arithmetic. Entry: 1
_The Elements of Geometry._--Legendre's name is most widely known on account of his _Eléments de géométrie_, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry. It first appeared in 1794, and went through very many editions, and has been translated into almost all languages. An English translation, by Sir David Brewster, from the eleventh French edition, was published in 1823, and is well known in England. The earlier editions did not contain the trigonometry. In one of the notes Legendre gives a proof of the irrationality of [pi]. This had been first proved by J. H. Lambert in the Berlin _Memoirs_ for 1768. Legendre's proof is similar in principle to Lambert's, but much simpler. On account of the objections urged against the treatment of parallels in this work, Legendre was induced to publish in 1803 his _Nouvelle Théorie des parallèles_. His _Géométrie_ gave rise in England also to a lengthened discussion on the difficult question of the treatment of the theory of parallels. Entry: LEGENDRE
The relation between the memory in dreams and in the hypnotic trance is curious: suggestions given in the trance may be accepted and then forgotten or never remembered in ordinary life; this does not prevent them from reappearing occasionally in dreams; conversely dreams forgotten in ordinary life may be remembered in the hypnotic trance. These dream memories of other states of consciousness suggest that dreams are sometimes the product of a deeper stratum of the personality than comes into play in ordinary waking life. It must be remembered in this connexion that we judge of our dream consciousness by our waking recollections, not directly, and our recollection of our dreams is extraordinarily fragmentary; we do not know how far our dream memory really extends. Connected with memory of other states is the question of memory in dreams of previous dream states; occasionally a separate chain of memory, analogous to a secondary personality, seems to be formed. We may be also conscious that we have been dreaming, and subsequently, without intermediate waking, relate as a dream the dream previously experienced. In spite of the irrationality of dreams in general, it by no means follows that the earlier and later portions of a dream do not cohere; we may interpolate an episode and again take up the first motive, exactly as happens in real life. The strength of the dream memory is shown by the recurrence of images in dreams; a picture, the page of a book, or other image may be reproduced before our eyes several times in the course of a dream without the slightest alteration, although the waking consciousness would be quite incapable of such a feat of visualizing. In this connexion may be mentioned the phenomenon of redreaming; the same dream may recur either on the same or on different nights; this seems to be in many cases pathological or due to drugs, but may also occur under normal conditions. Entry: DREAM
_Case_ 3. When neither a nor c is a square number, yet if the expression a + bx + cx² can be resolved into two simple factors, as f + gx and h + kx, the irrationality may be taken away as follows:-- Entry: 6
His mathematical discoveries were extended and overshadowed by his contemporaries. His development of the equation x
The most important publication, however, on the subject in the 18th century was a paper by J.H. Lambert,[26] read before the Berlin Academy in 1761, in which he demonstrated the irrationality of [pi]. The general test of irrationality which he established is that, if Entry: 4