On Numbers: A Piece of π

"It is probable that the number π is not even contained among the algebraical irrationalities, i.e., that it cannot be a root of an algebraical equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly." (Adrien-Marie Legendre, "Elements de geometrie", 1794)

"Whether any other Mathematician will appear, possessing sufficient leisure, patience, and facility of computation, to calculate the value of π to a still greater extent, remains to be seen: all that the Author can say is, he takes leave of the subject for the present […]" (William Shanks, “Contributions to Mathematics”, 1853)

"What good is your beautiful investigation regarding π? Why study such problems, since irrational numbers do not exist?" (Leopold Kronecker [letter to Ferdinand von Lindemann] 1882)

"The proof that π is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that π can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots." (Felix Klein, "Lectures on Mathematics", 1911)

"The mysterious and wonderful π is reduced to a gargle that helps computing machines clear their throats." (Philip J Davis, "The Lore of Large Numbers", 1961)

"This mysterious π which comes in at every door and window, and down every chimney." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

“Ten decimal places of π are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope.” (Simon Newcomb)

"If we take the geometrical relations, the thousandth decimal of π sleeps there, though no one may ever try to compute it." (William James, "The Meaning of Truth", 1909)

"The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i. e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted." (Ernest W Hobson, "Squaring the Circle", 1913)

"What does it matter whether π is rational or irrational? A mathematician faced with [this] question is in much the same position as a composer of music being questioned by someone with no ear for music. Why do you select some sets of notes and have them repeated by musicians, and reject others as worthless? It is difficult to answer except to say that there are harmonies in these things which we find that we can enjoy. It is true of course that some mathematics is useful. [...] But the so-called pure mathematicians do not do mathematics for such [practical applications]. It can be of no practical used to know that π is irrational, but if we can know it would surely be intolerable not to know. (Edward C Titchmarsh, "Mathematics for the General Reader", 1948)

"The mysterious and wonderful π is reduced to a gargle that helps computing machines clear their throats." (Philip J Davis, "The Lore of Large Numbers", 1961)

"Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number π." (William L Schaaf, "Nature and History of π", 1967)

"There is more to the calculation of π to a large number of decimal places than just the challenge involved. One reason for doing it is to secure statistical information concerning the 'normalcy' of π. A real number is said to be simply normal if in its decimal expansion all digits occur with equal frequency, and it is said to be normal if all blocks of digits of the same length occur with equal frequency. It is not known if π (or even √2, for that matter) is normal or even simply normal." (Howard Eves, "Mathematical Circles Revisited", 1971)

"The matter of the normalcy or non-normalcy of π will never, of course, be resolved by electronic computers. We have here an example of a theoretical problem which requires profound mathematical talent and cannot be solved by computations alone. The existence of such problems ought to furnish at least a partial antidote to the disease of computeritis, which seems so rampant today." (Howard Eves, "Mathematical Circles Revisited", 1971)

“The digits of π beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere” (Petr Beckmann, “A History of π”, 1976)

"Computing π is the ultimate stress test for a computer - a kind of digital cardiogram." (Ivars Peterson, Islands of Truth, 1990)

“The digits of π march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)

“π is not the solution to any equation built from a less than infinite series of whole numbers. If equations are trains threading the landscape of numbers, then no train stops at π.” (Richard Preston, “The Mountains of π”, The New Yorker, March 2, 1992)

"The story of π has been extensively told, no doubt because its history goes back to ancient times, but also because much of it can be grasped without a knowledge of advanced mathematics." (Eli Maor, "e: The Story of a Number", 1994)

"There’s a beauty to π that keeps us looking at it [...] The digits of π are extremely random. They really have no pattern, and in mathematics that’s really the same as saying they have every pattern." (Peter Borwein, 1996)

"When we think of π, let’s not always think of circles. It is related to all the odd whole numbers. It also is connected to all the whole numbers that are not divisible by the square of a prime. And it is part of an important formula in statistics. These are just a few of the many places where it appears, as if by magic. It is through such astonishing connections that mathematics reveals its unique and beguiling charm." (Sherman K Stein, "Strength in Numbers", 1996)

"The story of π reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics. A surprising amount of the most important mathematicians and a significant number of the most important mathematicians have contributed to its unfolding - directly or otherwise." (J Lennart Berggren et al, "π", 2004)

“A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say π.” (Daniel Tammet, “Thinking in Numbers: How Maths Illuminates Our Lives”, 2012)

"It turns out π is different. Not only is it incapable of being expressed as a fraction, but in fact π fails to satisfy any algebraic relationship whatsoever. What does π do? It doesn’t do anything. It is what it is. Numbers like this are called transcendental (Latin for 'climbing beyond'). Transcendental numbers - and there are lots of them - are simply beyond the power of algebra to describe." (Paul Lockhart, "Measurement", 2012)

"Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π." (Steven Strogatz, "Why π Matters" 2015)

"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the "i times π" power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Since it’s impossible to express an irrational number such as π as a fraction, the quest for a fraction equal to π could never be successful. Ancient mathematicians didn’t know that, however. As noted above, it wasn’t until the eighteenth century that the irrationality of π was demonstrated. Their labors weren’t in vain, though. While enthusiastically pursuing their fundamentally doomed enterprise, they developed a lot of interesting mathematics as well as impressively accurate approximations of π." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The geometric interpretation of e^iπ is rich with emblematic potential. You could see its suggestion of a 180-degree spin as standing for a soldier’s about-face, a ballet dancer’s half pirouette, a turnaround jump shot, the movement of someone setting out on a long journey who looks back to wave farewell, the motion of the sun from dawn to dusk, the changing of the seasons from winter to summer, the turning of the tide. You could also associate it with turning the tables on someone, a reversal of fortune, turning one’s life around, the transformation of Dr. Jekyll into Mr. Hyde (and vice versa), the pivoting away from loss or regret to face the future, the ugly duckling becoming a beauty, drought giving way to rain. You might even interpret its highlighting of opposites as an allusion to elemental dualities - shadow and light, birth and death, yin and yang." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"[…] the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.[…] when the three enigmatic numbers are combined in this form, e^iπ, they react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The most remarkable thing about π, however, is the way it turns up all over the place in math, including in calculations that seem to have nothing to do with circles." (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)

"Thus, while feelings may be the essence of subjectivity, they are by no means part of our weaker nature - the valences they automatically generate are integral to our thought processes and without them we’d simply be lost. In particular, we’d have no sense of beauty at all, much less be able to feel (there’s that word again) that we’re in the presence of beauty when contemplating a work such as Euler’s formula. After all, e^iπ + 1 = 0 can give people limbic-triggered goosebumps when they first peer with understanding into its depths." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"[…] the story of π is the deeply ironic tale of one thinker after another trying to nail down the size of a number that is fundamentally immeasurable. (Because it’s irrational.)" (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"There’s something so paradoxical about pi. On the one hand, it represents order, as embodied by the shape of a circle, long held to be a symbol of perfection and eternity. On the other hand, pi is unruly, disheveled in appearance, its digits obeying no obvious rule, or at least none that we can perceive. Pi is elusive and mysterious, forever beyond reach. Its mix of order and disorder is what makes it so bewitching." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"With its yin and yang binaries, pi is like all of calculus in miniature. Pi is a portal between the round and the straight, a single number yet infinitely complex, a balance of order and chaos. Calculus, for its part, uses the infinite to study the finite, the unlimited to study the limited, and the straight to study the curved. The Infinity Principle is the key to unlocking the mystery of curves, and it arose here first, in the mystery of pi." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Exploring π is like exploring the universe." (David Chudnovsky)

"God is in the details, for mathematicians have plunged deeper and deeper within π's digits with a religious fervor, hoping to find even a hint of understanding." (Ludwig Mies van der Rohe)

"π is more like exploring underwater. You’re in the mud, and every thing looks the same." (Gregory Chudnovsky)

"π is not just a collection of random digits. π is a journey, an experience; unless you try to see the natural poetry that exists in π, you will find it very difficult to learn." (Antranig Basman)

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Beyond the History of Mathematics III

“The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.” (Florian Cajori, “A History of Mathematics”, 1893)

 “This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time.” (Florian Cajori, “A History of Mathematical Notations”, 1928)

“There are no absolutes [...] in mathematics or in its history.” (Eric T Bell, The Development of Mathematics, 1940)

“Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution.” (Richard Curand [forward to Carl B Boyer’s “The History of the Calculus and Its Conceptual Development”, 1949])

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors.” (Richard von Mises, “Positivism: A Study in Human Understanding”, 1951)

“It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands.” (Philip J Davis, “The Mathematics of Matrices”, 1965)

“Students enjoy […] and gain in their understanding of today's mathematics through analyzing older and alternative approaches.” (Lucas N H Bunt, Phillip S Jones & Jack D Bedient, “The Historical Roots of Elementary Mathematics”, 1976)

“[…] how completely inadequate it is to limit the history of mathematics to the history of what has been formalized and made rigorous. The unrigorous and the contradictory play important parts in this history.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

“Like anything else, mathematics is created within the context of history […]” (William Dunham, “Journey Through Genius”, 1990)

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Beyond the History of Mathematics II

"I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history." (James W L Glaisher, [opening address] 1890)

 “In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure” (Hermann Hankel, “Die Entwicklung der Mathematik in den letzten Jahrhunderten, 1884)

"The true method of foreseeing the future of mathematics is to study its history and its actual state." (Henri Poincaré, "Science and Method", 1908)

"One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics." (Henri Poincaré)

“Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him.” (Alfred N Whitehead, “Mathematics as an Element in the History of Thought” in “Science and the Modern World”, 1925)

“Mathematicians study their problems on account of their intrinsic interest, and develop their theories on account of their beauty. History shows that some of these mathematical theories which were developed without any chance of immediate use later on found very important applications.” (Karl Menger, “What is calculus of variations and what are its applications?”, The Scientific Monthly 45, 1937)

“The history of mathematics shows that the introduction of better and better symbolism and operations has made a commonplace of processes that would have been impossible with the unimproved techniques.” (Morris Kline, “Mathematics in Western culture”, 1953)

“Although the study of the history of mathematics has an intrinsic appeal of its own, its chief raison d'être is surely the illumination of mathematics itself.” (Charles H Edwards Jr, “The Historical Development of the Calculus”, 1979)

“Mathematical research should be as broad and as original as possible, with very long range-goals. We expect history to repeat itself: we expect that the most profound and useful future applications of mathematics cannot be predicted today, since they will arise from mathematics yet to be discovered." (Arthur Jaffe, “Ordering the universe: the role of mathematics”, SIAM Review Vol 26. No 4, 1984)

"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way. (Carl B Boyer & Uta C Merzbach, “A History of Mathematics”, 1991)

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Negative Numbers: The Unimaginable

“Many persons rise up against these negative magnitudes, as if they were objects difficult to conceive, yet there is nothing at the same time more simple nor more natural.” (L'Abbé Deidier, 1739)

 "[negative numbers] darken the very whole doctrines of the equations and to make dark of the things which are in their nature excessively obvious and simple. It would have been desirable in consequence that the negative roots were never allowed in algebra or that they were discarded." (Francis Meseres, 1759)

“One must admit that it is not a simple matter to accurately outline the idea of negative numbers, and that some capable people have added to the confusion by their inexact pronouncements. To say that the negative numbers are below nothing is to assert an unimaginable thing.” (Jean le Rond d'Alembert, "Negatif”, Encyclopédie [1751 – 1772])

“[…] the algebraic rules of operation with negative numbers are generally admitted by everyone and acknowledged as exact, whatever idea we may have about this quantities. “ (Jean Le Rond d'Alembert, Encyclopédie, [1751 – 1772])

“It is very inaccurate to say that a negative number is less than 0, which is what many authors claim. A negative number is a positive number, but in another sense, and therefore relative. “ (Van Swinden, cca 1800)

„Every negative quantity standing by itself is a mere creature of the mind and [...] those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective.“ (Lazare Carnot, “Geometrie de Position”, 1803)

Negative Numbers: Direction

"[H]ere negation is […] contrariety […] that is to say, in the contrary direction. As the west is contrary of east; and the south the converse of north. Thus, of two countries, east and west, if one be taken as positive, the other is relatively negative. So when motion to the east is assumed to be positive, if a planets motion be westward, then the number of degrees equivalent to the planets motion is negative.” (Bhāskara II, "Bijaganita", 12th century)

“Magnitudes have more or less reality as their being takes them further from zero, and they have less reality when their non-being takes them further from this same zero. It became customary to call positive or true every magnitude which adds to zero, and negative or false every magnitude which takes away from this same zero.” (Jean Prestet, 1675) 

“It is evident that zero, or nothing, is the term between the positive and negative magnitudes that separates them one from the other. The positives are magnitudes added to zero; the negatives are, as it were, below zero or nothing; or to put it a better way, zero or nothing lies between the positive and negative magnitudes; and it is as the term between the positive and negative magnitudes, where they both begin.” (Charles-René Reyneau, 1714)

“From this it follows that the idea of positive or negative is added to those magnitudes which are contrary in some way. […] All contrariness or opposition suffices for the idea of positive or negative. […] Thus every positive or negative magnitude does not have just its numerical being, by which it is a certain number, a certain quantity, but has in addition its specific being, by which it is a certain Thing opposite to another. I say opposite to another, because it is only by this opposition that it attains a specific being (Bernard le Bouyer de Fontenelle, “Éléments de la géométrie de l'Infini“, 1727)

“It should be remarked that negative quantities are magnitudes opposite to positive quantities. […] With this notion of positive and negative quantities, it follows that both are equally real and that, consequently, negatives are not the negation or absence of positives; but they are certain magnitudes opposite to those which are regarded as positive (Dominique-François Rivard, “Élémens de Mathématique”, 1744)

“When two quantities equal in respect of magnitude, but of those opposite kinds, are joined together, and conceived to take place in the same subject, they destroy each other’s effect, and their amount is nothing.” (Colin MacLaurin, “A Treatise of Algebra”, 1748)

“If two quantities are in such a relation to each other that the one decreases just as much as the other one increases, and vice versa, then they are called opposite quantities. […] Such opposite quantities, considered for themselves, are quantities of a different kind, or are to be regarded as having different denominations. However, they are always situated under a common principal concept, and can in so far be considered as quantities of the same kind.” ” (Wenceslaus J G Karste, 1768)

“With respect to magnitude, a negative quantity is not distinct from a positive one at all, but it is distinct with respect to the operation which is to be executed with this quantity.” (Moses Mendelssohn)

“[…] direction is not a subject for algebra except in so far as it can be changed by algebraic operations. But since these cannot change direction (at least, as commonly explained) except to its opposite, that is, from positive to negative, or vice versa, these are the only directions it should be possible to designate. […] It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic. “ (Casper Wessel, „On the Analytical Representation of Direction“, 1787)

"The words positive and negative are general terms, that indicate the different states a quantity can be in, and that in special cases will have interpretations such as capital and debt, east and west, right and left, up and down, ascending and descending, winning and losing, etc. In each particular case it is up to us to choose which of the two states we wish to call positive, and thereby denote with the + sign, but once this is determined, we must consistently call the other state negative, and indicate it by the sign −." (Sylvestre-François Lacroix, "Beginselen der Stelkunst", 1821)

Negative Numbers: Minus Times Minus

“The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.” (Brahmagupta, “Brahmasphuṭasiddhanta”, cca. 628)
 
"The square of a positive, as also of a negative number, is positive; that the square root of a positive number is twofold, positive and negative. There is no square root of a negative number, for it is not a square.” (Bhaskara, “Lilavati”, 1150)

“And therefore lies open the error commonly asserted that minus times minus produces plus, lest indeed it be more correct that minus times minus produces plus than plus times plus would produce minus” (Cardano, “De Aliza Regulae”, 1570)

 “I see no other answer to this [concerning the proportion argument] than to say that the multiplication of minus by minus is carried out by means of subtraction, whereas all the others are carried out by addition: it is not strange that the notion of ordinary multiplications does not conform to this sort of multiplication, which is of a different kind from the others.” (Antoine Arnauld, “Nouveaux Elémens de Géométrie”, 1683)

“It is not necessary to search for any mystery here: it is not that minus is able to produce a plus as the rule appears to say, but that it is natural that, when too much has been taken away, one puts back the too much that has been taken away.” (Bernard Lamy, 1692)

„Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary. Hence they talk of two roots to every equation of the second order, and the learner is to try which will succeed in a given equation: they talk of solving an equation which requires two impossible roots to make it solvable: they can find out some impossible numbers, which, being multiplied together, produce unity. This is all jargon, at which common sense recoils; but, from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust, and hate the labour of a serious thought.“ (William Frend, “The Principles of Algebra”, 1796)

“I thought that mathematics ruled out all hypocrisy, and, in my youthful ingenuousness, I believed that the same must be true of all sciences which, I was told, used it. Imagine how I felt when I realized that no one could explain to me why minus times minus yields plus. […] That this difficulty was not explained to me was bad enough (it leads to truth and so must, undoubtedly, be explainable). What was worse was that it was explained to me by means of reasons that were obviously unclear to those who employed them.” (Stendhal, ”The Life of Henry Brulard”, 1835)

”There are elements of freedom in mathematics. We can decide in favor of one thing or another. Reference to the permanence principle (or another principle) is not a logical argument. We are free to opt for one or another. But we are not free when it comes to the consequences. We achieve harmony if we opt for a certain one (that minus times minus is plus). By making this choice we make the same choice as others in the past and present.” (Ernst Schuberth, “Minus mal Minus”, Forum Pädagogik, Vol. 2, 1988)

On Numbers: Large Numbers I

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"Further, it cannot escape anyone that for judging in this way about any event at all, it is not enough to use one or two trials, but rather a great number of trials is required. And sometimes the stupidest man - by some instinct of nature per se and by no previous instruction (this is truly amazing) - knows for sure that the more observations of this sort that are taken, the less the danger will be of straying from the mark." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"If thus all events through all eternity could be repeated, by which we would go from probability to certainty, one would find that everything in the world happens from definite causes and according to definite rules, and that we would be forced to assume amongst the most apparently fortuitous things a certain necessity, or, so to say, FATE." (Jacob Bernoulli, "The Art of Conjecturing", 1713)

"And thus in all cases it will be found, that although Chance produces Irregularities, still the odds will be infinitely great that in the process of time, those Irregularities will bear no proportion to the recurrency of that Order which naturally results from ORIGINAL DESIGN." (Abraham de Moivre, "The Doctrine of Chances", 1718)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

"Huge numbers are commonplace in our culture, but oddly enough the larger the number the less meaningful it seems to be." (Albert Sukoff, "Lotsa Hamburgers", Saturday Review of the Society, 1973)

"We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, [Nobel Prize Lecture], 1974)

"The logarithm is an extremely powerful and useful tool for graphical data presentation. One reason is that logarithms turn ratios into differences, and for many sets of data, it is natural to think in terms of ratios. […] Another reason for the power of logarithms is resolution. Data that are amounts or counts are often very skewed to the right; on graphs of such data, there are a few large values that take up most of the scale and the majority of the points are squashed into a small region of the scale with no resolution." (William S. Cleveland, "Graphical Methods for Data Presentation: Full Scale Breaks, Dot Charts, and Multibased Logging", The American Statistician Vol. 38 (4) 1984)

"The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t get our hand on or even begin to think about in any very definite way. So maybe all the action is really inaccessible and we’re just fiddling around. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." (Paul Hauffman, "The Man Who Loves Only Numbers", The Atlantic Magazine, Vol 260, No 5, 1987)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, "Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)

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5 Books 10 Quotes IV: On Complex Numbers IV

Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2007
	“A complex number is just a pair of real numbers, manipulated according to a short list of simple rules. Since a pair of real numbers is surely just as ‘real’ as a single real number, real and complex numbers are equally closely related to reality, and ‘imaginary’ is misleading.”
“The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence.”
David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002
	“Complex numbers are really not as complex as you might expect from their name, particularly if we think of them in terms of the underlying two dimensional geometry which they describe. Perhaps it would have been better to call them 'nature's numbers'. Behind complex numbers is a wonderful synthesis between two dimensional geometry and an elegant arithmetic in which every polynomial equation has a solution.”
“Ordinary numbers have immediate connection to the world around us; they are used to count and measure every sort of thing. Adding, subtracting, multiplying and dividing all have simple interpretations in terms of the objects being counted and measured. When we pass to complex numbers, though, the arithmetic takes on a life of its own. Since -1 has no square root, we decided to create a new number game which supplies the missing piece. By adding in just this one new element √-1. we created a whole new world in which everything arithmetical, miraculously, works out just fine.”
Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998
	“The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex.”
“When we try to take the square root of -1 (a real number), for example, we suddenly leave the real numbers, and so the reals are not complete with respect to the square root operation. We don’t have to be concerned that something like that will happen with the complex numbers, however, and we won’t have to invent even more exotic numbers (the ‘really complex’!) Complex numbers are everything there is in the two-dimensional plane.”
Jerry R Muir Jr., “Complex Analysis: A Modern First Course in Function Theory”, 2015
	“Complex analysis should never be underestimated as simply being calculus with complex numbers in place of real numbers and is distinguished from being so at every possible opportunity.”
“The upgrade from the real numbers to the complex numbers has both algebraic and analytic motivation. The real numbers are not algebraically complete, meaning there are polynomial equations such as x^2 = −1 with no solutions. The incorporation of  √-1 […] is a direct response to this.”
Tobias Dantzig, “Number: The Language of Science”, 1930
	“[…] extensions beyond the complex number domain are possible only at the expense of the principle of permanence. The complex number domain is the last frontier of this principle. Beyond this either the commutativity of the operations or the rôle which zero plays in arithmetic must be sacrificed.”
“And so it was that the complex number, which had its origin in a symbol for a fiction, ended by becoming an indispensable tool for the formulation of mathematical ideas, a powerful instrument for the solution of intricate problems, a means for tracing kinships between remote mathematical disciplines.” Previous Post <<||>> Next Post See also: More Quotes on Complex Numbers III More Quotes on Complex Numbers II More Quotes on Complex Numbers I Complex Numbers