On Complex Numbers VI

"We do not perceive any quantity such as that its square is negative!" (Bhāskara II, "Bijaganita", 12th century)

"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)

“Even zero and complex numbers are not excluded from the signification of a variable quantity.” (Leonhard Euler, “Introductio in Analysin Infinitorum” Vol. I, 1748)

“Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers” (Carl F Gauss, “Disquisitiones arithmeticae” [“Arithmetical Researches”], 1801)

"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)

“At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications. “ (Carl F Gauss, [letter to Bessel,] 1811)

"Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative .and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to, negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities - formerly and occasionally now, though improperly, called impossible-as opposed to real quantities are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities." (Carl F Gauss, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831)

“The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting.” (Bernhard Riemann, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse”, 1851)

“The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention […] And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning [...] It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression.” (William R Hamilton, “Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method… “, 1853) 

“The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.” (Bernhard Riemann, “Gesammelte Mathematische Werke”)

On Numbers: Zero

"When sunya [zero] is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by sunya becomes sunya." (Brahmagupta, 628)

"Every number arises from One, and this in turn from the Zero. In this lies a great and sacred mystery - in hoc magnum latet sacramentum: HE is symbolized by that which has neither beginning nor end; and just as the zero neither increases nor diminishes another number to which it is added or from which it is subtracted so does HE neither wax nor wane. And as the zero multiplies by ten the number behind which it is placed so does HE increase not tenfold, but a thousand fold - nay, to speak more correctly, HE creates all out of nothing, preserves and rules it  - omnia ex nichillo creat, conservat atque gubernat." ("Salem Codex", 12th century)

"The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing." (Lorenz Oken, "Elements of Physiophilosophy", 1847)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

"The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"In the history of culture the discovery of zero will always stand out as one of the greatest achievements of the human race." (Tobias Danzig, "Number: The Language of Science", 1930)

"The zero is the most important digit. It is a stroke of genius, to make something out of noting by giving it a name and inventing a symbol for it." (B L van der Waerden, "Science Awakening", 1962)

"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)

"[zero is] A mysterious number, which started life as a space on a counting board, turned into a written notice that a space was present, that is to say that something was absent, then confused medieval mathematicians who could not decide whether it was really a number or not, and achieved its highest status in modern abstract mathematics in which numbers are defined anyway only by their properties, and the properties of zero are at least as clear, and rather more substantial, than those of many other numbers." (David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", 1986)

"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and the framework of logic - and threatens to undermine the very basis of science. […] The universe begins and ends with zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"Mathematics is an activity about activity. It hasn't ended - has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)

"Zero was at the heart of the battle between East and West. Zero was at the center of the struggle between religion and science. Zero became the language of nature and the most important tool in mathematics. And the most profound problems in physics - the dark core of a black hole and the brilliant flash of the big bang - are struggles to defeat zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"In one of the largest calculations done to date, it was checked that the first ten trillion of these zeros lie on the correct line. So there are ten trillion pieces of evidence indicating that the Riemann hypothesis is true and not a single piece of evidence indicating that it is false. A physicist might be overwhelmingly pleased with this much evidence in favour of the hypothesis, but to some mathematicians this is hardly evidence at all. However, it is interesting ancillary information." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"One of the current ideas regarding the Riemann hypothesis is that the zeros of the zeta function can be interpreted as eigenvalues of certain matrices. This line of thinking is attractive and is potentially a good way to attack the hypothesis, since it gives a possible connection to physical phenomena. [...] Empirical results indicate that the zeros of the Riemann zeta function are indeed distributed like the eigenvalues of certain matrix ensembles, in particular the Gaussian unitary ensemble. This suggests that random matrix theory might provide an avenue for the proof of the Riemann hypothesis." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"The concept of zero is so familiar that it takes a great deal of effort to recapture how mysterious, subtle, and contradictory the idea really is." (William Byers, "How Mathematicians Think", 2007)

"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)

"However, in contrast to one, which is singularly straightforward, zero is secretly peculiar. If you pierce the obscuring haze of familiarity around it, you’ll see that it is a quantitative entity that, curiously, is really the absence of quantity. It took people a long time to get their minds around that." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Zero is not a point of non-existence. Zero is always a balance point of existents. The human understanding of 'zero' must undergo the most radical of all transformations. Most people, especially scientists, associate it with absolute nothingness, with non-existence. This is absolutely untrue. Or, to put it another way, we can define it in two ways: 1) nothing as non-existence, in which case it has absolutely no consequences but leads to all manner of abstract paradoxes and contradictions, or 2) nothing as existence, in which case it is always a mathematical balance point for somethings. It is purely mathematical, not scientific, or religious, or spiritual, or emotional, or sensory, or mystical. It is analytic nothing and whenever you encounter it you have to establish the exact means by which it is maintaining its balance of zero." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Numbers: From Zero to Infinity

“A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.” (Bhaskara II, “Bijaganita”, 12th century)

 “The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H. Lewes, “Problems of Life and Mind”, 1873)

“One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite.” (Victor Hugo, “The Toilers of the Sea”, 1874)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

“Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

“I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities.” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

"Each act of creation could be symbolized as a particular product of infinity and zero. From each such product could emerge a particular entity of which the appropriate symbol was a particular number." (Srinivasa Ramanujan)

“If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things.” (Robert Kaplan, “The Nothing that Is: A Natural History of Zero”, 2000)

“Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity.” (Charles Seife, “Zero: The Biography of a Dangerous Idea”, 2000)

“Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line - at least as the line is usually drawn - making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one.” (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)

Previous Post <<||>> Next Post

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)