On Complex Numbers XVI

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."  (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

 "√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"[…] i is not a real number-not ordered anywhere relative to the real numbers! In other words, it does not even have the central property of ‘numbers’, indicating a magnitude that can be linearly compared to all other magnitudes. You can see why i has been called imaginary. It has almost none of the properties of the small natural numbers-not subitizability, not groupability, and not even relative magnitude. If i is to be a number, it is a number only by virtue of closure and the laws of arithmetic." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The complex plane is just the 90° rotation plane-the rotation plane with the structure imposed by the 90° Rotation metaphor added to it. Multiplication by i is 'just' rotation by 90°. This is not arbitrary; it makes sense. Multiplication by-1 is rotation by 180°. A rotation of 180° is the result of two 90° rotations. Since i times i is -1, it makes sense that multiplication by i should be a rotation by 90°, since two of them yield a rotation by 180°, which is multiplication by -1." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Negative numbers posed some of the same quandaries that the imaginary numbers did to Renaissance mathematicians - they didn’t seem to correspond to quantities associated with physical objects or geometrical figures. But they proved less conceptually challenging than the imaginaries. For instance, negative numbers can be thought of as monetary debts, providing a readily grasped way to make sense of them." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty 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)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The fact that multiplying positive 4i times positive 4i yields negative 16 seems like saying that the friend of my friend is my enemy. Which in turn suggests that bad things would happen if i and its offspring were granted citizenship in the number world. Unlike real numbers, which always feel friendly toward the friends of their friends, the i-things would plainly be subject to insane fits of jealousy, causing them to treat numbers that cozy up to their friends as threats. That might cause a general breakdown of numerical civility." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the 'physical' universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Complex Numbers IX

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"The employment of the uninterpretable symbol √-1 the intermediate processes of trigonometry furnishes an illustration of what has been said. I apprehend that there is no mode of explaining that application which does not covertly assume the very principle in question." (George Boole, "Laws of Thought", 1854)

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers." (Richard Dedekind, "Stetigkeit und irrationale Zahlen", 1872)

"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900) 

 "It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)
[French: "On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités du domaine réel passe souvent par le domaine imaginaire." (Jacques Hadamard, Essai sur la psychologie de l'invention dans le domaine mathématique, 1945)]

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)

"Complex numbers seem to be fundamental for the description of the world proposed by quantum mechanics. In principle, this can be a source of puzzlement: Why do we need such abstract entities to describe real things? One way to refute this bewilderment is to stress that what we can measure is essentially real, so complex numbers are not directly related to observable quantities. A more philosophical argument is to say that real numbers are no less abstract than complex ones, the actual question is why mathematics is so effective for the description of the physical world." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)

On Complex Numbers VII

“The remark which you make concerning roots that can not be extracted, and containing imaginary quantities which when added together give none the less a real quantity, is surprising and entirely new. One would never have believed that √(1 + √-3) + √(1 - √- 3) would make √6, and there is something hidden in this which is incomprehensible.” (Christaan Huygens, [letter to Gottfried W Leibniz] cca. 1670)

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

“For it ought to be considered that both –b   and –c  , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b)  and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination.” (Thomas Simpson, “A Treatise of Algebra”, 1745) 

“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential."  (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748)

“Moreover, the whole method has the essential disadvantage that it occupies the mind with the distinction of a great number of cases that can be recognized only by inner intuition, and thus neutralizes an important part of that which algebra is supposed to accomplish, which is relieving the power of inner intuition. Finally, in such a treatment algebra loses a great part of the generality that it can obtain by the mutual connection of different problems, which becomes evident so easily when one uses isolated negative quantities. [...] Since imaginary quantities have to occur, science would certainly not win that much by avoiding negative quantities than it would lose in terms of clarity and generality.” (Johann P W Stein,  “Die Elemente der Algebra: Erster Cursus”, 1828) 

"Originally assuming the concept of the absolute integers, it extended its domain step by step; integers were supplemented by fractions, rational numbers by irrational numbers, positive numbers by negative numbers, and real numbers by imaginary numbers. This advance, however, occurred initially with a fearfully hesitant step. The first algebraists preferred to call negative roots of equations false roots, and it is precisely these where the problem to which they refer was always termed in such a way as to ensure that the nature of the quantity sought did not admit any opposite.” (Carl F Gauss, “Theoria residuorum biquadraticum. Commentatio secunda. [Selbstanzeige]”, Göttingische gelehrte Anzeigen 23 (4), 1831)

“[T]he notion of a negative magnitude has become quite a familiar one […] But it is far otherwise with the notion which is really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole of modern analysis and geometry, that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry: I use in each case the word imaginary as including real. This has not been, so far as I am aware, a subject of philosophical discussion or inquiry. […] considering the prominent position which the notion occupies-say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it.” (Arthur Cayley, [address before the meeting of the British Association at Southport] 1870) 

“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed. [...] But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)

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)