The History of Complex Numbers in Quotes:
"[...] as in the nature of things a negative (quantity) is not a square (quantity), it has therefore no square root." (Mahavira, cca. 850 AD)
"The square of an affirmative or of a negative is affirmative; and the square root of an affirmative quantity is two-fold: positive and negative. There is no square root of a negative quantity; for it is not a square." (Bhascara, 12th century)
"We do not perceive any quantity such as that its square is negative!" (Bhāskara II, "Bijaganita", 12th century)
"A second type of the false position makes use of roots of negative numbers. I will give an example: If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion. This, however, is closest to the quantity which is truly imaginary since operations may not be performed with it as with a pure negative number, nor as in other numbers. [...] This subtlety results from arithmetic of which this final point is, as I have said, as subtle as it is useless." (Girolamo Cardano, "Ars Magna", 1545)
"Someone could also ask what these impossible solutions are. I would answer that they are good for three things: for the certainty of the general rule, for being sure that there are no other solutions, and for its utility." (Albert Girard, "L'Invention nouvelle de l'Algébre", 1629)
"Thus we can give three names to the other solutions, seeing that there are some which are greater than nothing, other less than nothing, and other enveloped, as those which have like √- or √-3 or other similar numbers." (Albert Girard, "L'Invention nouvelle de l'Algébre", 1629)
"[…] neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3 - 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two [2+i and 2-i]], however, increased, diminished, or multiplied them as we just laid down, remain always imaginary." (René Descartes, "Gemetry", 1637)
"We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible." (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)
"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)
"And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
"But it is just that the Roots of Equation should be impossible, lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
"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)
"The nature, mother of the eternal diversities, or the divine spirit, are zaelous of her variety by accepting one and only one pattern for all things, By these reasons she has invented this elegant and admirable proceeding. This wonder of Analysis, prodigy of the universe of ideas, a kind of hermaphrodite between existence and non-existence, which we have named imaginary root?" (Gottfried W Leibniz, "De Bisectione Latereum", 1675)
"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, "Treatise of Algebra", 1685)
"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)
"From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised." (Gottfried W Leibniz, "Specimen novum analyses pro Scientia infinity circa summas et quadraturas", 1700)
"[…] even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts (notions ideales) which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra, for example, √-2." (Gottfried W Leibniz, [letter to Varignon], 1702)
"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)
"Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being." (Gottfried Leibniz, 1702)
"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)
"Even zero and complex numbers are not excluded from the signification of a variable quantity." (Leonhard Euler, "Introductio in Analysin Infinitorum" Vol. I, 1748)
"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)
"Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination." (Leonhard Euler, "Vollständige Anleitung zur Algebra", 1768-69)
"All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible." (Leonhard Euler, "Algebra", 1770)
"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)
"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." (Carl Friedrich Gauss)
"In the following I shall denote the expression √-1 by the letter i so that i*i=-1." (Leohnard Euler, "De formulis differentialibus angularibus" Vol. IV, 1794)
"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)
"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)
"[…] although the symbol √-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, and deserve, if any operations do, the name of reasoning." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The application of imaginary quantities to the theory of equations, has perhaps been made more extensively than to any other part of analysis. To consider the propriety of this application on the grounds of perspicuity and conciseness, a long discussion would be necessary. I may, however, be here permited merely to state my opinion, that impossible quantities must be employed in the theory of equations, in order to obtain general rules and compendious methods." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 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)
"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages."(Carl F Gauss, [letter to Bessel] 1811)
"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)
"The true meaning of √-1 reveals itself vividly before my soul, but it will be very difficult to express it in words, which can give only an image suspended in the air." (Carl F Gauss, [Letter to Peter Hanson] 1825)
"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)
"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)
"Imaginary quantities could be supposed to have an underlying object just as well […] as negative ones could" (Carl F Gauss, "Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)
"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)
"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", Göttingische gelehrte Anzeigen 23 (4), 1831)
"[geometrical representation of complex numbers] completely established the intuitive meaning of complex numbers, and more is not needed to admit these quantities into the domain of arithmetic." (Carl F Gauss, 1831)
"We have shown the symbol √-a to be void of meaning, or rather self-contradictory and absurd." (Augustus De Morgan, 1831)
"There seems to me to be something analogous to polarized intensity in the pure imaginary part; and to unpolarized energy (indifferent to direction) in the real part of a quaternion: and thus we have some slight glimpse of a future Calculus of Polarities. This is certainly very vague […]" (Sir William R Hamilton, "On Quaternions; or on a new System of Imaginaries in Algebra", 1844)
"Those who can, in common algebra, find a square root of -1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √-1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)
"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 Funktionen 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)
"Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics." (Bernhard Riemann, 1854)
"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)
"The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments." (Thomas Hill, "The Imagination in Mathematics", North American Review Vol. 85, 1857)
"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")
"[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)
"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)
"When we consider that the whole of geometry rests ultimately on axioms which derive their validity from the nature of our intuitive faculty, we seem well justified in questioning the sense of imaginary forms, since we attribute to them properties which not infrequently contradict all our intuitions." (Gottlob Frege, "On a Geometrical Representation of Imaginary forms in the Plane", 1873)
"[…] with few exceptions all the operations and concepts that occur in the case of real numbers can indeed be carried over unchanged to complex ones. However, the concept of being greater cannot very well be applied to complex numbers. In the case of integration, too, there appear differences which rest on the multplicity of possible paths of integration when we are dealing with complex variables. Nevertheless, the large extent to which imaginary forms conform to the same laws as real ones justifies the introduction of imaginary forms into geometry." (Gottlob Frege, "On a Geometrical Representation of Imaginary forms in the Plane", 1873)
"When we consider complex numbers and their geometrical representation, we leave the field of the original concept of quantity, as contained especially in the quantities of Euclidean geometry: its lines, surfaces and volumes. According to the old conception, length appears as something material which fills the straight line between its end points and at the same time prevents another thing from penetrating into its space by its rigidity. In adding quantities, we are therefore forced to place one quantity against another. Something similar holds for surfaces and solid contents. The introduction of negative quantities made a dent in this conception, and imaginary quantities made it completely impossible. Now all that matters is the point of origin and the end point; whether there is a continuous line between them, and if so which, appears to make no difference whatsoever; the idea of filling space has been completely lost. All that has remained is certain general properties of addition, which now emerge as the essential characteristic marks of quantity. The concept has thus gradually freed itself from intuition and made itself independent. This is quite unobjectionable, especially since its earlier intuitive character was at bottom mere appearance. Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition." (Gottlob Frege, "Methods of Calculation based on an Extension of the Concept of Quantity", 1874)
"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)
"[…] it is not immaterial to the cogency of our proof whether 'a + bi' has a sense or is nothing more than printer's ink. It will not get us anywhere simply to require that it have a sense, or to say that it is to have the sense of the sum of a and bi, when we have not previously defined what 'sum' means in this case and when we have given no justification for the use of the definite article." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"How are complex numbers to be given to us then […]? If we turn for assistance to intuition, we import something foreign into arithmetic; but if we only define the concept of such a number by giving its characteristics, if we simply require the number to have certain properties, then there is no guarantee that anything falls under the concept and answers to our requirements, and yet it is precisely on this that proofs must be based." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"Nothing prevents us from using the concept 'square root of-1'; but we are not entitled to put the definite article in front of it without more ado and take the expression 'the square root of -' as having a sense." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"What is commonly called the geometrical representation of complex numbers has at least this advantage […] that in it 1 and i do not appear as wholly unconnected and different in kind: the segment taken to represent i stands in a regular relation to the segment which represents 1. […] A complex number, on this interpretation, shows how the segment taken as its representation is reached, starting from a given segment (the unit segment), by means of operations of multiplication, division, and rotation." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"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)
"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)
"It is generally true, that wherever an imaginary expression occurs the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)
"If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -1 does not exist mathematically." (David Hilbert, [address to the International Congress of Mathematicians], 1900)
"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity." (Bertrand Russell, "Introduction to Mathematical Philosophy", 1919)
"[…] 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." (Tobias Dantzig, "Number: The Language of Science", 1930)
"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." (Tobias Dantzig, "Number: The Language of Science", 1930)
"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, "The Two Realities", 1930)
"[…] imaginary numbers made their own way into arithmetical calculation without the approval, and even against the desires of individual mathematicians, and obtained wider circulation only gradually and the extent to which they showed themselves useful." (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1945)
"The rules of algebra show that the square of any number, whether positive or negative, is a positive number: therefore, to speak of the square root of a negative number is mere absurdity. Now, Cardan deliberately commits that absurdity and begins to calculate on such 'imaginary' quantities.
One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament - which, of course, was, in the nineteenth century, established on solid and rigorous bases. 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)
"The sweeping development of mathematics during the last two centuries is due in large part to the introduction of complex numbers; paradoxically, this is based on the seemingly absurd notion that there are numbers whose squares are negative." (Emile Borel, 1952)
"[…] to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of quantum mechanics." (Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)
"Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Communications in Pure and Applied Mathematics 13 (1), 1960)
"It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex." (Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics" , Journal of Mathematical Physics Vol. 3, 1962)
"The origin and immediate purpose of the introduction of complex magnitudes into mathematics lie in the theory of simple laws of dependence between variable magnitudes expressed by means of operations on magnitudes. If we enlarge the scope of applications of these laws by assigning to the variables they involve complex values, then there appears an otherwise hidden harmony and regularity." (Heinz-Dieter Ebbinghaus et al., "Numbers", 1983)
"What could be more beautiful than a deep, satisfying relation between whole numbers. How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers." (Manfred Schroeder, "Number Theory in Science and Communication", 1984)
"The attitudes of mathematicians can be found not only in what they wrote, but in what they did not write. It is possible to divide mathematicians into those who gave complex numbers some kind of coverage, and those who sometimes or always ignored them." (Diana Willment, "Complex Numbers from 1600 to 1840" [Masters thesis], 1985)
"The lack of a visual representation for √-1 had a profound influence on attitudes to it, and complex numbers were not widely accented until after the invention of the Argand diagram." (Diana Willment, "Complex Numbers from 1600 to 1840" [Masters thesis], 1985)
"The square roots of negative numbers! If negative numbers were false, absurd or fictitious, it is hardly to be wondered at that their square roots were described as 'imaginary'." (David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", 1986)
"The original purpose and immediate objective in introducing complex numbers into mathematics is to express laws of dependence between variables by simpler operations on the quantities involved. If one applies these laws of dependence in an extended context, by giving the variables to which they relate complex values, there emerges a regularity and harmony which would otherwise have remained concealed." (Heinz-Dieter Ebbinghaus et al, "Numbers", 1990)
"The letter ‘i’ originally was meant to suggest the imaginary nature of this number, but with the greater abstraction of mathematics, it came to be realized that it was no more imaginary than many other mathematical constructs. True, it is not suitable for measuring quantities, but it obeys the same laws of arithmetic as do the real numbers, and, surprisingly enough, it makes the statement of various physical laws very natural." (John A Paulos, "Beyond Numeracy", 1991)
"The number ‘i’ is evidence that much real progress can result from the positing of imaginary entities. Theologians who have built elaborate systems on much flimsier analogies should perhaps take heart." (John A Paulos, "Beyond Numeracy", 1991)
"[…] the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood." (Harold S M Coxeter, "The Real Projective Plane" 3rd Ed, 1993)
"The dictum that everything that people do is 'cultural' licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. [...] It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is!" (Norman Levitt, "The Flight from Science and Reason", Science, 1996)
"At this stage you might be thinking that there is no justification for calling something of the form a+bi a number, even if you are prepared to countenance i = √-1 in the first place. But remember, it is not what numbers are that matters, but how they behave. Provided the complex numbers have a workable and useful (either in mathematics itself or possibly in a wider context) arithmetic, possibly forming a field, then they have as much right to be called 'numbers' as do any others." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 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." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)
"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"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." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)
"√-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 […]"
"[…] 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."
"[…] because imaginary time is at right angles to real time, it behaves like a fourth spatial direction. It can therefore have a much richer range of possibilities than the railroad track of ordinary real time, which can only have a beginning or an end or go around in circles. It is in this imaginary sense that time has a shape." (Stephen W Hawking, "The Universe in a Nutshell", 2001)
"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?" (Stephen W Hawking, "The Universe in a Nutshell", 2001)
"To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. […] Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don't need a physical realization […]" (Stephen W Hawking, "The Universe in a Nutshell", 2001)
"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." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)
"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." (David Mumford, Caroline Series & David Wright, "Indra’s Pearls: The Vision of Felix Klein", 2002)
"If we think of square roots in the geometric manner, as we have just done, to ask for the square root of a negative quantity is like asking: ‘What is the length of the side of a square whose area is less than zero?’ This has more the ring of a Zen koan than of a question amenable to a quantitative answer." (Barry Mazur, "Imagining Numbers", 2003)
"How is it that -1 can have a square root? The square of a positive number is always positive, and the square of a negative number is again positive (and the square of 0 is just 0 again, so that is hardly of use to us here). It seems impossible that we can find a number whose square is actually negative." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)
"When we get used to playing with these complex numbers, we cease to think of a + ib as a pair of things, namely the two real numbers a and b, but we think of a+ib as an entire thing on its own, and we could use a single letter, say:, to denote the whole complex number z = a+ib. It may be checked that all the normal rules of algebra are satisfied by complex numbers. In fact, all this is a good deal more straightforward than checking everything for real numbers. […] From this point of view, it seems rather extraordinary that complex numbers were viewed with suspicion for so long, whereas the much more complicated extension from the nationals to the reals had, after ancient Greek times, been generally accepted without question." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)
"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love."
"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)
"To have the courage to think outside the square, we need to be intrigued by a problem. This intrigue will encourage us to use our imaginations to find solutions which are beyond our current view of the world. This was the challenge that faced mathematicians as they searched for a solution to the problem of finding meaning for the square root of a negative number, in particular √-1." (Les Evans, "Complex Numbers and Vectors", 2006)
"Unfortunately, if we were to use geometry to explore the concept of the square root of a negative number, we would be setting a boundary to our imagination that would be difficult to cross. To represent -1 using geometry would require us to draw a square with each side length being less than zero. To be asked to draw a square with side length less than zero sounds similar to the Zen Buddhists asking ‘What is the sound of one hand clapping?’" (Les Evans, "Complex Numbers and Vectors", 2006)
"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." (Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2007)
"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." (Ian Stewart, "Why Beauty Is Truth: The History of Symmetry", 2007)
"How are we to explain the contrast between the matter-of-fact way in which √-1 and other imaginary numbers are accepted today and the great difficulty they posed for learned mathematicians when they first appeared on the scene? One possibility is that mathematical intuitions have evolved over the centuries and people are generally more willing to see mathematics as a matter of manipulating symbols according to rules and are less insistent on interpreting all symbols as representative of one or another aspect of physical reality. Another, less self-congratulatory possibility is that most of us are content to follow the computational rules we are taught and do not give a lot of thought to rationales." (Raymond S Nickerson, "Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs", 2009)
"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C. Baez, "Division Algebras and Quantum Theory", 2011)
"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." (Jerry R Muir Jr., "Complex Analysis: A Modern First Course in Function Theory", 2015)
"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." (Jerry R Muir Jr., "Complex Analysis: A Modern First Course in Function Theory", 2015)
"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."
"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)
"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)
"The existing definition of imaginary numbers is solely based on the fact that certain mathematical operation, square operation, would not yield certain type of outcome, negative numbers; hence such operational outcome could only be imagined to exist. Although complex numbers actually form the largest set of numbers, it appears that almost no thought has been given until now into the full extent of all possible types of imaginary numbers." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)
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