On Complex Numbers XII

"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)

"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)

"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)

"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)

"The symbol √-1 might arise from translating questions of which the statement involved a contradiction of ideas into algebraic language, and reasoning on them, as if they really admitted a solution." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"We have shown the symbol √-a to be void of meaning, or rather self-contradictory and absurd." (Augustus De Morgan, 1831)

"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)

"Now as far as the arithmetical signs for addition, multiplication, etc. are concerned, I believe we shall have to take the domain of common complex numbers as our basis; for after including these complex numbers we reach the natural end of the domain of numbers." (Gottlob Frege, [letter to Peano])