On Complex Numbers XI

"Certain authors who seem to have perceived the weakness of this method assume virtually as an axiom that an equation has indeed roots, if not possible ones, then impossible roots. What they want to be understood under possible and impossible quantities, does not seem to be set forth sufficiently clearly at all. If possible quantities are to denote the same as real quantities, impossible ones the same as imaginaries: then that axiom can on no account be admitted but needs a proof necessarily." (Carl F Gauss, "New proof of the theorem that every algebraic rational integral function in one variable can be resolved into real factors of the first or the second degree", 1799)

"[…] 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)

"By use of the symbol √-1 and of the forms proved to obtain in the combination of real quantities, a mode of notation is obtained, by which we may express sines and cosines, relatively to their arc." (Robert Woodhouse,"On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"The operations performed with imaginary characters, though destitute of meaning themselves, are yet notes of reference to others which are significant. They, point out indirectly a method of demonstrating a certain property of the hyperbola, and then leave us to conclude from analogy, that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its conclusion may, with all the strictness of mathematical reasoning, be proved of the hyperbola; but if from thence we would transfer that conclusion to the circle, it must be in consequence of the principle just now mentioned. The investigation therefore resolves itself ultimately into an argument from analogy; and, after the strictest examination, will be found without any other claim to the evidence of demonstration." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"√-1 is therefore not the sign of an arithmetic operation, nor of an arithmetic-geometric operation, but of a purely geometric operation. It is a sign of perpendicularity." (Adrien Q Buée, "Memoire sur les Quantités Imaginaires", Philosophical Transactions of the Royal Society, 1806)

"[…] 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)

"And when the idea of number was further extended so as to include 'complex' numbers, i.e. numbers involving the square root of −1, it was thought that real numbers could be regarded as those among complex numbers in which the imaginary part (i.e. the part which was a multiple of the square root of −1) was zero. All these suppositions were erroneous, and must be discarded, as we shall find, if correct definitions are to be given."(Bertrand Russell," Introduction to Mathematical Philosophy", 1919)

"Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A 'complex' number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number." (Bertrand Russell," Introduction to Mathematical Philosophy", 1919)