“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)
"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)
"It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretched out his arms for others." (Carl F Gauss, [Letter to Farkas Bolyai] 1808)
"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)
"The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers. […] Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae” [“Arithmetical Researches”], 1801)
"It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretched out his arms for others." (Carl F Gauss, [Letter to Farkas Bolyai] 1808)
"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)
“It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations.” (Carl F Gauss, 1817)
"We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori." (Carl F Gauss, 1830)
"[…] if number is merely the product of our mind, space has a reality outside our mind whose laws we cannot a priori completely prescribe" (Carl F Gauss, 1830)
"[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)
“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)
"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)
"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)
"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible." (Carl F Gauss)
“In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction.” (Carl F Gauss)
"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 F Gauss)
"We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori." (Carl F Gauss, 1830)
"[…] if number is merely the product of our mind, space has a reality outside our mind whose laws we cannot a priori completely prescribe" (Carl F Gauss, 1830)
"[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)
“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)
"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)
"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)
"I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible." (Carl F Gauss)
“In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction.” (Carl F Gauss)
"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 F Gauss)
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