An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 63-65. Real and imaginary differ in conception not in application.

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64. The contrasted epithets "real" and "imaginary" favor the
erroneous impression, which indeed has impeded the systematic introduction of imaginary numbers into analysis, that numbers of the first
kind possess a practical reality which those of the second have not.
Considering the arithmetical operations merely, without application
to physical quantities, fractions, irrational numbers, imaginaries, all
form legitimate extensions of the conception of number, that are
connected with the integer by determinate arithmetical operations.
In the applications of these operations on the other hand everything depends on the kind of numbers introduced at the outset in
framing a problem analytically. If ex. gr. in the case of discrete
quantities, from the way in which the problem is proposed only
integers are admissible, the proposed problem is seen to be impossible when the result is a fraction. Likewise in the result of
a calculation referring to physical quantities, a negative number will
have a meaning applicable to these quantities, only when from the
first the quantities were distinguished in the sense of positive and
negative. In analogy with this, even when the result of calculation
is imaginary, its meaning is no longer unreal, when the actual
quantities considered, are characterised not only by real, but also by
imaginary numbers. The simplest example of a representation of intuitive
quantities by imaginary numbers is the geometric interpretation, which
we shall deal with as we go on. "As mathematical science strives towards
doing away with exceptions to rules and towards contemplating different
propositions from one point of view, it is often compelled to enlarge its
conceptions or to establish new ones, and this nearly always denotes
a progress in the science. A great example of this is the introduction
of imaginary quantities into analysis." (v. Staudt, Beitrige zur Geometrie der Lage. Heft I. Vorwort.)
65. It follows from the definition of the imaginary unit that:
2 i. i = (/- 1)2 1.
Accordingly we understand by:
i- = i = -- i. i -- (i2) 1.
Hence follows by inversion:
1-. 1 1 -
i 7 i2? i3 7 4 -2ti) 7 e 4
that a real solution could be expressed only by means of square roots of negative quantities (Bombelli 1579). Since that time imaginary numbers have never
disappeared from analysis. The earliest who found their employment fruitful
was Euler. But it was the works of Gauss (1777-1855) and of Cauchy which first
manifested the importance of the complex number as a generalised conception
of number.