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.

Second Book.

Complex Numbers and functions of Complex Numbers.
First Chapter.
The complex number and the Operations of Arithmetic.
63. In a manner similar to that in which Subtraction, Division
and Evolution each required the conception of number to be extended
so as to include respectively, negative, fractional and irrational numbers,
the attempt to render all the seven operations of Arithmetic possible
on real numbers without any exception, requires the adoption into
analysis of a new conception, the complex number.
Regarding roots of positive quantities we have the propositions:
1. The root of a number is equal to the product of the roots of
its factors.
2. If the exponent of the root be a product inn, the root can
be reduced by taking the mth root of the nth root of the quantity or
inversely.
These propositions being extended to the even roots of negative
quantities, it will appear that the problem of evolution is solved in
all cases, as soon as we adopt the square root of negative unity into
the numerical system and define the arithmetical operations with it;
for we have
a a a a, a r Ga, bi d-.
I/-1 is called the imaginary unit and, after Gauss, briefly denoted by + i. In like manner as real positive and negative numbers arise
from - 1 by multiplication, division and involution, so positive and
negative imaginary numbers are obtained from 4- i:
+ (i + i) -== + 2i, + (2i +-i) =-3- 3i,. + (ai + i) - + (a + 1)i,
i - i = 0 O i = 0.
The most general imaginary number is: + cai, where a signifies an
arbitrary real number rational or irrational.*)
*) It being already known that all quadratic equations could not be solved by
means of real quantities, the introduction of imaginary numbers became unavoidable, when it was found in the irreducible case of solving cubic equations