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.

112

The complex number and the operations of arithmetic. Bk. II. ch. 1.

The combination of real and imaginary numbers in a sum, furnishes
the complex number. The complex number a + ib is a sum of a
positive or negative real units and of b positive or negative imaginary
units. Therefore the conception of the complex number comprehends
that of the real, when b = 0, and that of the imaginary, when a = 0.
To add two complex numbers: a + ib and a' + i b' is to carry
on this enumeration of the two units, that is: to form a new number,
of which the real units equal the sum of the real, and the imaginary equal the sum of the imaginary units in the two given numbers:
(a + ib) + (a' + ib') = (a + a') + i (b + b')
and in general
(a + ib) + (a'+ i b') +-...( ) (+a t -a' -.. a) -i (b+-b'... ).
Such addition complies with the fundamental proposition that the
summands may be interchanged.
Subtraction here also means the inverse of addition.
66. In like manner as real numbers can all be brought under
intuition by terminal points of lengths, measured off for simplicity
on a right line in the positive and negative sense from a certain
origin, so complex numbers are all figured as points of a surface,
for simplicity, of a plane.*) For, in the rectangular system of Cartesian coordinates (~ 15) each point of the plane is uniquely determined by two numbers x and y positive or negative, one of them x
being laid off as abscissa, the other y as ordinate, in a direction determined by the sign. If we combine the real numerical values x and y
in the complex number x + iy, we arrive at the proposition:
A system of coordinates and a unit measure being once adopted,
each point in the plane has its own complex number, and inversely
each complex number determines uniquely a point of the plane.
The points of the axis of abscisse belong to the real, those of the
axis of ordinates to the imaginary numbers.
The totality of points in the plane gives an intuition of the total
continuous domain of complex numbers.
67. This representation conducts at once to a new and very convenient form which complex numbers admit of. For if we introduce
*) Argand: Essai sur une maniere de representer les quantit6s imaginaires
dans les constructions geometriques, published anonymously 1806; extracts from
this appeared in 1813 in Gergonne's Annales vol. IV, and it was republished separately by Houel in 1874. As to the history of complex numbers, see Drobisch:
Berichte fiber die Verhandlungen der K. Sachs. Gesellsch. d. Wissensch., Vol. II;
Hankel: Theorie der complexen Zahlensysteme, pp. 71 and 81; Houiel: Th6orie le6 -mentaire des quantites complexes, p. 4; the importance of Gauss' work is left
somewhat too much in the background in this last treatise. The oldest notice on
the geometric interpretation is in the Novi Comm. Acad. Petrop. 1750.