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.

114 The complex number and the operations of arithmetic. Bk. II. ch. I.
on the circle with radius r. All numbers with the same amplitude
belong to points on a right line proceeding from the origin.
We found for calculating cos qp and sin g the convergent series:
cos - 1 + -- 1 + *- *.. Lim B - O,
sin + 9) -- --. -...R' Lim = -O.
If we take their sum, having multiplied the latter by i, we get:
(1_+- +p 4 ) + (P- + -5 +*-+ P )
Employing the properties of the powers of i, this assumes the form:
1 + i ~ (j 12 + 3 + - + 1 + +
The remainder of this series, a complex number, converges to zero
for all values of (p; accordingly the convergent series:
1 J_ (i)3 +(i) (im))
+l+-i+- + + (i+-3 (i- + - +. iininf.
expresses the complex quantity: cos (p + i sin lp as approximately as
can be desired. But this series is got from the exponential series
found in ~ 42, by writing icp for x; accordingly*) we denote it by
the symbol eip, and obtain in this notation the theorem:
Every complex number can be written in the form r eiP, vwhere e'ip
stands for the infinite series just defined.
In consequence of this definition we have:
ei= i, ei = — 1, et -ei, ez2t =1. e~i2k 1
68. Complex numbers form a group complete in themselves; that
is, every operation of arithmetic when applied to complex numbers
presents without exception a result which can be expressed by
a complex number. Before this can be shown, we must define what is
now to be understood by the operations of arithmetic; the definitions
must be framed so as to embrace those already given for real
numbers.
69. Sum and difference (~ 65):
(a + ib) + (a'+ ib') = (a a') + i (b + ' ),
or r (cos p -{ i sin 9q) + r'(cos qp'+ i sin p ')
= (r cos p + r'cos T') + i (r sin P -,- r'siln ').
In this second form we prove most easily the theorem: The modulus of
*) Euler: Introductio, I. Cap. VIII. These equations establish the connexion
stated in ~ 14 to exist between exponential and trigonometrical functions:
cos + i Sin 9= ei(, cos - isin cp e —i'P, cos C- os (e- + e-p), sinl = (ei(-e-).