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.

~ 65-67.

Geometrical interpretation.

113

instead of the system of Cartesian coordinates a polar system with
the same origin and its prime radius the axis of x, every point P of
the plane will be uniquely determined by its distance r from the origin:O~~~yr ~and by the angle (p taken in a determinate direction of rotation which
p the radius vector O P makes with
the axis of x. The length r is
A/^^r~ |throughout taken only in an abso/^" |lute sense, (p passes through all
^^ T \ values from 0 to 2; if 9p pass
a. xs beyond 2 r, previous points reFig. 5. peat themselves. Now x = r cos (p,
y = r sin (p; and conversely we obtain for each pair of values x and
y without ambiguity the values:
r += + 1/2 +- y, cos p 1/, sin == g / y2
These last two equations determine a value of (p without any
ambiguity except what arises from additional integer multiples of 2z.
Our present knowledge enables us to calculate it from the formula
tan cp -, whence Cp tan-' Y. But since the series for tan-' Y
(~ 48) always represents an angle between -- C and + i, we
must put: for x > 0, y > 0 p tan-'1 -
x
forx<0, y > 0 p= + tan-1 y,
for x < 0, y < 0 p — t + tan-1,
X X
for x > O < O 'p=2 + tan-', or, tan-l Y
Every complex number a + ib can therefore be written in the form
r(cos p + -i sin 'p) or more briefly re*). After Argand (1814) we
call the quantity r = + /a -I- b+ that represents the absolute magnitude of'a + ib the modulus, or (after Weierstrass, Journal f. M.
vol. 52) the absolute amount; the quantity (p (after Cauchy,
who also called it the argument) we call the amplitude of the
complex number. Since cos (p and sin (p do not vanish together, a
complex number only vanishes when its modulus r = /a2 + b-2 - 0.
This involves both a - 0 and b = 0. All complex numbers with the
same modulus r are figured by points equally distant from the origin
*) This representation of the complex number is found in Euler: Introductio, I.
Cap. VIII; as a general representation of all complex numbers it underlies Gauss'
first Memoir (1799).
IARNACK, Calculus. 8