An introduction to the modern theory of equations, by Florian Cajori.

SOME ELEMENTARY PROPERTIES OF EQUATIONS 23

22. Graphic Representation of Complex Numbers. In the construction of graphs of polynomials y=f(x) we assumed a
horizontal and a vertical axis, and from this point of intersection measured off values of x parallel to the horizontal axis
and values of y parallel to the p
vertical axis. A similar plan is
commonly adopted for the repre-/
sentation of complex numbers/
or imaginaries. If z= x iy, /
where x and y are real numbers, /0 _
either + or -, rational or irra- Q
tional, then x and y are laid off
parallel to the horizontal and
vertical axis, respectively. If
x = OQ, y = QP, then z is represented in magnitude and direction by OP. The length of OP
is called the modulus of z, and is equal to Vx 2+ y2.* The
direction of z is indicated by the angle 0, which is called the
amplitude or argument of z.
Since x = p cos 0, y = p sin 0, we have
z = x + iy = p(cos 0 + i sin 0).
This graphic representation of complex numbers is due to
Caspar Wessel (1797).
23. Addition and Subtraction of Complex Numbers. Let OP
=a + ib and OP' =' + ib', then, OP+ OP' =(a+a') +i(b+b').
Draw P'S parallel and equal to OP, then OT=a+a', TS=b+b',
and OS= OP+ OP'.
* This graphic representation is of great help to the mathematician. But
attention should be called to the fact that the statement, that to every irrational number there corresponds a line of definite length, is no longer considered self-evident nor demonstrable; it involves the geometric postulate:
"If all points of the line fall into two classes in such a manner that each
point of the first class lies to the left of each point of the second class, then
there exists one point, and only one, which brings about this separation."
See the Encyklopadie d. Math. Wiss., I A 3, No. 4.