Useful Identities Among Complex Numbers
|(1)||Re(z + w) = Re(z) + Re(w) and Im(z + w) = Im(z) + Im(w).|
|z + w||= (Re(z) + i·Im(z)) + (Re(w) + i·Im(w))|
|= (Re(z) + (Re(w)) + i·(Im(z) + Im(w)),|
which is exactly what (1) claims.
The two functions Re(z) and Im(z) are in fact linear:
Re(tz + sw) = tRe(z) + sRe(w),|
Im(tz + sw) = tIm(z) + sIm(w),
for any real t and s. In particular,
Re(z - w) = Re(z) - Re(w),
Im(z - w) = Im(z) - Im(w).
It follows from (2) that the z' is also linear, i.e., for any real t and s, we have
(tz + sw)' = tz' + sw'.
As every linear function, Re(z), Im(z) and z' are continuous. For example,
||z' - w'|||= |(z - w)'||
|= |z - w|,|
such that if z and w are close, so are z' and w'.
Re(zw) = Re(z)Re(w) - Im(z)Im(w),
Im(zw) = Re(z)Im(w) + Im(z)Re(w).
|(3)||(zw)' = z'w'.|
Since z' = Re(z) - i·Im(z),
Re(z) = (z + z') / 2 and|
Im(z) = (z - z') / 2i.
For any four complex numbers u, v, w, z, the following non-trivial identity can be verified by direct inspection:
|(5)||(u - v)(w - z) + (u - z)(v - w) = (u - w)(v - z).|
|(6)|||u - v|·|w - z| + |u - z|·|v - w| ≥ |u - w|·|v - z|.|
The equality here is achieved only if
UV·WZ + UZ·VW = UW·VZ,
where according to our stipulations, UV, VW, WZ, and UZ are the sides and UV and VZ the diagonals of a convex cyclic quadrilateral UVWZ. We owe the more general assertion (6) to a discovery of L. Euler. However, the inequality (6) expressed in geometric terms
UV·WZ + UZ·VW ≥ UW·VZ,
is known as Ptolemy's inequality. Ptolemy's identity implies cyclicity of a quadrilateral and is thus a necessary and sufficient condition for a quadrilateral to be cyclic.
Euler's theorem (6) led (1936) the Romanian mathematician D. Pompeiu to a generalization of van Schooten's theorem, about 300 years after the latter's discovery. If ΔUVZ is equilateral, then for any W, not on the circumcircle of ΔUVZ, there exists a triangle with side lengths UW, VW, ZW. Indeed, (6) implies that
|(7)||UV·WZ + UZ·VW ≥ UV·VZ,|
with the equality only for W on the circumcircle of ΔUVZ. Since the triangle is equilateral,
WZ + VW ≥ VZ.
- T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
- C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
- Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
- E. Landau, Foundations of Analysis, Chelsea Publ, 3rd edition, 1966
Copyright © 1996-2018 Alexander Bogomolny