Real and Complex Products of Complex Numbers
Complex numbers, as an ordered pair of real numbers, can be identified with either points or vectors in the plane. The Argand diagram, with its axes and the origin, explicitly associates complex numbers with points. The operations of addition and multiplication highlight the implicit vector link. This is especially obvious in the case of addition, which is illustrated graphically with the parallelogram rule.
Two uncommon borrowings from the vector algebra have been demonstrated in a recent book by T. Andreescu to provide powerful tools for solving complex number problems.
Real Product
The real product is a complex number incarnation of the scalar product (and this is what it is called in Gardiner, Bradley, p. 239]):
(1)  z.w = (z'w + zw')/2. 
It is important (although difficult) to distinguish between the symbols for the common symbol of multiplication
Assuming z = z_{1} + iz_{2} and w = w_{1} + iw_{2},

so that
(2)  z.w = z_{1}w_{1} + z_{2}w_{2} 
which we may call the real form of the real product, for it shows immediately that the real product is always a real number. Since
z.w = Re(z'w) = Re(zw'), 
which is not as symmetric as (2), but may be computationally useful. There is also a symmetric derivation:

The real product has the following properties:
 z.z = z^{2}.
 z.w = w.z.
 r(z.w) = (rz).w = z.(rw), for any real r.
 z.(u + v) = z.u + z.v.
 z.w = 0 iff OZ ⊥ OW, where O, Z, W are the points with coordinates 0, z, w.
 (zu).(zv) = z^{2}(u.v).
Maxwell's theorem serves a nice application for #5.
Complex Product
The complex product is the incarnation of the vector product in complex terms (in [Gardiner, Bradley, p. 239] the product is designated exterior):
(3)  z×w = (z'w  zw')/2. 
We can verify that:

which says that z×w is purely imaginary and justifies the terminology.
The complex product have the following properties:
 For z,w ≠ 0, z×w = 0 iff z = sw for some real s.
 z×w =  w×z. (anticommutativity)
 z×(u + v) = z×u + z×v.
 s(z×w) = (sz)×w = z×(sw), for any real s.
 z×w = 0 iff O, Z, W are collinear, where O, Z, W are the points with coordinates 0, z, w.
... to be continued ...
References
 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)
 A. D. Gardiner, C. J. Bradley, Plane Euclidean Geometry: Theory and Problems, UKMT, 2005
 Liangshin Hahn, Complex Numbers & Geometry, MAA, 1994
Complex Numbers
 Algebraic Structure of Complex Numbers
 Division of Complex Numbers
 Useful Identities Among Complex Numbers
 Useful Inequalities Among Complex Numbers
 Trigonometric Form of Complex Numbers
 Real and Complex Products of Complex Numbers
 Complex Numbers and Geometry
 Plane Isometries As Complex Functions
 Remarks on the History of Complex Numbers
 Complex Numbers: an Interactive Gizmo
 Cartesian Coordinate System
 Fundamental Theorem of Algebra
 Complex Number To a Complex Power May Be Real
 One can't compare two complex numbers
 Riemann Sphere and Möbius Transformation
 Problems
Contact Front page Contents Algebra
Copyright © 19962018 Alexander Bogomolny
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