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 z·w (= zw) and the symbol for the real product z.w. Usually the context is indicative of which one is used and, in addition, the symbol is never omitted (as in zw) for the real product.

Assuming z = z1 + iz2 and w = w1 + iw2,

 
2z.w =[(z1w1 + z2w2) + i(z1w2 - z2w1)] +
[(z1w1 + z2w2) + i(-z1w2 + z2w1)] =
 2(z1w1 + z2w2),

so that

(2) z.w = z1w1 + z2w2

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)' = zw', we may obtain the same from the definition:

  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:

 
(z.w)'= (z'w + zw')'/2
 = (z''w' + z'w'')/2
 = (zw' + z'w)/2
 = z.w.

The real product has the following properties:

  1. z.z = |z|2.
  2. z.w = w.z.
  3. r(z.w) = (rz).w = z.(rw), for any real r.
  4. z.(u + v) = z.u + z.v.
  5. z.w = 0 iff OZ ⊥ OW, where O, Z, W are the points with coordinates 0, z, w.
  6. (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:

 
(z×w)'= (z'w - zw')'/2
 = (z''w' - z'w'')/2
 = (zw' - z'w)/2
 = -z×w,

which says that z×w is purely imaginary and justifies the terminology.

The complex product have the following properties:

  1. For z,w ≠ 0, z×w = 0 iff z = sw for some real s.
  2. z×w = - w×z. (anticommutativity)
  3. z×(u + v) = z×u + z×v.
  4. s(z×w) = (sz)×w = z×(sw), for any real s.
  5. 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

  1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
  2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
  3. A. D. Gardiner, C. J. Bradley, Plane Euclidean Geometry: Theory and Problems, UKMT, 2005
  4. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
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