Useful Identities Among Complex Numbers

The basic properties of complex numbers follow directly from the defintion.

(1) Re(z + w) = Re(z) + Re(w) and Im(z + w) = Im(z) + Im(w).

Indeed

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:

(2) 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'.

By definition,

Re(zw) = Re(z)Re(w) - Im(z)Im(w),
Im(zw) = Re(z)Im(w) + Im(z)Re(w).

Also,

(3) (zw)' = z'w'.

Since z' = Re(z) - i·Im(z),

(4) 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).

By the triangle inequality, and the multiplicative property of the absolute (e.g. (u - v)(w - z) = |u - v|·|w - z|) we obtain

(6) |u - v|·|w - z| + |u - z|·|v - w| ≥ |u - w|·|v - z|.

The equality here is achieved only if (u - v)(w - z) / (u - z)(v - w) is a positive real number. This requirement is equivalent to saying that (u - v)/(u - z) : (w - v)/(w - z) is a negative real number. (In passing, this is the cross-ratio (uwvz).) The ratio is real iff the four points u, v, w, z are concylic. For (u - v)(w - z) / (u - z)(v - w) to be negative, i.e., for the inequality (6) to hold, the points should follow in the indicated order: u, v, w, z. In the case of equality, we recognize the famous Ptolemy's theorem: if the points U, V, W, Z correspond to complex numbers u, v, w, z,

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.

Remark

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, UV = UZ = VZ and, from (7),

WZ + VW ≥ VZ.

If we assume that, of the three segments WZ, VW, VZ, VZ is the largest, the other two required inequalities (see Euclid I.20 and I.22), VZ + VW ≥ WZ and WZ + VZ ≥ VW, will follow by default.

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. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
  4. E. Landau, Foundations of Analysis, Chelsea Publ, 3rd edition, 1966
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