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

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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