# 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'.

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.

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

The equality here is achieved only if *concylic*. For

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,

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),

### 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) - Liang-shin Hahn,
*Complex Numbers & Geometry*, MAA, 1994 - E. Landau,
*Foundations of Analysis*, Chelsea Publ, 3^{rd}edition, 1966

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