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
- Remarks on the History of Complex Numbers
- Complex Numbers: A Dynamic Tool
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One Can't Compare Two Complex Numbers
- Problems
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).
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Indeed
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| z + w | = (Re(z) + i·Im(z)) + (Re(w) + i·Im(w)) |
| | = (Re(z) + (Re(w)) + i·(Im(z) + Im(w)), |
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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),
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for any real t and s. In particular,
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Re(z - s) = Re(z) - Re(w),
Im(z - w) = Im(z) - Im(w).
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It follows from (2) that the z' is also linear, i.e., for any real t and s, we have
As every linear function, Re(z), Im(z) and z' are continuous. For example,
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| |z' - w'| | = |(z - w)'| |
| | = |z - w|, |
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such that if z and w are close, so are z' and w'.
By definition,
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Re(zw) = Re(z)Re(w) - Im(z)Im(w),
Im(zw) = Re(z)Im(w) + Im(z)Re(w).
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Also,
Since z' = Re(z) - i·Im(z),
| (4) |
Re(z) = (z + z') / 2 and
Im(z) = (z - z') / 2i.
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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).
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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|.
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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 that (u - v)/(u - z) : (w - v)/(w - z) is a negative real number. It is satisfied iff the four points u, v, w, z (in this order) are cocylic. 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,
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
is known as Ptolemy's inequality.
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,
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with the equality only for W on the circumcircle of ΔUVZ. Since the triangle is equilateral, UV = UZ = VZ and, from (7),
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
- 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 Analisys, Chelsea Publ, 3rd edition, 1966
Copyright © 1996-2008 Alexander Bogomolny
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