Useful Inequalities Among Complex Numbers

By definition, for a complex number z = x + yi,

|z|2 = x2 + y2 = Re(z)2 + Im(z)2.

From here,

|z|2 ≥ Re(z)2 and |z|2 ≥ Im(z)2.

And, finally,

|z| ≥ |Re(z)| and |z| ≥ |Im(z)|.

The above help prove the triangle inequality in a formal manner.

|z + w|2 = (z + w)·(z + w)'   = (z + w)·[z' + w']   = zz' + [zw' + z'w] + ww'   = |z|2 + 2Re[zw'] + |w|2   ≤ |z|2 + 2|zw'| + |w|2   = |z|2 + 2|z||w| + |w|2   = (|z| + |w|)2.

Since both |z + w| and |z| + |w| are non-negative,

|z + w| ≤ |z| + |w|.

The equality holds if one of the numbers is 0 and, in a non-trivial case, only when Im(zw') = 0 and Re(zw') is positive. This is equivalent to the requirement that z/w be a positive real number.

Let's apply the triangle inequality in a round-about way:

|z| = |(z - w) + w|   ≤ |z - w| + |w|.

In other words,

|z| - |w| ≤ |z - w|

holds for any z and w. In particular, it holds for w and z (i.e., after the exchange of the symbols):

|w| - |z| ≤ |w - z| = |z - w|.

We can combine the two inequalities in one:

||w| - |z|| ≤ |z - w|.

We conclude from the latter inequality that the absolute value function f(z) = |z| is continuous: if two complex numbers z and w are close, so are their absolute values. The converse is not true. If z = (1, 0) and w = (-1, 0), |z| = |f| = 1 and ||z| - |w|| = 0, which could not be smaller. However, |z - w| = |(2, 0)| = 2.

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 Analisys, Chelsea Publ, 3^rd 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
   • Central and Inscribed Angles in Complex Numbers
8. Plane Isometries As Complex Functions
9. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and 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
    • Product of Diagonals in Regular N-gon
    • Sum of the Nth Roots of Unity
    • Napoleon's Relatives
    • Distance between the Orthocenter and Circumcenter
    • Two Properties of Flank Triangles - A Proof with Complex Numbers
    • Midpoint Reciprocity in Napoleon's Configuration
    • Orthocenter in Complex Plane

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