Useful Inequalities Among Complex Numbers
By definition, for a complex number z = x + yi,
z^{2} = x^{2} + y^{2} = 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.

Since both z + w and z + w are nonnegative,
z + w ≤ z + w. 
The equality holds if one of the numbers is 0 and, in a nontrivial 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 roundabout way:

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
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)
 Liangshin Hahn, Complex Numbers & Geometry, MAA, 1994
 E. Landau, Foundations of Analisys, Chelsea Publ, 3^{rd} edition, 1966
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
 Plane Isometries As Complex Functions
 Remarks on the History of Complex Numbers
 Complex Numbers: an Interactive Gizmo
 Cartesian Coordinate System
 Fundamental Theorem of Algebra
 Complex Number To a Complex Power May Be Real
 One can't compare two complex numbers
 Riemann Sphere and Möbius Transformation
 Problems
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