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
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Copyright © 19962018 Alexander Bogomolny