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, 3rd edition, 1966
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