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 Inequalities Among Complex Numbers
By definition, for a complex number z = x + yi,
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|z|2 = x2 + y2 = Re(z)2 + Im(z)2.
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From here,
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|z|2 ≥ Re(z)2 and |z|2 ≥ Im(z)2.
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And, finally,
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|z| ≥ |Re(z)| and |z| ≥ |Im(z)|.
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The above help prove the triangle inequality in a formal manner.
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| |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. |
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Since both |z + w| and |z| + |w| are non-negative,
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:
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| |z| | = |(z - w) + w| |
| | ≤ |z - w| + |w|. |
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In other words,
holds for any z and w. In particular, it holds for w and z (i.e., after the exchange of the symbols):
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|w| - |z| ≤ |w - z| = |z - w|.
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We can combine the two inequalities in one:
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
- 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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