Among any two integers or real numbers one is larger, another smaller. But you can't compare two complex numbers.
We must be cautious with this kind of general statements. It's actually possible to define the relationship "<" between complex numbers. Let's agree that
(a + ib) < (c + id), provided either a < c or a = c and b < d. |
This is what is called the lexicographic order. This is how words are ordered in dictionaries. This relation, exactly like in the case of integers or real numbers, exhibits the following properties:
|
Also, since a real number x can be identified with the complex number
Still it's not very useful. To see what the problem is let's turn to the multiplication. The following is true for real numbers:
(1) | If a < b and 0 < c then ac < bc |
Note that i may be written as 0 + i1 while 0 = 0 + i0. Therefore, by definition,
The derivation is quite formal and does not use any particular properties of the lexicographic order. The crucial thing was that i satisfied the inequality
The law by which for any two numbers a and b, either
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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