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
Contact Front page Contents Did you know? Algebra
Copyright © 19962018 Alexander Bogomolny71733547