classification of complex numbers
The set $\u2102$ of all complex numbers^{} and many of its subsets may be
partitioned (classified) into two subsets by certain criterion of the numbers.
A. F i r s t c l a s s i f i c a t i o n :
Complex numbers contain
 1.
 2.
Algebraic numbers contain

1.
algebraic integers^{} ( algebraic numbers)

2.
algebraic fractions (fractional algebraic numbers)
Algebraic integers contain
 1.

2.
nonrational integers
Transcendental numbers contain

1.
real transcendental numbers

2.
imaginary transcendental numbers
B. S e c o n d c l a s s i f i c a t i o n :
Complex numbers contain

1.
real numbers (http://planetmath.org/RealNumber) (the set $\mathbb{R}$)

2.
imaginary numbers (i.e. nonreal complex numbers)
Real numbers contain

1.
rational numbers^{} (the set $\mathbb{Q}$)
 2.
Rational numbers contain

1.
integers (http://planetmath.org/Integer) (the set $\mathbb{Z}$)
 2.
Imaginary numbers contain

1.
pure imaginary numbers (with real part^{} 0)

2.
other imaginary numbers (with real part $\ne 0$)
One can also combine the criterions of A and B; thus e.g. the irrational numbers consist of the algebraic irrational numbers and the irrational numbers.
In , any of the sets $\mathbb{R}$, $\mathbb{Q}$ and $\mathbb{Z}$ may be partitioneded into positive numbers, negative numbers and 0 (http://planetmath.org/Null).
Numbertheoretically, the set $\mathbb{Z}$ consists of four of integers:
${1}^{\mathrm{o}}$ the number 0,
${2}^{\mathrm{o}}$ the units of $\mathbb{Z}$ (only $+1$ and $1$),
${3}^{\mathrm{o}}$ the prime numbers^{} ($\pm 2,\pm 3,\pm 5,\pm 7,\pm 11,\mathrm{\dots}$),
${4}^{\mathrm{o}}$ the composite numbers^{} ($\pm 4,\pm 6,\pm 8,\pm 9,\pm 10,\mathrm{\dots}$)
Title  classification of complex numbers 

Canonical name  ClassificationOfComplexNumbers 
Date of creation  20130322 16:56:49 
Last modified on  20130322 16:56:49 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  11 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 11R04 
Related topic  NegativeNumber 
Related topic  Number 