classification of complex numbers

The set $ℂ$ of all complex numbers and many of its subsets may bepartitioned (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. algebraic numbers

2. transcendental numbers

Algebraic numbers contain

1. algebraic integers ( algebraic numbers)

2. algebraic fractions (fractional algebraic numbers)

Algebraic integers contain

1. rational integers

2. non-rational integers

Algebraic fractions contain

1. rational fractions

2. non-rational fractions

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 $ℝ$)

2. imaginary numbers (i.e. non-real complex numbers)

Real numbers contain

1. rational numbers (the set $\mathbb{Q}$)

2. irrational numbers

Rational numbers contain

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

2. fractional numbers

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 andthe irrational numbers.

In , any of the sets $ℝ$,$\mathbb{Q}$ and $\mathbb{Z}$ may be partitioneded into positive numbers, negative numbers and 0 (http://planetmath.org/Null).

Number-theoretically, 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  ($\pm 4,\pm 6,\pm 8,\pm 9,\pm 10,\mathrm{\dots }$)