The set $\mathbb{C}$ 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. algebraic numbers
2. transcendental numbers

Algebraic numbers contain

1. algebraic integers (entire 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

Complex numbers contain

1. real numbers (the set $\mathbb{R}$
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 (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 $\neq 0$

$$ $$ One can also combine the criterions of A and B; thus e.g. the irrational numbers consist of the algebraic irrational numbers and the transcendental irrational numbers.

In addition, any of the sets $\mathbb{R}$ $\mathbb{Q}$ and $\mathbb{Z}$ may be partitioneded into positive numbers, negative numbers and 0.

Number-theoretically, the set $\mathbb{Z}$ consists of four types 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 ($\pm2,\,\pm3,\,\pm5,\,\pm7,\,\pm11,\,\ldots$ ,
$4^\mathrm{o}$ ; the composite numbers ($\pm4,\,\pm6,\,\pm8,\,\pm9,\,\pm10,\,\ldots$