# classification of complex numbers

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. 1.
2. 2.

Algebraic numbers contain

1. 1.

algebraic integers ( algebraic numbers)

2. 2.

algebraic fractions (fractional algebraic numbers)

Algebraic integers contain

1. 1.
2. 2.

non-rational integers

Algebraic fractions contain

1. 1.
2. 2.

non-rational fractions

Transcendental numbers contain

1. 1.

real transcendental numbers

2. 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. 1.

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

2. 2.

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

Real numbers contain

1. 1.

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

2. 2.

Rational numbers contain

1. 1.

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

2. 2.

Imaginary numbers contain

1. 1.

pure imaginary numbers (with real part 0)

2. 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 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).

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  ($\pm 2,\,\pm 3,\,\pm 5,\,\pm 7,\,\pm 11,\,\ldots$),
$4^{\mathrm{o}}$  the composite numbers  ($\pm 4,\,\pm 6,\,\pm 8,\,\pm 9,\,\pm 10,\,\ldots$)

Title classification of complex numbers ClassificationOfComplexNumbers 2013-03-22 16:56:49 2013-03-22 16:56:49 pahio (2872) pahio (2872) 11 pahio (2872) Topic msc 11R04 NegativeNumber Number