Tool to understand sets of numbers N, Z, Q, R, I, C. Number sets are groups of numbers constructed by mathematicians in order to define them and classify them.

Number Sets - dCode

Tag(s) : Arithmetics, Notation System

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A set of numbers is a mathematical concept that allows different types of numbers to be placed in various categories, sometimes included between them.

The classical representation of usual sets is $$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} $$

In mathematics, there are multiple sets: the natural numbers N (or ℕ), the set of integers Z (or ℤ), all decimal numbers D or $ \mathbb{D} $, the set of rational numbers Q (or ℚ), the set of real numbers R (or ℝ) and the set of complex numbers C (or ℂ). These 5 sets are sometimes abbreviated as *NZQRC*.

Other sets like the set of decimal numbers D or $ \mathbb{D} $, or the set of pure imaginary numbers I or $ \mathbb{I} $ are sometimes used. There are also sets of transcendantal numbers, quaternions, or hypercomplex numbers, but they are reserved for advanced mathematical theories, NZQRC are the most common sets.

The sign `∈` (Unicode 2208) means *element of* or *belongs to*.

__Example:__ $ 2 \in \mathbb{N} $ is read *2 is an element of the set N*

There is also the sign `∊` (Unicode 220A) which is the same but smaller.

The sign `∉` (Unicode 2209) means *is not an element of* or *does not belong to*.

__Example:__ $ -2 \notin \mathbb{N} $

The sign `⊂` (Unicode 2282) means *is included in* or *is a subset of*

In maths, N is the set of natural numbers

__Example:__ 0, 1, 2, 3, 4, 5, … 10, 11, …, 100, … $ \in \mathbb{N} $

$ \mathbb{N}^* $ (N asterisk) is the set of natural numbers except 0 (zero), it is also referred as $ \mathbb{N}^{+} $

NB: Some (old) textbooks indicate the letter `W` instead of `N` for this set, `W` stands for *Whole numbers*

The set N is included in sets Z, D, Q, R and C.

Z is the set of integers, ie. positive, negative or zero.

__Example:__ …, -100, …, -12, -11, -10, …, -5, -4, -3, -2, - 1, 0, 1, 2, 3, 4, 5, … 10, 11, 12, …, 100, … $ \in \mathbb{Z} $

$ \mathbb{Z}^* $ (Z asterisk) is the set of integers except 0 (zero).

The set Z is included in sets D, Q, R and C.

The set N is included in the set Z (because all natural numbers are part of the relative integers). Any number in N is also in Z.

D is the set of decimal numbers (its use is rare and mainly limited to Europe)

$$ \mathbb {D} = \left\{ \frac{a}{10^{p}} , a \in \mathbb{Z}, p \in \mathbb {N} \right\} $$

All decimals in D are numbers that can be written with a finite number of digits (numbers containing a dot and a finite decimal part).

__Example:__ -123.45, -2.1, -1, 0, 5, 6.7, 8.987654 $ \in \mathbb{D} $

The numbers using suspension points `…` for their decimal writing therefore have an infinite number of decimal places and therefore do not belong to the set D.

The set D is included in sets Q, R and C.

The sets N and Z are included in the set D (because all integers are decimal numbers that have no decimal places). Any number in N or Z is also in D.

Q is the set of rational numbers, ie. represented by a fraction a/b with a belonging to Z and b belonging to Z * (excluding division by 0).

__Example:__ 1/3, -4/1, 17/34, 1/123456789 $ \in \mathbb{Q} $

The set Q is included in sets R and C.

Sets N, Z and D are included in the set Q (because all these numbers can be written in fraction). Any number in N or Z or D is also in Q.

R is the set of real numbers, ie. all numbers that can actually exist, it contains in addition to rational numbers, non-rational numbers or irrational as $ \pi $ or $ \sqrt{2} $.

Irrational numbers have an infinite, non-periodic decimal part.

__Example:__ $ \pi $, $ \sqrt{2} $, $ \sqrt{3} $, … $ \in \mathbb{R} $

$ \mathbb{R}^* $ (R asterisk) is the set of non-zero real numbers, so all but 0 (zero), also written $ \mathbb{R}_{\neq0} $

$ \mathbb{R}_+ $ (R plus) is the set of positive (including zero) real numbers, also written $ \mathbb{R}_{\geq0} $

$ \mathbb{R}_- $ (R minus) is the set of negative (including zero) real numbers, also written $ \mathbb{R}_{\leq0} $

$ \mathbb{R}_+^* $ (R asterisk plus) is the set of non-zero positive real numbers, also written $ \mathbb{R}_{>0} $

$ \mathbb{R}_-^* $ (R asterisk minus) is the set of non-zero negative real numbers, also written $ \mathbb{R}_{<0} $

The set R is included in the set C.

Sets N, Z, D and Q are included in the set R. Any number in N or Z or D or Q is also in R.

I is the set of (pure) imaginary numbers, that is to say complex numbers without real parts, the square roots of negative real numbers are pure imaginaries.

__Example:__ $ i \in \mathbb{I} $ with $ i^2=-1 $

The set I is included in the set C.

C is the set of complex numbers, a set created by mathematicians as an extension of the set of real numbers to which are added the numbers comprising an imaginary part.

__Example:__ $ a + i b \in \mathbb{C} $

Sets N, Z, D, Q, R and I are included in the set C. Any number in N or Z or D or Q or R or I is also in C.

The empty set is noted `Ø`, as its name indicates it is empty, it does not contain any number.

Constructible numbers are all numbers that can be geometrically drawn through a straightedge and compass construction.

__Example:__ $ \sqrt{2} $ is a constructible number, but $ \pi $ is not.

Algebraic numbers are a set of numbers that can be calculated as a root of a polynomial with rational coefficients.

Transcendent numbers are a set of numbers that cannot be calculated as a root of a polynomial with rational coefficients (so not algebraic).

Among the real or complex numbers, the majority are transcendental numbers.

Irrational numbers are a set of numbers that cannot be written as a fraction (i.e. all numbers that are not in $ \mathbb{Q} $)

Some books define the sets `E` for *even* numbers and `O` for *odd* numbers. This is not a standard notation.

The links between the different sets are represented by inclusions: $$ N \subset Z \subset D \subset Q \subset R \subset C $$

The subset symbol `⊆` is that of inclusion (broad sense), A ⊆ B if every element of A is an element of B.

The subset symbol `⊂` or `⊊` is that of proper inclusion (strict sense), A ⊂ B if every element of A is an element of B and A ≠ B.

The letter Q was chosen for the word Quotient.

If an element belongs to $ \mathbb{X}^n $ where $ X $ is a set and $ n $ an integer, then it is a tuple of numbers (containing $ n $ numbers).

__Example:__ The point P (a, b) of the 2D plane belongs to $ \mathbb{R}^2 $.

__Example:__ The point P (a, b, c) has integer coordinates, it belongs to the 3D grid $ \mathbb{Z}^3 $.

A set of numbers is written with the `mathbb` tag: \mathbb{Z} for $ \mathbb{Z} $

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Cite as source (bibliography):

*Number Sets* on dCode.fr [online website], retrieved on 2023-12-03,

- Number Sets Calculator
- What is a set of numbers? (Definition)
- What are common number sets?
- What does the symbol ∈ mean?
- What is the N number set?
- What is the Z number set?
- What is the D number set?
- What is the Q number set?
- What is the R number set?
- What is the I number set?
- What is the C number set?
- What is the Ø empty set?
- What is a constructible number?
- What is an algebraic number?
- What is a transcendental number?
- What are irrational numbers?
- What are E and O number sets?
- What are the inclusions of sets?
- Why the letter Q for Rationals?
- What does R^2 mean (or other power) of a set?
- How to write a number set in LaTeX?

set,number,integer,natural,rational,real,complex,imaginary,nzqrc

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