Number Sets

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.

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

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

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

In maths, N is the set of natural numbers

$ \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.

$ \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).

$ \mathbb{D}_+ $ (D plus) is the set of positive decimal numbers.

$ \mathbb{D}_+^* $ (D star plus) is the set of non-zero positive decimal numbers.

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).

$ \mathbb{Q}_+ $ (Q plus) is the set of positive rational numbers.

$ \mathbb{Q}_+^* $ (Q star plus) is the set of nonzero positive rational numbers.

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.

$ \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.

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.

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.

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).

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