Number Sets Finder/Checker ℕ,ℤ,ℚ,ℝ,ℂ - N Z Q R C - Online Calculator (2024)

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.

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.

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.

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.

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 $.