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Natural numbers set: ℕ = {0,1,2,3...}

Integer numbers set: ℤ = {0,1,-1,2,-2,3,-3...}

Rational numbers set: ℚ = {p/q | p,q ∈ ℤ, q ≠ 0}

Real numbers set: ℝ = {p,α012... | p ∈ ℤ, αi ∈ {0,1,2...9}i, i ∈ ℕ}

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ

Irrational numbers set: ℝ \ ℚ

Examples of irrational numbers are: π, e, √2, √3

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b−1 for every nonzero element b.

Subtraction: a − b = a + (−b)

Division: a / b = a · b−1

Formally, a field is a set F together with two binary operations on F called addition and multiplication.

A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F.

The result of the addition of a and b is called the sum of a and b, and is denoted a + b.

The result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b.

Addition and multiplication must satisfy some properties called field axioms: commutativity, associativity, identity, inverse, distributivity.

Commutativity of addition: a + b = b + a, ∀ a,b ∈ F

Commutativity of multiplication: a · b = b · a, ∀ a,b ∈ F

Associativity of addition: a + (b + c) = (a + b) + c, ∀ a,b,c ∈ F

Associativity of multiplication: a · (b · c) = (a · b) · c, ∀ a,b,c ∈ F

Additive identiy: a + 0 = a, ∀ a ∈ F

Multiplicative identity: a · 1 = 1, ∀ a ∈ F

Additive inverse: a + (-a) = 0, ∀ a ∈ F

Multiplicative inverse: a · a-1 = 1, ∀ a ≠ 0 ∈ F

Distributivity of multiplication over addition: a · (b + c) = a·b + a·c, ∀ a,b,c ∈ F

(ℚ, +, ·) is a field

(ℚ, +, ·, ≤) is an ordered field

∀ a,b ∈ ℚ, a ≤ b

(ℝ, +, ·) is a field

(ℝ, +, ·, ≤) is an ordered field

∀ a,b ∈ ℝ, a ≤ b

The real numbers can be defined synthetically as an ordered field satisfying the completeness axiom.

Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line; this contrasts with the rational numbers, whose corresponding number line has a “gap” at each irrational value.