What is a real number?

[admin]

Let's introduce some Mathematical Statements and the [url=https://en.wikipedia.org/wiki/Quantifier_(logic)]quantifiers[/url] before diving into math deeply.
∀ - "for each", "for all"
∃ - "there exists some"
⇒ - "result" or "​conclusion"
⇔ - "equivalence"
def - "definition"
: - "such that"
Examples:

∀ a>0 ∃ x>0 : x²=a
Translation of this expression into human language looks like
for each a>0 there exists some x>0 such that x²=a
Let's consider some real numbers a>0:
1)a=4 then x=√4=2
2)a=2 then x=√2
3)a=0.5 then x=√0.5
................
these statements will be true for any real number a>0 which you can pick up

a>b>0 ⇒ a² > b²
Let's consider some real numbers a>b>0:
1)a=3,b=2
3>2 ⇒ 9>4
2)a=5.2,b=4
5.2>4 ⇒ 5.2²>16
..............

[admin]

Axioms for the Real numbers
The algebraic axioms

a+b=b+a	a·b=b·a
(a+b)+c=a+(b+c)	(a·b)·c=a·(b·c)
a+0=a	a·1=a
∀ a ∃ (−a) : a + (−a) = 0	∀ a ≠ 0 ∃ aˉ¹ : a · aˉ¹ = 1

We can rewrite algebraic axioms as one mathematics statement:
a · (b + c) = a · b + a · c

The order axioms
a > b , b > c ⇒ a > c

a > b ⇒ ∀ c, a + c > b + c

a > 0, b > 0 ⇒ a · b > 0