What is a real number?

Post Reply
virtualmod
Site Admin
Posts: 2928
Joined: Wed Dec 11, 2019 8:31 am

What is a real number?

Post by virtualmod »

Let's introduce some Mathematical Statements and the quantifiers before diving into math deeply.
∀ - "for each", "for all"
∃ - "there exists some"
⇒ - "result" or "​conclusion"
⇔ - "equivalence"
def - "definition"
: - "such that"
Examples:
-------------------------------------------------
\(∀ \: a>0 \: ∃ \: x>0 : x^2=a\)
Translation of this expression into human language looks like
for each a>0 there exists some x>0 such that \(x^2=a\)
Let's consider some real numbers a>0:
1)a=4 then x=\(\sqrt{4}\)=2
2)a=2 then x=\(\sqrt{2}\)
3)a=0.5 then x=\(\sqrt{0.5}\)
................
these statements will be true for any real number a>0 which you can pick up
------------------------------------------------
a>b>0 ⇒ \(a^2 > b^2\)
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 ⇒ \(5.2^2>16\)
................
----------------------------------------------
virtualmod
Site Admin
Posts: 2928
Joined: Wed Dec 11, 2019 8:31 am

Re: What is a real number?

Post by virtualmod »

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 \(∀ \: a ≠ 0 \: ∃ \: a^{−1} : a · a^{−1} = 1\)
We can rewrite algebraic axioms as one mathematic 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
Post Reply