Hello everyone. I would like to start a math topic series here with the hope of helping who needs.
I will start with constructing real numbers. Many prefer starting with natural numbers but since i am not an expert of set theory i will construct the rest number sets with the help of real number.
Real numbers is not just a set. It is a structure that includes addition, multiplication and inequality. Not to dive into much complexity and confuse readers i also prefer adding zero and one in this structure. We will discuss zero and one much more detailed when we start doing some group theory.
Real numbers is a structure, 
, that satisfies the following axioms. (Note that, if not mentioned spesifically, every element in the structure satisfies these axioms)
A1: 
A2: 
A3: For every 
there exists 
such that 
(A3) Observe that by using A1 and A2 for a given , is unique. And we will denote that as 
.
A4: 
Now axioms for multiplication. We will denote 
as 
M1: 
M2: 
M3: For a given different than zero, there exist 
such that 
(Also is unique and we will denote 
)
M4: 
AM: 
and 
Observe followings;
i) 
ii) 
, where is not equal to zero.
iii) 
iv)If 
v) 
vi) 
And last part of axioms.
O1: 
O2: 
O3: 
O4: 
AO: 
MO: 
Note that rational number structer also satisfies these axioms. So we need a specific axiom that characterize real numbers.
Definition: Let 
be a subset of Real numbers. If, for every element 
in , there exists a real number, say 
, such that 
then we say is upper bound of A
Sup axiom : Any proper(not empty) subset of real numbers, that has upper bound, has a least upper bound.
We call that least upper bound a supremum or sup. So;
. Please generate some examples. Note that, supremum of a set should not necessarily be in that set.
So we say, a sturcture is a Real Number structure if it satisfies these axioms.
Next time i am planning to show an example of an arbitrary irrational number is belonging to this system by using these axioms.
Thats all for now. Thanks for reading :)