<The Detail of This Chapter.pdf>
the first event: 1, 2, 3,...
then, we know that counting means +1
addition, subtraction, multiplication, division
subtraction lead to 0 and negatives
division lead to fraction
the closeness of operation=>the set of number: natural number, integral number, rational number.
<from the demand of closeness to completeness, we introduce the notion of infinity.>
【complete induction】
how to denote number? base and power
how to solve algebraic equation?
the complete of order=>real number set
any two elements of set R, a>b, or b>a,
【the base of analysis】
【the power of set】
but, when we need describe something like a point on a plane, or a solution of a special equation, such a set is not enough.
complex number: if x^{2}=-1, whether x > or < any real number?
【complex number】
vector: is (3, 6) > or < (9, 2)?
the completeness of field: algebraic equation
【algebraic field】
the completeness of linear space: the structure of linear space
【linear space】
so, when we create mathematics from NATURE, we get algebra at first!
in succession, we need add more detail into the algebraic "coordinatisation" to realize or represent our understanding about NATURE.
function
the second event: to describe the causalities and the relationships that come from NATURE.
algebraic function, transcend function based on real or complex set or any other algebraic set.
equation
realization of function
algebra is the primary method that we grasp NATURE mathematically. and then, because geometry is one of the basic gates that we enter NATURE, we usually get useful notions from geometry as a part of mathematics.