Chapter 1
what is mathematics
This chapter will try to answer this question from some critical notions, not a panorama.
1.1 Complete Induction
if we can allege some properties about the elements of a finite set, how can we extend it to the natural number set?
we introduce complete induction as a principle.
(1)if the property is satisfy on a special natural number i;
(2)if: the property is satisfy on any natural number n ⇒ it is satisfy on n+1.
then, we can allege the property is satisfy on the natural set.
1.2 Module
the basic counting method
a natural classify method: natural map a≡b(modc) or, we take it as a map:
modc :a 7→b then, we make the map to act on operations of a: if
modc:x7→b;modc :y 7→d;
then
modc :x±y 7→b±d;
1
2 CHAPTER 1. WHAT IS MATHEMATICS modc :xy7→bd.
⇒ any polynomial is invariant under congruence.
the notion of module→ prime number.
Fermat Theorem
ap−1−1≡0(modp) Euler Theorem
ak−1≡0(modb)
1.3 Base and Power
an induction definite: any integer a=nc+b, n=ic+d,..., b, d, ... are always
< c, then any integer a can always be expressed by {0,1, ...b, ...d, ..., c.}, the number of the elements of the set is c+1.
1.4 Rational Field 1.5 Real number
the completeness of order: real set Example
n < n+ 1;
1 + 1
n+ 1 <1 + 1 n; (1 + 1
n)n <(1 + 1
n+ 1)n+1; (1 + 1
n)n <3;
question: when n→ ∞, what does (1 +n1)n means?
let xn = (1 +n1)n, suppose x∞ = pq, then we can say x1 < x2 < ... < xn <
...x∞ <3 is order complete in national set.
but, x∞6= pq
xn = 1n+...+ 1
nn = 1 + 1 + 1
2!(1− 1 n...
so, if we demand the completeness of order, get real set.
1.6. COMPLEX NUMBER 3
1.6 Complex Number
the completeness of field: algebraic equation
1.7 Vector
the completeness of linear space: the structure of linear space
1.8 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.
Example a method that describe a big positive number x:
x= 10r
r is little enough, so we can use r instead of x, and write r =log(x).
(we often use e instead of 10). From x to r, for any x, there is always one r, that r = log(x). then we say that this is a function. and more than that, they are one to one corresponds.
1.9 Equation
realization of function
