 |
| Selasa, 29 Desember 2009 |
| The research of mathematics |
1) Formal mathematics/Axiomatic mathematics/pure mathematics
Mathematics is a deductive system consist of definition, axioms, and theorem in which there is no contradiction inside. It is very easy to establish mathematics system.
Background
The stability of criteria of finding a solution of problems
The important
To formulate the aim of research of mathematics
To develop method of number teory
For example : analyze, syntetic, deductive, phenomenologi, hermonitica.
To collect data or literature
Discussion
Mathematics is a deductive system consist of definition, axioms, and theorem in which there is no contradiction inside. It is very easy to establish mathematics system.
Research in mathematics education has two main purposes, one pure and one applied:
• Pure (Basic Science): To understand the nature of mathematical thinking, teaching, and learning;
• Applied (Engineering): To use such understandings to improve mathematics instruction.
I got a theorem from number teory to proof. I try to proof and the result is above.
Theorem:
If p is a prime number and d|p-1, congruent of x^d-1= 0(mod p) have exactly d solution.
Solution
Proof :
According to Fermat’s theorem, whether P is prime number and (a,p)=1, so a^(p-1)= 1(mod p). It have meaning congruent of x^(p-1)= 0 (mod p) have exactly (p-1) solution:
For example d|(p-1), so
x^(p-1) = (x^d-1)( x^(p-1-d)+ x^(p-1-2d)+...+1)
= (x^d-1)f(x)
According lagrange theorem**, f(x) = 0 (mod p) have (p-1-d) solutions. For example x=a is a solution of x^(p-1-1) = 0 (mod p) which non solution of f(x) = 0 (mod p), so a is solution of x^d-1=0(mod p).
Because of 0 = a^(p-1)-1 =(a^d-1)f(a)(mod p)
Because p prime number and p f(a), so p|(ad-1)
So, x^d-1 = 0 (mod p) have minimally p-1-(p-1-d) = d solution
According to lagrange theorem x^d-1 = 0 (mod p) have maximally d solutions. So, congruent of that have exactly d solutions.
foot note:
**lagrange theorem: if p prime number and f is a polynomial n degrees, congruent of f(x)= 0 (mod p) have maximal n solutions.
References
Sukirman, Drs. 2006. Numbers teory. Adhi publisher: Yogyakarta.
http://www.google.com |
posted by sOFFia aNisa H.A.C (08305141004) @ 02.46  |
|
|
|
|
 |
|
|
Posting Komentar