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| 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  |
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| Senin, 28 Desember 2009 |
| The research of mathematics |
A. Marsigit’s Opinion
The aim of the research of mathematics is to examine and develop mathematics.
The kinds of the nature of mathematics instead of:
1) Formal mathematics/Axiomatic mathematics/Pure mathematics.
2) Applied mathematics.
3) School mathematics/concret mathematics/real 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.
Example:
Definition : x1,x2,x3,... Є R
Axiom
xi + xi+1 Є R
xi . xi+1 Є R
theorem
xi + xi+2 Є R
proof:
xi + xi+1 Є R
xi + xi+2 Є R
theorem:
xi + xi+2 Є R
formal mathematics is tell about numbers theory, group theory, ring teory, field teory, Euclidean geometry, non Euclidean geometry.
2) Applied mathematics
Applied mathematics using appropriate method to make a model airfield.
3) School mathematics/concrete mathematics
Mathematics phenomenon have two kinds, such as abstraction and idealization. Abstraction instead of form and shape. Idealization instead of straight(about perfectly). According to Ebbute Straker (1995), there are four parts of school mathematics: pattern/relationship, problem solving, investigation, communication.
To identify problems, we must know about mathematics knowledge, mathematics system, mathematics characteristic. When we make the research of mathematics, we need supporting factors, such as:
- The history of mathematics
- The works of mathematics
- The philosophy of mathematics
- Indepth study of mathematics
Background
The stability of criteria of finding....
the rational
The important
To formulate the aim of research
To develop method
For example : analyze, syntetic, deductive, phenomenologi, hermonitica.
To collect data or literature
Discussion
Solution
References |
| posted by sOFFia aNisa H.A.C (08305141004) @ 21.03 |
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