SRP: Relativitetsteori og Komplekse tal i Fysik og Matematik

STX 3.g

Abstact

This paper deals with complex numbers and their applications within the physics. Complex number’s field (C) is an extension to the real number’s fields (R) and it contains a pair of real numbers: (a,b) is most commonly written as a + ib. These numbers are defined as organized by two operations of addition and multiplication: or and or The numbers can also be considered as position vectors with the coordinate (a,b) in a twodimensional coordinate system which is called the complex plane, where the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis. The length of such a position vector is called modulus and the angle between the position vector and the real axis is called the argument. The number 0 has got no argument.

If r is the modulus and v is the argument the complex number can now be written as . The different notations have different advantages. For instance involution is easiest latter notation by using de Moivre’s formula: where v is the argument, r is the modulus and z is the symbol of a complex number. This paper also deals with the special theory of relativity, which is evolved by Albert Einstein. This theory contains the definitions of relative movement, inertial systems, the ether and the transformation of Lorentz.

In 1908 Hermann Minkowski expressed the special theory of relativity geometrically and added time as a fourth dimension. As a fourth dimension the time is considered as an imaginary dimension of space, where complex numbers are an advantage because of the imaginary dimension.

1 - Indledning

Normalt vil der umuligt kunne findes et tal, der kvadreret giver et negativt tal og dermed kan der ikke findes kvadratrødder til negative tal, selvom negative tals kvadratrødder kan være nyttige. Ved indførelse af de ”imaginære” tal er det gjort muligt at finde løsninger til ligninger, som ikke før Sidst regnes højre side af lighedstegnet: ibaibaibaibabbbbaaaazzzz ibaibabbaaibaibabbaazzzz ibaibabbaaibaibabbaazzzz ibababbaaibababbaazzzz ibaibaibaibazzzz Nu har vi regnet begge sider af lighedstegnet og mangler nu kun én ting, for at gøre beviset færdig, nemlig at tjekke, om det stemmer overens med det, vi ønskede at bevise: Vi kan nu se, at det ønskede er bevist - q.e.d. Det er nu vist, at C er et talområde, men det skulle også være en såkaldt ring.