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T h e r e is no suggestion t h a t a n y t h i n g more t h a n a finite or a d e n u m e r a b l e n u m b e r of points is defined. We use rods and small spheres to show the discrete c o o r d i n a t e system. Such a c o o r d i n a t e s y s t e m will be called n-dimensional if no more t h a n n coordinate 'rods' intersect at any coordinate point. 5-9 have a limited extension. We can either e x t e n d t h e m a r b i t r a r i l y f a r - - b u t not infinitely f a r - - b e y o n d their limits or we can t u r n t h e m into dosed systems.

They cannot be more for each other than a tool or a source of inspiration. To see how concepts of pure mathematics were transformed into concepts of physics consider the definition of the continuum with the help of the real numbers on the numbers axis. A one-dimensional continuum has the same topology as the real numbers. It is usual to map any real number x by means of the exponential function e i y z onto the unit circle in the complex plane. The resulting map { e iyx } is called the character group of the topologic group of real numbers.

T h e r e is also a l e ~ first-order difference q u o t i e n t ,~lA(O) Ao = A ( O ) - A ( O - 1) (5) T h e left first-order difference q u o t i e n t is not likely e n c o u n t e r e d if 0 is interp r e t e d as time. If 0 is a spatial variable one needs the right first-order difference q u o t i e n t for a b o u n d a r y condition on the left side a n d the left one for a b o u n d a r y c o n d i t i o n on the right side. 2-2 shows the t h r e e difference q u o t i e n t s a n d explains the use of 'left' and 'right'.

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