1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4
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1 . Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest? - Yes, he ca. There is a simple solutio: to ask the guests to chage room. Every guest has to move from his room to the room ext door. More precisely if deotes the umber of his room, he has to move to the room with umber. Ad thus the room umber will be free ad available for the ew guest. We have used the map: f : N N,.. Situatio. The Hotel is full ad each guest has oe fried comig. Ca the mager accommodate all these ew guests? - Yes, he ca. There is still a simple solutio: to ask the guests to chage room. Every guest has to move from his room to the room with double umber. More precisely if deotes the umber of his room, he has to move to the room with umber. Ad thus the rooms with odd umbers will be free ad available for the ew guest. The fried of the perso i room will be accommodated i room ad himself i room. We have used the map: f : N N,.. Situatio. The Hotel is full ad each guest has 9 frieds comig. Ca the mager accommodate all these ew guests? - Yes, he ca. There is still a simple solutio: to ask the guests to chage room. Every guest has to move from his room to the room with the umber which is 0 times the umber of the room he had previously. More precisely if deotes the umber of his room, he has to move to the room with umber 0. Ad thus the rooms with umbers which are ot multiple of 0 will be free ad available for the ew guests. The frieds of the perso i room will be accommodated i room 0(, 0(,, 0( 9, ad himself i room 0. We have used the map: f : N N, 0.. Situatio. The Hotel is full ad each guest has ifiite may frieds comig. Ca the mager accommodate all these ew guests? - It depeds o the kid of ifiity. If it is possible to label the frieds of the guest i room by for each, the yes it is possible! Here is oe solutio: label the frieds
2 of guest by (,, (,, (,, ad give the rooms followig the ew rule: the fried of the perso i room umbered (,j will be accommodated i room ( (j-(jj, ad himself i room ( (j-(j. We have used the map: f : NN N, ( (j-(jj I have said that the aswer depeds o the kid of ifiity ivolved. If oe guest had as may frieds as there are real umbers betwee 0 ad, tha it would be too may guests for the maager.. A ifiitely deep well Suppose you have ifiite may balls umbered,,,, ad a dwell where you ca put the balls. We suppose we ca do the operatios as fast as we wat. Let us suppose we do it each time twice as fast as the previous time: the first operatio betwee o clock ad :0, the secod betwee :0 ad :, the third betwee : ad ::0, ad so o. At oo, that is :00, we ll have doe ifiite may operatios... First procedure. Rule: Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball 0. Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball 0. Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball 0.. Operatio. Put the balls 0(- to 0 i the well ad extract the ball 0. Ad so o.. Questio: how may balls are i the well at oo?
3 .. Secod procedure. Rule: Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball. Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball. Operatio. Put the balls to 0 (that is the balls,,,,, 6, 7, 8, 9 ad 0 i the well ad extract the ball.. Operatio. Put the balls 0(- to 0 i the well ad extract the ball. Ad so o.. Questio: how may balls are i the well at oo?.. Third procedure. Rule: Operatio. Put the balls to 0 i the well ad extract a ball at radom. Operatio. Put the balls to 0 i the well ad extract a ball at radom. Operatio. Put the balls to 0 i the well ad extract a ball at radom.. Operatio. Put the balls 0(- to 0 i the well ad extract a ball at radom. Ad so o.. Questio: What is the probability that the well is empty at oo?
4 . Decimal developmet of fractios.. A ice proof Let multiply both sides by 0 thus the ad fiely x = 0, x = 9, x = 9 0, = 9 x 0 x x = 9 or 9 x = 9 x =.. Write the decimal developmet of fractios with umerator. = 0, ; = 0,... ; = 0, ; = 0, ; = 0, ; = 0, = 0, ; = 0,... ; = 0, ; = 0, Describe what happes. Why is the period of the developmet of at most? ca it happe (that the period is equal to? We call period the miimum umber of digits which are repeated at ifiity. For istace: = 9, has a period equal to 6: the sequece 68 is repeatig itself to ifiity. 6 Do you kow or ca you imagie umbers whose developmet ever becomes periodic? Whe the developmet is periodic after some decimals, ca you fid back the fractio? p a Put = 0,... 0 q Example. = 0,7... = 0,7 0,... 0 p q, we have =,... thus the quotiet of the divisio of 0 p by q is ad the remaider should be p sice the period is, ad we should get o with the same operatio at each iteratio. p 7 6 Thus 0 p = q p or 9 p = q ad =. Fiely a = 0,7 = = =. q
5 Example. a = 0, The period is 6, thus 0 p = q r 0r = q r 0r 0r 0r 0r = q r = 8q r = q r = 7q p multiply the last but oe relatio by 0, the oe before by 00,, the first oe by The ad thus p 87 a = = =. q p = 00000q 0000q 000q 800q 0q 7q p p = 87q Other method : use the fodametal formula of aalysis ad thus = x x x x x... x... For example : p p p p 0 p = 0 (0 (0 (0... (0 p 0, , = 9, 0,068( Exercice. Fid ad d relatively prime, such that = 0, Exercice. Fid ad d relatively prime, such that = 0, I base 9, d d. Developmet i bases ad Itegers: 0,,, times three, times three, times three, times three, times three, times three, times three times three, or simpler. 0,,, 0,,, 0,,, 00, 0, 0, 0,,, 0,
6 Example: 0 meas i decimal writig 89 0 =. ad the other way roud : startig with 89, divide it by the remaider is, that we keep as last digit, the quotiet is 96 that we divide by, ad so o. Real positive umbers betwee 0 ad : Example. 0,0 meas =. As i the base 0, we have 0,0 =0, The fractios with deomiator k have two developmets. Ad ow begis my story about Cator ad his marvelous set.
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