ࡱ> 241#` bjbj\.\. .>D>D |||| +$hI   "   : ,~ Pm>|." 0+,R~~,  +|| Mathematical Thinking Due November 30 3:3; 1-8, 15-16 Shakeem up. What did Georg Cantor do that shook the foundations of infinity? Detecting digits. Heres a list of three numbers between 0 and 1; 0.12345 0.24242 0.98765 Whats the first digit of the first number? Whats the second digit of the second number? Whats the third digit of the third number? Delving into digits. Consider the real number M=0.12345678910111213141516. Describe in words how this number is constructed. Whats its 14th digit? Whats the 25th digit? Whats the 31st digit? Undercover friend. Your friend gives you a list of three, five-digit numbers, but she only reveals one digit in each: 3???? ?8??? ??2?? Can you describe a five-digit number you know for certain will not be on her list? If so, give one; if not, explain why not. Underhanded friend. Now your friend shows you a new list of three, five-digit numbers, again with only a few digits revealed: 6???? ?5??? ????? Can you describe a five-digit number you know for certain will not be on her list? If so, give one; if not, explain why not. Dodge Ball. Revisit the game of Dodge Ball from Chapter 1: Fun and Games. Play it several times with several people. Get the strategy down, and then explain to your opponents the underlying principle. Record the results of the games. Dont dodge the connection (S). Explain the connection between the Dodge Ball game and Cantors proof that the cardinality of the reals is greater than the cardinality of the natural numbers. Solution is on page 729. (S) means solutions at back of book and (H) means hints at back of book. So that means that 15 and 16 have hints at the back of the book. Cantor with 3s and 7s. Rework Cantors proof from the beginning. This time, however, if the digit under consideration is 3, then make the corresponding digit of M an 7; and if the digit is not 3, make the associated digit of M a 3. The first digit (H). Suppose that, in constructing the number M in the Cantor diagonalization argument, we declare that the first digit to the right of the decimal point of M will be 7, and then the other digits are selected as before (if the second digit of the second real number has a 2, we make the second digit of M a 4; otherwise, we make the second digit a 2, and so on). Show by example that the number M may, in fact, be a real number on our list. Ones and twos (H). Show that the set of all real numbers between 0 and 1 just having 1s and 2s after the decimal point in their decimal expansions has a greater cardinality than the set of natural numbers (So, the number 0.112111122212122211112..is a number in this set, but 0.1161221212122122.is not, because it contains digits other than just 1s and 2s.) 7Dm   & ( 0 1 2 D < O L W : Y Tfh3W hh{h+h*/h+H* h+5h+ &67l m 0 1 2 ; < K ^gd+ hh^h`hgd+ & Fgd+gd+K L 9 : ST & Fgd+h^hgd+ & Fgd+gd+ ,1h/ =!"#$% @@@ NormalCJ_HaJmH sH tH DAD Default Paragraph FontRiR  Table Normal4 l4a (k(No List &67lm012;<KL9:S T 0000 0 000000 000 000000 000000 00 0000 00 00 00 K   7? ;< 33 pA\B-QN^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.-QpA                  3W 3mq,+@;;` ;; @UnknownGz Times New Roman5Symbol3& z Arial"qhWfkf& & !24 2QHX)?+2Mathematical ThinkingComputing ServicesComputing Services  Oh+'0  (4 T ` lxMathematical ThinkingComputing Services Normal.dotComputing Services1Microsoft Office Word@G@zz@& ՜.+,0 hp  UNC Charlotte  Mathematical Thinking Title  "#$%&'(*+,-./03Root Entry FP@5Data 1TableWordDocument.SummaryInformation(!DocumentSummaryInformation8)CompObjq  FMicrosoft Office Word Document MSWordDocWord.Document.89q