Mathematical Thinking

Due November 30

3:3; 1-8, 15-16

1. Shake’em up. What did Georg Cantor do that “shook the foundations of infinity?”

He showed that infinity comes in different sizes.

1. Detecting digits. Here’s a list of three numbers between 0 and 1;

0.12345

0.24242

0.98765

What’s the first digit of the first number? What’s the second digit of the second number? What’s the third digit of the third number?  1,4, and 7.

1. Delving into digits. Consider the real number M=0.12345678910111213141516…. Describe in words how this number is constructed. What’s its 14th digit? What’s the 25th digit? What’s the 31st digit?

This is an interesting problem which we discussed at length in class. The 14th digit is the fifth digit we encounter when writing the two-digit numbers, and that is 1.

The 25th digit is the 16th digit we encounter when writing the two-digit numbers and that two digit number is the 8th two digit number, which is 17. Therefore the answer is 7. The 31st digit is the 22nd digit we encounter when writing the two-digit numbers and that two digit number is the 11th two digit number, which is 20. Therefore the answer is 0. We spend some time in class finding the 2000th digit of M.

1. 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.

Sure, just be sure your first digit isn’t 3 and second digit isn’t 8 and the third digit isn’t 2, like 11111.

1. 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.

You cannot describe a five-digit number that is not in the list because you know nothing about the third number.

1. 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.

1. Don’t dodge the connection (S). Explain the connection between the Dodge Ball game and Cantor’s proof that the cardinality of the reals is greater than the cardinality of the natural numbers.

Winning at Dodge Ball (dodging) requires an understanding of coordinates like Cantor’s argument.

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.

1. Cantor with 3’s and 7’s. Rework Cantor’s 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.

1. 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.

If the first number in the list is 0.72222222222… and all the other numbers in the list

Do not have a 2 in the diagonal entry, the number M matches the first number.

1. Ones and twos (H). Show that the set of all real numbers between 0 and 1 just having 1’s and 2’s 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 1’s and 2’s.)

The proof works just like Cantor’s theorem. If the diagonal digit is 1, make the digit 2 and if its 2, make it 1.