The first test for this course takes place in class on Friday, February 14. The test counts for 15% of the final grade for the course. You can expect a three or four page test (with room for answers). The test will be a mix of calculation problems, short answer questions such as definitions, and longer essay-type questions. Essay questions can range from very specifc questions with a definite right answer to more general conceptual questions.
We began the course with the very general question of whether mathematics is invented or discovered, and we looked at one of the very earliest proofs, the fact that there are an infinite number of prime numbers. We then looked at a couple of mathematical puzzles about counterfeit coins, based on one of the puzzles in the first chapter of the textbook. After that introduction, we covered some material from Sections 2.1, 2.2, 2.3, 2.6, 2.7, 3.1, and 3.2 in the textbook. All of this material is fair game for the test. Here are some of the things that you should be familiar with:
Any integer bigger than 1 is either prime or can be factored into a product of primes.
There is no largest prime number: given any N there is a prime number bigger than N.
How to find a prime bigger than N: Consider (2×3×4×...×N) + 1.
How the preceding proves that there are infinitely many prime numbers.
Using a balance can give 3 possible results.
Using a balance once cannot distinguish among more than 3 cases. (Why?)
Using a balance twice cannot distinguish among more than 9 cases. (Why?)
Ideas from the solutions to counterfeit coin problems.
The pigeonhole principle and its applications.
Infinite sequences of numbers.
The notation ai.
Understanding notations like Fn-1 and An+1.
Defining sequences by simple formulas: An = 2n-1, or TN = 1+2+3+...+N.
Defining sequences by "recursive" formulas: A1 = 1 and An = 2An-1 - 1 for n>1.
The Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Recursive definition of the Fibonacci numbers: F1 = 1, F2 = 2, and Fn = Fn-1 + Fn-2 for n>2.
The ratios Fn/Fn-1 approach the number φ as n gets bigger.
The "square" numbers 1, 4, 9, 16, 25, 36, 49, ..., and the geometric meaning of square number.
The "triangular" numbers 1, 3, 6, 10, 15, 21, ..., and the geometric meaning of triangular number.
The Goldbach Conjecture: Every even number greater than 2 can be written as a sum of two primes.
Twin primes: two prime numbers that differ by 2 like 11 and 13, or 17 and 19.
The Twin Prime Conjecture: There is an infinite number of twin primes.
The Golbach Conjecture and the Twin Primes Conjecture have not yet been proved or disproved.
The positive integers, the natural numbers, and the integers.
Rational numbers and irrational numbers.
Writing numbers as decimals.
How to convert repeating decimals into rational form m/n.
A rational number has a decimal expansion that terminates or ends in a repeating pattern.
A decimal number that never repeats is irrational.
Any irrational number can be approximated arbitrarily closely by a rational number.
Sets.
Cardinality of a set.
What it means to "count".
One-to-one correspondence.
What it means for two infinite sets to have the same cardinality.
Many infinite sets have the same cardinality (integers, natural nubmers, rational nubmers,...).
1. You have ten coins. One is counterfeit and weighs less than the real coins. All the real coins weigh the same. Explain why you can't be sure of finding the counterfeit by using a balance just two times.
2. Explain why (5×7×11×17×19)+1 is not evenly divisible by 11.
3. I have 25 jobs to do in a 24 hour day. Each job takes one hour. Explain why I'll have to do two things at once. (What principle are you applying, and how?)
4. How likely is it that there are two people in this class (25 people) who are the same height to withing one foot? To within one inch? To withing one millimeter?
5. Write the first 6 terms of the number sequence defined by A1 = 3, and An = An-1 + n for n>1.
6. Write the first 6 terms in the sequence defined by B1=0, B2=2, and Bn = 2×Bn-1 - Bn-2 for n>2.
7. Find a formula that defines the sequence 1, 8, 23, 68, 263, 768, ..., 2363
8. Explain how the triangular numbers get their name.
9. State the Goldbach Conjecture.
10. Can 11 be written as the sum of two prime numbers?
11. Write 0.3211111111111... as a rational number m/n. Do the same for 3.7676767676...
12. Suppose that you have a decimal number that contains no digits except for 3 and 7, and that it contains only a finite number of 7's. (That is, it does not contain any 0, 1, 2, 4, 5, 6, 8, or 9.) Can the number be irrational? Explain.
13. Write down an irrational number that contains only the digits 3 and 7,
14. We know that π is irrational. Show that π/k is irrational for every positive integer k. (Hint: What if π/k = m/n for some integers m and n?)
17. Show that there are infinitely many irrational numbers. (Hint: See the previous problem.)
16. Show that the set of positive integers has the same cardinality as the set of negative integers.
17. If you take one item out of a finite set, the set that is left has a smaller cardinality. Does the same thing happen if you take one item out of an infinite set? Explain.
18. What is a "one-to-one correspondence"? Give a simple, specific example.
19. What does it mean for a set to be "finite"? What does it mean for a set to be "infinite"?
20. Is mathematics created or is it discovered? What do you think? Explain.