Final exam information for 55930 (M 302 INTRODUCTION TO MATHEMATICS, RUSIN, D) Date & Time: TUESDAY, DECEMBER 17, 9am-12 Noon Location: GSB 2.126 The final exam is comprehensive. I want you to be impressed with yourself because of how much we did this semester -- look below at how many topics you have been introduced to! The test questions will be similar to those on previous exams. As in the last test I will let you consult with your peers but only until 11am; after that I need you to be quiet and not consulting anyone else's paper. Most of these topics appear in the recommended textbook; you can also find more information than you could possibly need with a quick google search of any of these terms. I will limit the questions to the following topics (not all of which will fit on the final exam!) 0. Be prepared to play Set and to remember how it works. :-) 1. Mathematical communication: Be prepared to see me use mathematical language: * the quantifiers "for all" and "there exists" * special notations for "implies" and for sums * referring to the n-th (or (n+1)st, or ... ) terms in a squence * unions and intersections of sets of things * why "it" is a dirty word in math communication 2. Sequences of integers: You should be able to recognize patterns involving consecutive integers, perfect squares (1,4,9,16,...) , and primes (2,3,5,7,11,...) You should be prepared to compute sums like 1+2+3+4+...+99+100 using a formula (a theorem) rather than a calculator. 3. Counting problems: If I ask how many things are there of such-and-such a type, I expect to see an explanation that involves adding, subtracting, multiplying, or dividing (for appropriate reasons). Probably a "tree diagram" will be useful. You have noticed that similar reasoning applies to probability problems, too. 4. Modular arithmetic: what it means, how to express it in words or symbols, and how to do computations (addition, subtraction, and multiplication only). For exponentiations you should be prepared to use Fermat's Little Theorem. Please remember the application of modular arithmetic to veriftying that 11-digit ISBNs are correct. You may bring and use a calculator. 5. Encryption: We discussed several types of encryption that all used modular arithmetic. It will suffice to know the method that encrypts symbols from an N-letter alphabet (29, or 31, or ...) by multiplying by a Secret Multiplier, and how to decrypt such messages (same method, but using a different, Super-Secret Multiplier that you can discover with a little arithmetic -- remember for example why a Secret Multiplier of 8 led to a Super-Secret Multiplier of 11 when N=29). 8. Working with integers in other bases: We have worked with numbers written in base 10 (duh), base 2, base 3, and base 16; we used other bases too but I won't include those on the test. Be prepared to convert numbers between bases, to do arithmetic in different notations, and to explain how the notations work (e.g. in base 16, "A30" means ten two-hundred- and-fifty-sixes plus three sixteens). 7. Sets of numbers: integers, rationals, irrationals, reals. You should be able to convert back and forth between the two ways to represent rational numbers: as fractions or as eventually-repeating decimals. (Nothing too complicated.) 8. Cardinalities of sets: Finite cardinalities are too boring, and the really big cardinalities will just give you a headache, but make sure you understand the difference between the countably infinite sets and the uncountably infinite sets. You might want to re-think Cantor's Diagonalization Process and what it's good for, and to review how we used all those really long hotels to demonstrate something or other. Also make sure you understand what "having the same cardinality" and "being topologically equivalent" mean: the way they are defined is kinda similar but there is a difference, right? 9. Geometry: We mostly did our geometry on 3-dimensional objects but took every opportunity to use our 2-dimensional tools, especially the Pythagorean theorem. Be prepared to compute straight-line distances in 3-space, as well as bug-crawling distances. Be familiar with the five Platonic Solids, especially the tetrahedron, cube, and octahedron. 10. Topology: Be familiar enough with the idea of topological equivalence that you can tell which of those pictures in my last homework set are equivalent and which are not. You might also want to review what equivalence means for other kinds of objects -- we spoke in class about why closed disks (including their boundary) and open disks (not including their boundary) are not equivalent; and we discussed why letters like H and I are equivalent to each other but not to K or B. Be able to move comfortably around surfaces that are presented as glueing diagrams: spheres, tori, Moebius strips, projective planes, and Klein bottles ought to be enough. 11. Probability: I will probably ask you how likely some event is, that involves flipping coins or drawing cards. Be prepared also to discuss the expected value of some process that yields numbers (e.g. monetary payoffs)