# MATH 100 SECTION 1 SECOND BLOCK: CORE MATERIAL

BASIC CONCEPTS: apportionment and voting. For a list of
basic concepts

see the lists of “Key Concepts” for Chapters 1 and 4 of Tannenbaum (on pages

29 and 144). You are not of course responsible for the material in section 1.6

on ranking (so you can ignore the key concepts on page 29 that have the word

“ranking” in them).

BASIC CONCEPTS: rational and irrational numbers.

rational number, irrational number, repeating decimal, nonrepeating decimal,

even and odd integers, right triangles

BASIC FACTS

1. The fundamental theorem of arithmetic (= the unique factorization theorem):

Every composite number can be written as a product of primes in

exactly one way (except for the order of the factors).

2. the Pythagorean theorem (page 58 of Stein)

3. theorems 1, 3, 4, 5, 6 on pages 63–66 of Stein

4. the rationality or irrationality of sums and products of rational and
irrational

numbers

5. a product of odd integers is odd (so if a product of integers is even, then

at least one must be even)

6. For any natural number N, we have

7. In an election with N alternatives, the total number of pairwise comparisons

is .

8. with t candidates the number of distinct possible preference ballots is

t! = 1 × 2 × 3 × · · · × t

9. The Arrow impossibility theorem

10. the Balinski and Young impossibility theorem

11. history of apportionment in the house of representatives

SOME BASIC COMPUTATIONS

Finding the decimal form of a rational number and writing a repeating decimal

as a quotient of integers.