1. WHOLE NUMBERS; INTEGERS Whole numbers: Z 0,+ = the natural numbers  {0}. Integers:

2. Properties: Z 0,+ is closed under addition and multiplication. Z 0,+ is not closed under subtraction and division. Z is closed under addition, subtraction, and multiplication. Z is not closed under division.

3. Multiples; Divisors (Factors) Let m, n  Z. m is a multiple of n if and only if there is an integer k such that m = k n. n is a divisor of m if and only if m is a multiple of n.

4. Greatest Common Divisor; Least Common Multiple Let a, b  Z, with a, b  0. The greatest common divisor of a and b, denoted gcd(a,b), is the largest integer that divides both a and b. The least common multiple of a and b, denoted lcm(a,b), is the smallest positive multiple of both a and b.

5. THEOREM: Let a, b  Z, a, b  0. Then and

6. THE RATIONAL NUMBERS The set of rational numbers, Q, is given by NOTE:

7. “CLOSURE” The set of rational numbers, Q, is closed under addition, subtraction, and multiplication; Q  {0} is closed under division.

8. ARITHMETIC

9. Field Axioms Addition (+): Let a, b, c be rational numbers. A1. a + b = b + a (commutative) A2. a + (b + c) = (a + b) + c (associative) A3. a + 0 = 0 + a = a (additive identity) A4. There exists a unique number ã such that a + ã = ã +a = 0 (additive inverse) ã is denoted by – a

10. Multiplication (·): Let a, b, c be rational numbers: M1. a  b = b  a (commutative) M2. a  (b  c) = (a  b)  c (associative) M3. a  1 = 1  a = a (multiplicative identity) M4. If a  0, then there exists a unique â such that a  â = â  a = 1 (multiplicative inverse) â is denoted by a -1 or by 1/a.

11. D. a(b + c) = ab + ac (a + b)c = ac + bc Distributive laws (the connection between addition and multiplication).

12. DECIMAL REPRESENTATIONS Let be a rational number. Use long division to divide p by q. The result is the decimal representation of r.

13. Examples

14. ALTERNATIVE DEFINITION. The set of rational numbers Q is the set of all terminating or (eventually) repeating decimals.

15. Repeating versus Terminating Decimals Problem: Given a rational number The decimal expansion of r either terminates or repeats. Give a condition that will imply that the decimal expansion: a. Terminates; b. repeats.

16. Answer: The decimal expansion of r terminates if and only if the prime factorization of q has the form

17. Converting decimal expansions to fractions Problems: Write in the form

20. Locating a rational number as a point on the real number line.

21. Distribution of the rational numbers on the real line. Let a and b be any two distinct real numbers with a < b. Then there is rational number r such that a < r < b. That is, there is a rational number between any two real numbers. The rational numbers are dense on the real line.