MATHEMATICS 2000 NUMBER THEORY NOTES AXIOMS FOR THE SET Z OF INTEGERS Axiom 1: Z is closed under addition: For all numbers a, b, if a and b are in Z then a + b ∈ Z. Axiom 2: Addition is associative on Z: For all a, b, c ∈ Z, a + (b + c) = (a + b) + c. Axiom 3: There is a special element 0 in Z which is an identity element for addition: For all a ∈ Z, a + 0 = 0 + a = a. Axiom 4: Every element a in Z has an inverse element −a for addition: For every a ∈ Z, there exists −a ∈ Z such that a + (−a) = 0 = (−a) + a. Axiom 5: Addition is commutative on Z: For all a, b ∈ Z, a + b = b + a. Axiom 6: Z is closed under multiplication: For all a, b ∈ Z, ab ∈ Z. Axiom 7: Multiplication is associative on Z: For all a, b, c ∈ Z, a(bc) = (ab)c. Axiom 8: There is a special element 1 in Z which is an identity element for multipli- cation: For all a ∈ Z, a · 1 = 1 · a = a. Axiom 9: Multiplication on Z is commutative: For all numbers a, b if a ∈ Z and b ∈ Z, then ab = ba. Axiom 10: Multiplication distributes over addition in Z: For all a, b, c ∈ Z, a(b + c) = ab + ac. DEFINITIONS: An algebraic structure of a set of objects with an operation + which satisfies all of axioms 1 through 4 is called a GROUP. A group which also satisfies axiom 5 is called a COMMUTATIVE GROUP. A set of objects with two operations + and × which satisfies axioms 1 through 7 plus 10 is called a RING. When all 10 axioms are satisfied the set is called a COMMUTATIVE RING WITH IDENTITY. A commutative ring with identity in which also every element except 0 has an inverse for × is called a FIELD. 1 DEFINITIONS FOR THE SET Z OF INTEGERS DEFINITION: We say that an integer n is even if n = 2k for some integer k. DEFINITION: We say that an integer n is odd if n = 2k + 1 for some integer k. DEFINITION: We say that two integers a and b have the same parity if a and b are both even or both odd. DEFINITION: We say that an integer a divides an integer b, written as a|b, if b = na for some integer n. DEFINITION: For all integers a, b and integer n ≥ 2, we say that a is congruent to b modulo n, written as a ≡ b(mod n), if n|(a − b). DEFINITION: A number x is called a rational number if it can be expressed as x = a/b for some integers a and b, with b 6= 0. FACTS ABOUT THE SET R OF REALS 1. ∀a ∈ R, a2 ≥ 0. 2. ∀a ∈ R, an ≥ 0 if n is a positive even integer. 3. If a < 0 and n is a positive odd integer then an < 0. 4. Let a, b ∈ R, ab > 0 if and only if both a > 0 and b > 0 or both a < 0 and b < 0. 5. Let a, b, c ∈ R if a > b and c > 0, then ac > bc and a/c > b/c. 6. Let a, b, c ∈ R if a > b and c < 0, then ac < bc and a/c < b/c. 7. ∀x ∈ R the absolute value of x, written |x|, is defined as: ( x x ≥ 0 |x| = . −x x < 0 2.