Appendix A Number Axioms P. Danziger 1 Number Axioms 1.1 Groups Definition 1 A Group is a set S together with a binary operation (*) on S, denoted a b such that for all a, b and c S 0. (Closure) 1. (Associativity) a b S (a b) c = a (b c) 2. (Identity) There is a distinct element of S, called 1 a 1 = a 3. (Inverse) For every element a S there exists an special element a 1 S such that a a 1 = 1 Note that symbols used are arbitrary, we could have used a multiplicative notation ab, or a b in which case the identity would be 1 and the inverses denoted a 1 or an additive notation a + b, in which case the identity would be 0 and the inverse of a is denoted as a. Though additive and multiplicative notations are the most common, we could use also some other notation, so long as the axioms are satisfied and we have an agreed symbol for the identity and a notation for inverse. If the Group also satisfies 4. (Commutativity) a + b S it is called an Abelian Group or a Commutative Group. Example 2 1. Addition over the integers Z. This is a commutative group, 0 is the identity and a is the inverse of the integer a. Note that addition over the Natural numbers, N is not a group, since it satisfies neither Closure nor Inverse. 1
2. Multiplication over the rationals, Q {0}. This is a commutative group, 1 is the identity and a 1 is the inverse of the rational a. Note that multiplication over the integers, Z is not a group, since it satisfies neither Closure nor Inverse. 3. Note that multiplication over the integers Z is not a group as it doesn t satisfy Axiom 3 Existence of Identity. For example 3 1 = 1/3, but 1/3 Z 4. Similarly, the natural numbers are not a group with addition. For example 3 Z. 5. C 4, the cyclic group of order 4. C 4 = {ι, a, a 2, a 3 }, where a 4 = ι. Operations take place in the obvious multiplicative sense, so a a = a 2 etc. C 4 ι a a 2 a 3 ι ι a a 2 a 3 a a a 2 a 3 ι a 2 a 2 a 3 ι a a 3 a 3 ι a a 2 C 4 certainly satisfies Closure and Associativity, ι is the identity and given a i, its inverse is a 4 i. This is a commutative group. 6. Z 4 = {0, 1, 2, 3}, where 3 + 1 = 0. Addition Modulo 4. Operations take place in the obvious additive sense, so 2 + 1 = 3 etc. The sums wrap around when they go over 4 (arithmetic mod 4). Here 0 is the identity and a is 4 a. Z 4 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 Note that the previous two examples are essentially the same, we have just used different symbols and notation. We call such similar systems isomorphic. Definition 3 Given two groups G and H, we say G is isomorphic to H, denoted G = H, if there exits a bijection f : G H, such that a b = c in G if and only if f(a) f(b) = f(c) in H, i.e. f preserves the group operation. For example C 4 = Z4. 2
5. Dihedral group D 4 = {ι, a, a 2, b, ab, a 2 b}, where a 3 = b 2 = ι. This group has 6 elements. D 4 ι a a 2 b ab a 2 b ι ι a a 2 b ab a 2 b a a a 2 ι ab a 2 b b a 2 a 2 ι a a 2 b b ab b b ab a 2 b ι a a 2 ab ab a 2 b b a a 2 ι a 2 b a 2 b b ab a 2 ι a Given the axioms of a Group, we can prove some basic Theorems which give the basic algebraic properties of Groups. Since both addition and multiplication (without 0) are groups we can get two forms. Theorem 4 Let a, b and c be members of a Group, then the following hold: 1. (Cancellation) If a b = a c, then b = c. In particular, for addition this means that if a + b = a + c, then b = c and for multiplication if ab = ac, then b = c. 2. (Uniqueness of Solution) Given a and b there is exactly one c such that a c = b In particular, given a and b, for addition this means that there is exactly one c such that a + c = b, we call this c, b a. Note that b a = b + ( a) For Multiplication (a, b 0) this means that there is exactly one c such that ac = b, we call this c b/a. Note that b/a = ba 1. 3. (Double Inverse) (a 1 ) 1 = a and ( a) = a Finally, we have some special rules which relate specifically to the a b notation for rationals. Theorem 5 For any pair of rationals a b and c d the following hold: 1. (Addition of Fractions) 2. (Multiplication of Fractions) 3. (Division of Fractions) a b + c d ad + bc =, (b, d 0). bd a b c d = ac, (b, d 0). bd a b c d = ad, (b, c, d 0) bc 3
1.2 Fields Definition 6 Given a set with a commutative group +, with identity 0 and another commutative group, defined on S {0} with identity 1; it is called a field if it also satisfies the distributive axiom: (Distributivity) For all elements a, b, c of a field F Example 7 1. The Rationals Q. 2. The Reals R. 3. Integers modulo a prime. a(b + c) = ab + ac and (a + b)c = ab + ac Given the axioms of a Field, we can prove some basic Theorems which give the basic algebraic properties of Fields. Theorem 8 Let a, b and c be members of a Field, then the following hold: 1. 0a = a0 = 0. 2. If ab = 0 then either a = 0 or b = 0. 3. ( a)b = (ab). 4. a(b c) = ab ac. 1.3 Ordered Fields On certain fields we may define an ordering. To do this we define a certain subset numbers to be positive. Ordered Fields are Fields, so they obey the Field axioms above and in addition they obey the following order Axioms relating to positive numbers. Let a and b be members of a field then: 1. If a and b are positive, then so are a + b and ab. 2. For every non-zero number a, either a is positive or a is positive, but not both. 3. 0 is not positive. Of course theses axioms are actually satisfied by the natural numbers, the integers, the rationals and the real numbers. Any field which satisfies the order axioms is called an ordered field. We may now define the terms <, > and negative from these order axioms. 4
Definition 9 a < b means that b + ( a) is positive. If a < 0 we say that a is negative. a > b means that b + ( a) is negative. a b means that either a > b, or a = b. From these definitions and axioms it is possible to prove some basic Theorems which give the algebraic properties of <. Theorem 10 Let a, b and c be members of an ordered field, then the following hold: 1. (Trichotomy) Exactly one of the three relations a > b, a < b or a = b holds. 2. (Transitivity) If a < b and b < c then a < c. 3. If a < b then a + c < b + c. 4. If a < b and c > 0 then ac < bc. 5. If a 0 then a 2 > 0 6. 1 > 0 7. If a < b and c < 0 then ac > bc. 8. If a < b then a > b. Note that this means that if a < 0 then a > 0. 9. If ab > 0 then either both a and b are positive, or both are negative. 10. If a < c and b > d then a + b < c + d. 1.4 Least Upper Bound and the Reals But what of the real numbers? Up to this point we have defined only the rational numbers. In order to generate the real numbers we need one more axiom which relates to the infinitesimal or continuous nature of the reals. In order to state this axiom we need another definition: For any set of (rational) numbers S, x is said to be a least upper bound of S if for every element a of S, x a and x is less than every other number which satisfies this property. It should be noted that while all the elements of S may be rational the least upper bound may not be. There are examples of sequences of rational numbers which tend (monotonically) to a given non rational number. Thus, though all the elements of the sequence are rational, the least upper bound is not. These sequences of rational numbers can be used to define the irrational number to which they tend. In order to formalise this notion we introduce one final axiom. 5
LUB Least Upper Bound Axiom Any nonempty set of numbers that is bounded above has a least upper bound. This axiom was first introduced by J.W.R. Dedekind (1831-1916) in 1872, though he may have had the idea as early as 1858. Dedekind came to this idea by considering the correspondence of the real numbers to geometric lines, and then considering the notion of continuity. He came up with the following idea. For every division of the rational numbers into two classes such that all members of the first class are less than all the members of the second class there is one and only one (real) number which produces this cut. These cuts are known as Dedekind cuts. Thus 2 produces a unique cut of the rationals, which is not producable by any other number. All the rationals on one side have square less than two, on the other they have square greater than two. To get the above axiom consider the class which is less than the cut, this class has least upper bound equal to the cut. 6