Real Analysis/Rational Numbers
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Before we can build up the notion of an ordered field, we first need some basic concepts from algebra.
Groups play an important role in mathematics. They describe the most basic structures in algebra. The subject of group theory studies the nature and structure of general groups. In this book we will mostly be concerned with groups you are already familiar with, so this section is just to set some standard terminology. We begin by defining a binary operation, which is often thought of as either multiplication or addition, depending on the context. To avoid confusion we will denote this operation by * while discussing general groups, but in specific cases we will generally use + or ·
Definition: A binary operation on a set S is a function from S×S→S
Definition: A group is a set G together with a binary operation on G that satisfies the following axioms.
- G is closed under the binary operation. That is, for all x, y in G, x*y is in G.
- The binary operation is associative. That is, for all x, y, and z in G, x*(y*z)=(x*y)*z.
- There exists an identity element, which we denote by e, that satisfies e*x=x*e=x for all x in G.
- For all x in G there exists an inverse element, which we denote by x-1, so that x*x-1=x-1*x=e.
- The integers together with the binary operation of addition are a group.
- The rational numbers with the binary operation of addition are a group.
- The non-zero rational numbers with the binary operation of multiplication are a group.
- The set together with the binary operation of multiplication is not a group.
- The set with the binary operation given by multiplication is a group.
- The set with a binary relation given by , , , and is a group. If one thinks of as a shorthand for even, and as a short hand for odd, these are the familiar rules from childhood "An even number plus an even number is again an even number", etc.
It is often useful to talk about when two groups are basically the same. It may happen that two groups have a different underlying set, and have a different binary operation, but behave exactly the same algebraically. When this happens the two groups are called isomorphic.
Definition The groups (G,*) and (H,⊗) are said to be isomorphic if there is a bijective function φ:G→H that satisfies the following two properties:
- φ(eG)=eH, where eG is the identity element in G and eH is the identity element in H;
- φ(x*y)=φ(x)⊗φ(y) for all x and y in G.
The set of integers and the operation of addition form a group, multiplication lacks inverses. If we allow multiplication and addition to operate on we can define a set where every element except zero has a multiplicative inverse. This is the set of rational numbers.
The next standard extension adds the possibility of quotients or division, and gives us the rational numbers (or just rationals) , which includes the multiplicative inverses of of the form fractions such as , as well as products of the two sets of the form such as . The rationals allow us to use arbitrary precision, and they suffice for measurement.
The rational numbers can be constructed from the integers as equivalence classes of order pairs (a,b) of integers such that (a,b) and (c,d) are equivalent when ad=bc using the definition of multiplication of integers. These ordered pairs are, of course, commonly written . One can define addition as (a,b)+(c,d)=(ad+bc,bd) and multiplication as (ac,bd) all using the definition of addition and multiplication of integers.