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Just a start on the Algebraic properties of rationals, I hope to write about extensions (very very basic Galois theory) and their role in solving polynomials. This edit can probably be cleaned up.
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology.
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology.
==Algebraic Properties==
The algebraic implications of the rational numbers are both interesting and far reaching. The rational numbers represent a specific example of a [[field (mathematics)|Field of Fractions]] over the [[Integers]]. The [[integers]] constitute a [[ring]], there are two defined operations - addition and multiplication, which follow a certain set of rules (or as Mathematicians would say - have a certain structure). Specifically, the integers are an integral domain, a special type of ring with no zero divisors (There is no integer <math>a,b\neq 0</math> such that <math>a*b = 0</math>) and where the multiplication is commutative. The integers, however, are not a [[field (mathematics)|field]] because not every integer has an integer multiplicative inverse (in fact, only the number 1 and -1 do) .
As mentioned above, we can define an [[equivalence class|equivalence relation]] on (<math>\mathbb{Z}\times\mathbb{Z}\backslash\{0\}</math>) (A set of pairs of integers, where the second number in the pair cannot be 0 - thus avoiding the issue of dividing by zero) where two tuples <math>(a,b)</math> and <math>(c,d)</math> (which can be written in the more familiar form <math>\frac{a}{b}</math> and <math>\frac{c}{d}</math>) are equivalent if <math>ad-bc = 0</math>. For example, the two fractions <math>\frac{1}{4}</math> and <math>\frac{2}{8}</math> are equivalent because <math>2*4 - 1*8 = 0</math>. When two pairs of numbers (or fractions) are equivalent we call them members of the same equivalence class. The Field of fractions is the [[set]] of these equivalence classes, it is a simple exercise to show that they constitute a field. Along with the equivalence class, we define the operations of addition (<math>\frac{a}{b} + \frac{c}{d} = \frac{ad+cb}{bd}</math>) and multiplication (<math>\frac{a}{b}*\frac{c}{d} = \frac{ac}{bd}</math>), these operations are intentionally designed to mimic the addition and multiplication of fractions often taught in gradeschool, however keep in mind that the underlying ring does not necessarily have to be the integers (this same formulation constructs a field for ''any'' ring). Since the [[integers]] are ''naturally embedded'' in its field of fractions (we define the trivial mapping that <math>a\in \mathbb{Z} \rightarrow (a,1) \equiv \frac{a}{1} \in \mathbb{Z}\times \mathbb{Z}\backslash\{0\}</math> we only need to show the existence of a multiplicative inverse for each member of the set (all the other rules are satisfied by the ring and integral domain structure of the integers). Since for every fraction <math>\frac{a}{b}</math> we can construct a multiplicative inverse <math>\frac{b}{a}</math>, we satisfy all the conditions for a [[field]] and complete the proof. If the field of fractions is taken over a bona-fide field we will just get back the same field (technically, a new field but which is isomorphic to the original) - so the field of fractions over the rationals is just the rationals again.
Therefore, the rational numbers (or following the definition above, the field of fraction over the integers) can be though of as a "completion" of the integers in the sense that we construct a field "from the ashes" of the ring of integers. It is also important to note that the rational numbers is the smallest field in which the integers can be embedded, in fact, for any arbitrary ring, its field of fractions constitute the smallest field into which the ring can be embedded. Since both the integers and the rationals are infinite in size, we define the notion of smallest to mean that there doesn't exist a subfield <math>F \subset \mathbb{Q}</math> such that <math>\mathbb{Z} \subset F</math>.
The concept of a field of fraction may seem heavy-handed for something as intuitive as rational numbers, however when we are dealing with very abstract rings that look very little like the integers we have been acquainted with since kindergarten its field of fractions may look very bizarre , it is useful that we can rely on our intuition of rational numbers to understand very abstract things.
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Calculus]]