rational number

From Anarchopedia

The rational numbers are the set of all fractions:

Q = {0, 1, -1, 1/2, -1/2, 2, -2, ...}

They are an Abelian group under addition, and, if {0} is removed from the set, they form an Abelian group under multiplication as well. Thus, the rational numbers form a field.

To make this concrete, we can construct a model for the rational numbers. First, for our purposes here, let us define

I = integers = { ... , -2, -1, 0, 1, 2, ... }

P = Positive integers = { 1, 2, 3, 4, ... }

Let the "<", "+", "x" operations on P be defined by restriction of the same operations on I.

Next, we form a set of ordered pairs of integers:

Failed to parse (Cannot write to or create math temp directory): \mathbb{Q}_0 \equiv \{(x,y) | x \in \mathbb{I}, y \in \mathbb{P} \}

Define an equivalence relation on the set:

Failed to parse (Cannot write to or create math temp directory): (x,y) \sim (z,w)\ \equiv\ \exists k \in \mathbb{I} \left ( x = kz \and y = kw \right )

This just says that all "equivalent" fractions, which differ only by a common factor in the numerator and denominator, are actually equal. We then form the set

Failed to parse (Cannot write to or create math temp directory): \mathbb{Q} = \mathbb{Q}_0 / \sim

This is similar to discarding everything from the set except the "reduced fractions" -- those for which the numerator and denominator have no common divisor.

If we use [q] to represent the equivalence class of q, where Failed to parse (Cannot write to or create math temp directory): q \in \mathbb{Q}_0 , then by using the relation "<" defined on the integers, we can define "<" in Q as the relation:

Failed to parse (Cannot write to or create math temp directory): [(x,y)] < [(z,w)]\ \equiv\ xw < yz

Multiplication is also easy to define:

Failed to parse (Cannot write to or create math temp directory): [(x,y)] \cdot [(z,w)] \equiv [(xz,yw)]

And addition is nearly as simple:

Failed to parse (Cannot write to or create math temp directory): [(x,y)] + [(z,w)] \equiv [(xw + zy, yw)]

It is straightforward to prove the axioms which describe the rational numbers within this model, thus showing that it is, indeed, a model for the rationals.

Finally, as an aside, we exhibit a very small theorem. Define the set of all reduced fractions:

Failed to parse (Cannot write to or create math temp directory): F = \{(x,y) \in \mathbb{Q}_0\ | \ \not\exists k \in \mathbb{I} \ (k \neq x \and x = ky) \}

Then given any element Failed to parse (Cannot write to or create math temp directory): (x,y) \in \mathbb{Q}_0 , we will show that there exists a reduced fraction, Failed to parse (Cannot write to or create math temp directory): f \in F , such that [f] = [(x,y)]. Consider the set of all denominators for elements of Failed to parse (Cannot write to or create math temp directory): \mathbb{Q}_0

which are equivalent

to (x,y):

Failed to parse (Cannot write to or create math temp directory): D = \{w \in \mathbb{P} \ |\ \exists z \in \mathbb{I} \left ([(z,w)] = [(x,y)] \right )\}

Then Failed to parse (Cannot write to or create math temp directory): D \subset \mathbb{P} . But P is well ordered so D must have a smallest element. Call that element W, and the corresponding numerator Z. Then (Z,W) must be a reduced fraction which is equal to (x,y).

The rational numbers, in addition to being important in and of themselves, are one of the building blocks we use to construct the real numbers.