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Rational numbers (ℚ) are a foundational element in mathematics, forming a field with unique structural characteristics. They are countable, densely ordered, and serve as a model for countable ordered sets. ℚ is also extendable to p-adic numbers, highlighting its importance in number theory. The text explores ℚ's algebraic structure, order properties, countability, topological aspects, and its extension to p-adic numbers.

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## Definition and Properties of Rational Numbers

### Mathematical Field

Rational numbers are a mathematical field with specific axioms, allowing for the operations of addition and multiplication

### Additive and Multiplicative Inverses

Each rational number has an additive inverse and, with the exception of zero, a multiplicative inverse

### Unique Field Structure

The field of rational numbers has no non-trivial field automorphisms, making it unique in its structure

## Ordered and Dense Nature of Rational Numbers

### Ordered Field

Rational numbers form an ordered field, allowing for comparison between any two numbers

### Density of Rational Numbers

Rational numbers are infinitely dense, with an infinite number of numbers between any two given rational numbers

### Mapping onto Rational Numbers

Any countable, densely ordered set without endpoints can be mapped onto the rational numbers while preserving the order

## Countability and Unique Representation of Rational Numbers

### Countably Infinite Set

The set of rational numbers can be put into a one-to-one correspondence with the set of natural numbers

### Square Lattice Representation

Rational numbers can be represented as points on a square lattice, but this representation includes duplicates

### Calkin-Wilf Tree and Stern-Brocot Tree

These trees are structured to produce each rational number exactly once, providing a unique representation of rational numbers

## Topological Aspects of Rational Numbers

### Dense Subset of Real Numbers

Rational numbers form a dense subset of real numbers, meaning there are rational numbers in any interval of real numbers

### Topological Properties

Rational numbers possess an order topology, a subspace topology, and a metric topology, making them a topological field

### Incompleteness of Rational Numbers

Rational numbers are not locally compact and are not complete with respect to the absolute difference metric, with the real numbers serving as their completion

## Extension of Rational Numbers to p-adic Numbers

### Alternative Metrics

Rational numbers can be extended beyond the usual absolute value metric by introducing alternative metrics, such as the p-adic metrics

### p-adic Number Field

The completion of the p-adic metric on rational numbers yields the p-adic number field

### Ostrowski's Theorem

This theorem shows that any non-trivial absolute value on rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value, highlighting the significance of rational numbers in number theory

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