Terminology
The term rational in reference to the set Q refers
to the fact that a rational number represents a ratio of two integers. In mathematics, the
adjective rational often means that the underlying field considered is the field Q of
rational numbers. Rational
polynomial usually, and most correctly, means a
polynomial with rational coefficients, also called a "polynomial over the
rationals". However,
rational function does not mean
the underlying field is the rational numbers, and a rational algebraic curve is notan algebraic curve with rational coefficients.
Source: wikipidia
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Getting the rationals from the
integers
A
diagram showing a representation of the equivalent classes of pairs of integers Wikipedia.
Our next task is to define the set of rational numbers
from the integers using equivalence classes of pairs of integers.
The idea is clear: we think of a pair of integers (p,
q) as the fraction p/q and use an equivalence relation to identify fractions
that should have the same values.
The Rational Numbers by Ron Freiwald: pdf
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The field Q
In this section we are going to construct the rational
number from the integers. Historically, the positive rational numbers came first:
the Babylonians, Egyptians and Greeks knew how to work with fractions, but
negative numbers were introduced by the Hindus hundreds of years later. It is
possible to reflect this in the build-up of the rationals from the natural
numbers by first constructing the positive rational numbers from the naturals,
and then introducing negatives (Landau proceeds like this in his Foundations of
Analysis). While being closer to history, this has the disadvantage of getting
a ring structure only at the end.
The Field Q of Rational Numbers by Franz Lemmermeyer:
pdf
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More understanding:
Constructing Integers and Rational Numbers by Jeff
Thunder: pdf
The Rational Numbers: pdf
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