Real analysis/Rational Numbers

A Field

The set of integers $\mathbb{Z}$ and the operation of addition $+$ form a group, multiplication $\times$ lacks inverses. If we allow multiplication and addition to operate on $\mathbb{Z}$ we can define a set where every element except zero has a multiplicative inverse. This is the set of rational numbers.

Rational Numbers

The next standard extension adds the possibility of quotients or division, and gives us the rational numbers (or just rationals) $\mathbb{Q}$, which includes the multiplicative inverses of $\mathbb{Z}\setminus \{0\}$ of the form $\frac{1}{z}$ fractions such as $\frac{1}{2}$, as well as produts of the two sets of the from $\frac{z_1}{z_2}$ such as $\frac{64}{7},\frac{17}{16\times 10^5}$. 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 $\frac{a}{b}$. 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.

Exercises

1. Prove that there is no rational number whose square is 2.

See Also

• Abstract Algebra
  • Fields