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Section 3.2 The rational numbers

In \(\ZZ\text{,}\) we can add, subtract, and multiply, but we cannot divide. That is, the equation \(2x = 1\) has no solution in \(\ZZ\text{.}\) This issue can be summarized thus: \(\ZZ\) is not a field:

Definition 3.2.1. Fields.

A field \(F\) is a set along with distinct elements \(0,1\in F\) and maps

\begin{equation*} + \colon F \times F \to F \text{ and } \times \colon F \times F \to F \end{equation*}

such that the following axioms hold.

  1. Associative addition. For every \(x,y,z\in F\text{,}\) we have \((x+y)+z = x+(y+z)\text{.}\)

  2. Associative multiplication. For every \(x,y,z\in F\text{,}\) we have \((x\times y)\times z = x\times (y\times z)\text{.}\)

  3. Distributivity. For every \(x,y,z \in F\text{,}\) we have \(x(y+z) = xy+xz\)

  4. Additive identity. For every \(x\in F\text{,}\) we have \(x+0=x\text{.}\)

  5. Multiplicative identity. For every \(x\in F\text{,}\) we have \(x\times 1=x\text{.}\)

  6. Additive inverses. For every \(x \in F\text{,}\) there exists an element \(-x \in F\) such that \(x+(-x)=0\text{.}\)

  7. Multiplicative inverses. For every \(x \in F\text{,}\) if \(x \neq 0\text{,}\) there exists an element \(1/x \in F\) such that \(x\times(1/x)=1\text{.}\)

  8. Commutative addition. For every \(x,y\in F\text{,}\) we have \(x+y = y+x\text{.}\)

  9. Commutative multiplication. For every \(x,y\in F\text{,}\) we have \(x\times y = y\times x\text{.}\)

What we will do is define the smallest field that contains \(\ZZ\text{.}\) The basic idea is that we will want numbers of the form \(m/n\text{,}\) where \(m\) and \(n\) are integers, where \(n\neq 0\text{.}\)

Definition 3.2.2. Rational numbers.

Consider the equivalence relation \(\sim\) on \(\ZZ\times\NN\) in which \((m_1,n_1)\sim(m_2,n_2)\) if and only if \(m_1n_2=m_2n_1\text{.}\) A rational number is an equivalence class under this equivalence relation. We will write \(m/n\) on \(\frac{m}{n}\) for the rational number given by the equivalence class of \((m,n)\text{.}\) We will write \(\QQ\) for the set of rational numbers.

Definition 3.2.3.

Let \(\frac{m}{n},\frac{p}{q}\in\QQ\text{.}\)

We define the sum

\begin{equation*} \frac{m}{n}+\frac{p}{q} = \frac{mq+np}{nq}\text{.} \end{equation*}

We define the product

\begin{equation*} \frac{m}{n}\times\frac{p}{q}=\frac{mp}{nq}\text{.} \end{equation*}

Because \(\QQ\) is defined as a set of equivalence classes, it is necessary to confirm that these operations are well-defined; check this! With addition and multiplication defined above, \(\QQ\) is a field.