18
Aug
Rational Number - Definition
Before we prove that there are infinitely many rational numbers, let us recall the definition of Rational Numbers.
A number that is of the form p/q where p and q are integers and q≠0 is called a rational number.
\( \begin{aligned} \frac{p}{q}\ \ \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0 \end{aligned}\)
Proof - Rational Numbers are Infinitely Many
Theorem: Prove that there are infinitely many rational numbers.
Proof: Let us assume that there are finitely many Rational Numbers.
\( \begin{aligned} &\Rightarrow \text{∃ a greatest rational number, say } x\\\\ &\Rightarrow x = \frac{p}{q} \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0 \end{aligned}\)
Let us get a new number say y by adding 1 to x.
\( \begin{aligned} y&=x+1\\\\ &=\frac{p}{q}+1\\\\ &=\frac{p+q}{q} \end{aligned}\)
\( \begin{aligned} &Now \ \ p+q \in\mathbb{Z}\\\\ &\Rightarrow y = \frac{p+q}{q} \ \ \text{is also a rational number}\\\\ &Now \ \ y > x \end{aligned}\)
This is a contradiction that x is the greatest rational number.
This means our assumption is wrong that there are finitely many rational numbers.
Hence we can conclude that there are infinitely many rational numbers.
