### 7 - Rational Numbers and Irrational Numbers

Real numbers that are of the form $\frac{m}{n}$ where $m$  and $n$  are integers, are called rational numbers.

Note that all integers are rational numbers.

The decimal representation of any rational number has finite number of digits after the decimal point or a block of consequtive digits reapeat for ever. There are simple ways to demonstrate how such a decimal number can be expresses as fraction $\frac{m}{n}$.

However not all real numbers are rational numbers. It can be easily shown that the number whose square is 2 is not a rational number. The real numbers that are not rational numbers are called irrational numbers.

There are infinitely many irrational numbers. As a matter of fact, there are 'more' irrational numbers than rational numbers.

Still the set of all rational numbers is densely spread over the set of all real numbers. What this means is that if you have an irrational number, then you can always find a rational number in close proximity to the irrational number. This close proximity between a given irrational number and the rational numbers sorrounding to it can be as small as desired.

Let $x$  be an irrational number and let $\epsilon$ be a very small positive real number, that is very close to the number 0. Then if we consider the interval $\left(x-\epsilon ,x+\epsilon \right)$ , then we will find infinitely many rational numbers in that interval. And, that is so even if $\epsilon$   is as small as we want.

What it all means is that if we have an irrational number, then we can always say it is approximatedly equal to a rational number. And, this approximation can be made as small as we desire.

Another way of thinking of it is considering the decimal representation of the irrational number and saying it is approximately equal to the decimal number that is obtained by trunncating the decimal representation of the irrational number to a finite number of digits.

In terms of Real Analysis, given any real number, in particular an irrational number $q$  , we can find an infinite sequnce of rational numbers ${\left\{{r}_{k}\right\}}_{k=1}^{k=\infty }$  such that $\underset{k\to \infty }{\mathrm{lim}}{r}_{k}=q$. What this means is that the infinite sequnce of rational numbers ${r}_{1}$ , $r$ , ${r}_{3}$ , $\cdot \cdot \cdot$   as we keep proceeding along the infinite seqence.