Real Numbers, Sequences and Series: Part 7

Exercise.

Discover (and justify) an essential difference between the decimal expansions of rational and irrational numbers.

Giving a decimal expansion of a real number means that given $n \in N$ , we can find $a_{0} \in Z$ and $0 \leq a_{1}, \ldots, a_{n} \leq 9$ such that

$|x-\sum_{k=0}^{n}\frac{a_{k}}{10^{k}}|< \frac{1}{10^{n}}$

In other words if we write

$x_{n}=a_{0}+\frac{a_{1}}{10}+\frac{a_{2}}{10^{2}}+\ldots +\frac{a_{n}}{10^{n}}$

then $x_{1}, x_{2}, x_{3}, \ldots, x_{n}, \ldots$ are approximate values of x correct up to the first, second, third, …, nth place of decimal respectively. So when we write a real number by a non-terminating decimal expansion, we mean that we have a scheme of approximation of the real numbers by terminating decimals in such a way that if we stop after the nth place of decimal expansion, then the maximum error committed by us is $10^{-n}$ .

This brings us to the question of successive approximations of a number. It is obvious that when we have some approximation we ought to have some notion of the error committed. Often we try to reach a number through its approximate values, and the context determines the maximum error admissible. Now, if the error admissible is $\varepsilon >0$ , and $x_{1}, x_{2}, x_{3}, \ldots$ is a scheme of successive is approximation of a number x, then we should be able to tell at which stage the desired accuracy is achieved. In fact, we should find an n such that $|x-x_{n}|<\varepsilon$ . But this could be a chance event. If the error exceeds $\varepsilon$ at a later stage, then the scheme cannot be a good approximation as it is not “stable”. Instead, it would be desirable that accuracy is achieved at a certain stage and it should not get worse after that stage. This can be realized by demanding that there is a natural number $n_{0}$ such that $|x-x_{n}|<\varepsilon$ for all $n > n_{0}$ . It is clear that $n_{0}$ will depend on $varepsilon$ . This leads to the notion of convergence, which is the subject of a later blog.