_{1}

^{*}

It is widely held that irrational numbers can be represented by infinite digit-sequences. We will show that this is not possible. A digit sequence is only an abbreviated notation for an infinite sequence of rational partial sums. As limits of sequences, irrational numbers are incommensurable with any grid of decimal fractions.

Strictly monotonic sequences do not assume their limit. Rarely the terms of the sequence and its limit are confused. But this situation changes dramatically when sequences of partial sums of series are involved. It is customary in textbooks to identify the infinite sum over all terms of a series and the limit of this series [

In the following we will see that this is imprecise and point out an important consequence. A limit is not defined by the infinite sequence of partial sums because the sequence cannot be given in the necessary completeness. Only a finite formula can determine both the terms of the sequence of partial sums and the limit as well.

Theorem A non-terminating series of decimal fractions does not determine a real number.

Corollary A non-terminating digit sequence does not determine a real number.

Proof. The limit of a strictly monotonic sequence is not among its terms. Strictly monotonic sequences like

The same distinction has to be observed with series. There must not be a difference in the mathematical contents whether the partial sums are written separately like

or are written in one line with interruptions

or without interruptions

The infinite sequence of digits d_{n} is completely exhausted by all terms of the Cauchy-sequence of rational partial sums of decimal fractions. The intended meaning as a sequence of rational partial sums according to Eq-

uation (1) can be expressed also by

abbreviation: All partial sums are written in one and the same line without adding the limit. Equation (2) is the same because writing or not writing parentheses must not change the result. In all cases none of the

Digits are simply too coarse-grained to represent irrational limits of Cauchy-sequences.

To represent

This leads us to the often asserted double-representation of periodic rational numbers. For all

is 1. Since all

The usual proof for

As a result we can state that an infinite digit sequence like

partial sums of decimal fractions, also called an infinite series

eventually becoming constant).

A periodic decimal fraction has as its limit a rational number. A non-periodic decimal fraction has as its limit an irrational number. But it is not this number. In case of periodic decimal fractions it is possible, by changing the basis, to obtain a terminating digit sequence. Irrational numbers have no decimal expansion, no representation by digits or bits, not even by infinitely many. They are incommensurable with every rational-measure expanded by digits or bits. An irrational number requires a generating formula F in order to calculate every digit of the infinite digit sequence S and in addition to calculate the limit. The formula F may be interpreted as the number as well as the limit. It may be involved or as simple as “0.111…” which is a finite formula (consisting of eight symbols) allowing to obtain every digit of the infinite sequence converging to 1/9.

The implication

The mathematical facts discussed above also apply to all sequences of digits or bits appearing in the folklore version of Cantor’s diagonal argument [