It is common for us to write numbers as finite sequences of digits. Here, we don’t care about the actual symbol (like ) we use for a digit but only that we have finitely many of them. So for now, we assume that we have many digits and, for the sake of simplicity, we identify them with the natural numbers where is called the base of a specific positional number system. In our daily lives we use base 10 systems, a binary computer uses a base 2 system.

Formally, a sequence of digits  of length  is an -tuple of digits, i.e. an element of the set . To simplify the presentation, we write an -tuple shortly as .

Up to now a sequence of digits is just a bunch of symbols and does not have a particular meaning. The following definition defines a function, called the unsigned interpretation that associates a natural number to each finite digit sequence.

is called the most significant digit and the least significant digit.

Given two digit sequences and , we can concatenate them and denote this by juxtaposing them . We abbreviate the digit sequence by .

The definition of matches our intuition and coincides with the way we use numbers in our daily life. However, we might be interested in demonstrating that indeed each natural can be represented by a digit sequence. This is established by the next lemma.

Figure 1.1.4: This figure gives some intuition for Lemma 1.1.3. A digit sequence of a base  can be seen as a path in a tree in which each inner node has  children. The path shown in bold is the number . Each layer of tree edges corresponds to a position, less significant digits being further down. One can also see that using positional notation is very efficient because to represent a number  one only needs  many digits.