ABSTRACT
10.1 The integers, like the natural numbers, are commonly used in everyday life. Everybody having to payback some debt has quickly understood the negative side of the existence of negative numbers. The set of integers will provide us with a first concrete example of an infinite group. We construct the set ℤ of integers from the set ℕ of natural numbers and endow it with usual addition and multiplication as well as an ordering compatible with these operations. Observe that here the second operation, called multiplication, is really controlled by addition because any product turns out to be a sum of a suitable number of times one of the factors. This is because ℤ will be a cyclic group (see Chapter 14). The presence of the multiplication provides extra information, for example via the definition of divisors and multiples, and the group structure interacts with this, e.g. see Proposition 10.23, and the role played by the greatest common divisor or the least common multiple.
