In the beginning were the natural numbers: 0, 1, 2, 3, and so on. Known as ℕ, the land that brought them together seemed a model of stability, since adding or multiplying two natural numbers always produced another natural number.
The land of integers
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Of course, the need to calculate an expression such as 4 – 9 had led to an expansion of the borders. This had given rise to ℤ, the land of integers. In addition to the natural numbers, this new land contained their opposites: –1, –2, –3… They had proved highly compatible with the original inhabitants, so after a few teething troubles, caused mainly by the rule of signs, the newcomers were accepted for good. They found a place in weather reports giving winter temperatures, in lifts descending to underground car parks, and on certain bank statements (where, in this last case, they were not always welcome).
From then on, just as with addition and multiplication, subtraction never led beyond the borders. The country's system of government could change along with its population: until then an additive monoid and a multiplicative semigroup, it became a commutative ring.
Once stability had been achieved for subtraction, the inverse operation of addition, a similar demand was made for division, the inverse operation of multiplication. Yet it soon became clear that dividing 7 by 3, or 9 by 5, could not be done within the borders of ℤ. The few more favourable cases, such as dividing 8 by 4 or 6 by 2, did nothing to calm the unrest, since division usually did not come out evenly.
