The impossiblility of certain subtractions is an ancient crossroads in numeration. One choice is to stick to natural arithmetic, ignoring such meaningless inquiries. The other is to set aside practical implications and build a new and more complete system of unnatural numbers, closed under all possible operations. The latter route is the revolutionary one: we leave arithmetic behind and begin to do mathematics.
Since no existing number satisfies:
We can invent one that does, and add it by decree to the system of natural numbers we already have. It's a number not found in nature, but in our imaginations; call it i. So:
We claim that this imaginary quantity can be added it to a "real" one to get an real result, since it must be that:
In fact, this is philosophically dangerous ground, debated hotly in Europe through the seventeenth century. Nevertheless we can't stop there. By the rules of multiplication, there must also exist an infinitude of other imaginary values:
All of these new values arise inevitably from the single innovation of i; all must become part of our system.
This infinite extension of the natural numbers forms the integers. The integers include all the old natural numbers as well as all multiples of i. Formally, an integer is:
where a and b are any of the familiar natural numbers. An integer can be entirely natural, such as:
which is written 2. It can be entirely imaginary, such as:
which represents the value 13i. Or it can mix natural and imaginary:
But regardless of the form they take, all these are now legitimate values in our new system.