Three mathematical realms come together in Euler's formula: the arithmetic, the source of i; the analytic, based on e; and the trigonometric, centered on π. Arithmetic is the place to begin, building up to complex numbers from those that are the oldest, the purest, and the simplest possible.
The counting numbers (the integers from zero) are called natural because they are intimately associated with counting actual things. They begin with zero (for our purposes) and go upward by 1:
These are real numbers in the most literal sense; each one has direct concrete examples in the real world, embracing both the many (they never run out) and the few (they include a succinct symbol for nothing at all).
Inextricably associated with the natural numbers is the act of addition and its extension, multiplication. Addition is the hinge of the natural numbers. We can freely add any two of them and be sure of always arriving at another natural number. There are no gaps to be filled, and no edge to fall off. The natural numbers are closed under addition, a complete system and a perfect roadmap for counting things in the world.
There is grace and a kind of perfection in this system, admirable for its robustness and simplicity. It has been invented many times and in different ways, and has sustained itself over the centuries with no further need for elaboration. It is only on closer inspection that troubling questions arise.
Because the natural numbers are closed under addition, this kind of question always has an answer. But to the inquisitive, it's easy and often useful to ask a slightly different question:
This looks like a question of addition, but it's really a wholly new operation: subtraction, the complement of addition. The question of "what with five makes eight?" is really:
Subtraction arises naturally out of addition, but it's a different creature entirely. For unlike addition, it leads to questions with no answer:
"What takes one to make nothing?" There is no number in the system that satisfies this; the natural numbers are not closed under subtraction. Depending on the type of person you are, this is either a minor inconvenience, or a fundamental crisis.