ordered field
Imagine a number system where you can add, subtract, multiply, divide (except by zero), and also line every element up on a number line so it always makes sense to ask which of two numbers is bigger. An ordered field is exactly such a system: it has all the usual arithmetic and a notion of less-than that plays nicely with that arithmetic.
Formally, an ordered field is a field together with a total order such that two compatibility rules hold: if a is less than b then a plus c is less than b plus c for every c, and if a and b are both positive then their product is positive. From these you can derive every familiar sign rule, such as that a negative times a negative is positive, and that you may add inequalities or multiply them by a positive quantity.
The rational numbers and the real numbers are both ordered fields. The complex numbers are a field but cannot be made into an ordered field at all, because in any ordered field a square is never negative, yet i squared equals minus one. So being an ordered field is a real restriction, not an automatic feature of every number system.
In any ordered field, from 0 < 1 one can prove 1 < 1 + 1, then 1 + 1 < 1 + 1 + 1, and so on, so the field contains a copy of the natural numbers in strictly increasing order.
Order plus arithmetic forces a built-in copy of the naturals.
Being an ordered field is not enough to pin down the reals: the rationals are an ordered field too. What singles out the reals among ordered fields is completeness, the absence of gaps.