We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seein

We can informally derive other types of number by extending operations on our existing set of natural numbers, seeking a "closure" for that operation, and seeing if we get consistent results. So extend the concept of differences between natural numbers, by subtracting a large number from a smaller number and defining the result as belonging to a new class of number. Similarly, get fractions by defining non-integer ratios between integers, real numbers by defining non-fractional limits of infinite sums of fractions, imaginary numbers of defining non-real roots of polynomials with real co-efficients, and so on. Each time, we have an operation that works for some subset of our numbers, so we look at what does not work, and see if we can make it work anyway by defining the result of such an operation as a new type of number. And then we repeat with a different operation. But there are things we cannot consistently "extend", like division by zero, or 0^0, so we leave just those out.