Folha: 'A phone number is not just a number!'
Reproduction of Marcelo Viana's column in Folha de São Paulo.
I believe that, for most people, a number is what we write with the symbols 1, …, 9, and 0. But that can't be right: the Romans (and many others) used numbers for centuries, and they didn't know the Indo-Arabic numerals we use today. What differentiates numbers from other forms of communication, like words, is not how we represent them, but what we can do with them, or better yet, what they can do for us.
To begin, (real) numbers can be ordered from smallest to largest. They share this characteristic with words, which can also be ordered, for example, alphabetically. But numbers are much more powerful, as they are capable of generating new numbers through mathematical operations—addition, subtraction, multiplication, division, etc.—that no one else possesses. This is what characterizes what it means to be a number. Thus, there are many things we write with the digits 0, 1, …, 9, but that doesn't make them worthy of being called "numbers."
Phone numbers are a good example. Ordering them makes no sense: if yours is 25295000, and mine is 25295001, that doesn't mean your line was created before mine, much less that my phone number is higher than yours. Worse, if we add the two numbers together, the result means nothing and may not even be a valid phone number!
Throughout history, the way we have developed increasingly larger sets of numbers from the natural numbers has always been to broaden the scope of operations to meet specific needs. Fractions so that it is always possible to divide (except by zero: there it's simply not possible!). Negative numbers so that it is always possible to subtract. Irrational numbers and, later, imaginary numbers, so that square roots always exist.
A crucial point is that extensions of the concept of number cannot be done haphazardly; they need to preserve the fundamental properties of the operations. For example, we learned in school that the multiplication of natural numbers is commutative: the result does not depend on the order of the factors. Well, commutativity continues to hold true for the multiplication of fractions, real numbers, and even complex numbers.
Every now and then someone writes to me saying they've discovered that the rule "minus times minus equals plus"—the product of two negative numbers is positive—is wrong, and proposing to develop (with funding from IMPA…) the "correct theory" of negative numbers. Well, a good theory is one in which the fundamental properties of addition and multiplication of natural numbers, such as commutativity and distributivity, continue to hold true for all integers. And these properties imply that "minus times minus equals plus," period.