By means of somewhat lengthy reasoning, you can find out how to do arithmetic with integers. But are the regularities observed about the whole number system (the rules in Box 3–1) still valid? Going through the cases again will show that they are. So not only has the number system been extended from the whole numbers to all integers, but the arithmetic in the larger system looks very similar to arithmetic in the original one in the sense that these laws are still valid.
The extension of whole numbers to integers is an example of the axiomatic method in mathematics: basing a mathematical system on a short list of key properties.
Something much more dramatic is also true. One can show that, if the goal is to extend addition and multiplication from the whole numbers to the integers in such a way that the laws of arithmetic of Boxes 3–1 and 3–2 remain true, then there is only one way to do it. And the rules in Box 3–3 describe how it has to work. Recipes laboriously constructed by means of some sort of concrete interpretation of negative numbers are all completely dictated by this short list of rules of arithmetic. This uniqueness is a striking exhibition of the power of these rules—that they capture in a few general statements a large chunk of people’s intuition about arithmetic. The extension of whole numbers to integers is an example of the axiomatic method in mathematics: basing a mathematical system on a short list of key properties. Its most famous success is the Elements of Euclid for plane geometry. Since Euclid’s time, axiomatic schemes have been constructed to cover most areas of mathematics.
Another rather striking thing has happened during this extension from whole numbers to (all) integers. The reason for making the extension was to
Box 3–2 Additional Properties of Addition
Additive identity. Adding zero to any number gives that number. For example, 3+0=3 and 0+3=3. In general, m+0=m, and 0+m=m.
Additive inverse. Every number has an additive inverse, also called an opposite. The opposite is the unique number that, when added to that number, gives zero. For example, the opposite of 3 is –3 because 3+–3=0; the opposite of –4 is 4 because –4+4=0. In general, –s is the unique solution m for s+m=0.