Folding numbers

This chapter contains some serious number theory. But, even if you aren't a number-theorist or do not have a special interest in number theory, we would encourage you to give this chapter some attention because it leads to surprising results that we could not have discovered had we not begun this exploration by folding paper! For example, the innocent-looking statement that 3 × 7 = 21 usually leads to two equivalent statements; namely that 21 ÷ 3 = 7 and 21 ÷ 7 = 3. But it turns out that, in an important and interesting sense, the first statement is very special while the second is merely one example of an infinite number of such statements.

Thus, even if you choose not to study the proofs of the theorems in this chapter in depth, you might get quite a bit of pleasure by seeing what the theorems say and looking at some examples of them.

Let us emphasize this point further. Many people – especially students – believe that a proof should be committed to memory, and try to do so. This is wrong. The purpose of a displayed proof is to convey the meaning of the statement of a theorem, and its significance. If it does not do that, it has failed to play its proper role in the student's education, however accurately it may be recorded in the student's memory.