One of my intuitions re numbers is that analysts worry about the high-order bits of numbers, while number theorists worry about the low-order bits. For example, some quick compositeness tests look only at the low-order digits of numbers. The usual GCD algorithm worries about reducing the length of the numbers, but the so-called "binary algorithm" (see Knuth if you need to know more about this) worries about reducing the number of non-zero bits in the numbers. If you really want to blow your student's mind, introduce her to p-adic numbers, which dramatically change the definition of "closeness".