EXAMPLE 12 Doubling on a “Stone Age” Calculator

Imagine that you have a calculator that keeps only the last two digits of a number. It has a special key marked [DBL] that doubles the number in the display and keeps only the last two digits. For example, [DBL] applied to 52 gives 04 (not 104). Let’s start with two numbers that are as close together as can be on this calculator, such as 37 and 38. As we push the [DBL] key over and over again, will the result stay close?

We get 37, 74, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, … 38, 76, 52, 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, … Already, by the fourth iteration, the two sequences are far apart. The function used in this iterated function system is

where the mod notation of modular arithmetic (introduced in Chapter 17 on page 715) means to take the remainder when 2x is divided by 100.