## 2.2 The Random Number Cycle

Pursuant to the above discussion, it is useful to present pseudorandom sequences in the context of their cyclical structure. Almost all random number generators have as their basis a sequence of pseudorandom integers (there are exceptions). The integers or ``fixed point'' numbers are manipulated arithmetically to yield floating point or ``real'' numbers. The random number cycle can be presented in terms of either integers or real numbers. Here, for clarity, we confine ourselves to integers.

Figure 2: Illustration of Random Number Cycle.

In Figure 2, we illustrate a random number cycle representing a sequence of 12 integers. Each black dot represents a distinct integer. Our convention will be to start at the top of the cycle (i.e. 12 o'clock), traverse the cycle clockwise, and finish at the integer just to the left of the start. The nature of the cycle illustrates that: (1) the sequence has a finite number of integers, (2) the sequence gets traversed in a particular order, and (3) the sequence repeats if the period of the generator is exceeded (i.e., the cycle can be traversed more than once). Furthermore, the integers need not be distinct; that is, they may repeat. We shall address this point subsequently. These all are properties of pseudorandom sequences of integers. In subsequent sections, we shall illustrate these aspects with specific examples.