Stated simply, a number system is a set of objects (often numbers), operations, and the rules governing those operations. One example is our familiar real number system, which uses base ten numbers and such operations as addition and multiplication. Another example is the binary number system, which uses binary addition and multiplication.
Gaining an understanding of the real number system's elements, operations, and rules is inherently difficult. One important reason for this is that the system has an infinite or unlimited number of elements. Although we use this system every day, we usually don't think much about it when we use it.
Before we begin to analyze the real number system, we will first examine a finite number system -- its elements (which, unlike the real number system, are limited in number), its operations, and the rules that govern it. You will see that this system follows some (but not all) of the same rules as the real number system.
To begin, suppose that when you add or multiply whole numbers, you only need to keep track of the units digit. Only the units digit of the original numbers affects the answer, and you record only the units digit of your answer.
Thus, we can think of this as a system that includes only the digits 0, 1, . . ., 9 and, for now, only the operations of addition and multiplication. Let's see what patterns emerge as we explore this finite number system.
In units digit arithmetic, 9 + 5 = 4 (as opposed to our regular system, in which 9 + 5 = 14), because we are only interested in the units digit. Fill in the addition table below, using units digit arithmetic.