Session 1, Part A:
A Simpler Number System (70 minutes)

In This Part: Addition in Units Digit Arithmetic | Inverse, Identity, and Closure
Multiplication in Units Digit Arithmetic | Multiplicative Inverse and Identity

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.

Problem A1

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.

 + 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Problem A2

 a. What patterns do you observe in this addition table? b. What do you think is responsible for the diagonal patterns you see?

 Pick two numbers, one from the top horizontal row and the other from the vertical column on the far left, and add them. See where you land on the table. Then consider what happens if you move one up from the original number in the vertical column and one to the right in the horizontal row.    Close Tip Pick two numbers, one from the top horizontal row and the other from the vertical column on the far left, and add them. See where you land on the table. Then consider what happens if you move one up from the original number in the vertical column and one to the right in the horizontal row.

 Video Segment In this video segment, Rhonda and Monique discuss patterns they noticed in the finite system. Next, the whole class begins to think about why those patterns occur. Watch this segment after you have completed Problems A1 and A2, and then compare your findings with those of the onscreen participants. Did you notice any additional patterns? Can you explain why they occur? If you are using a VCR, you can find this segment on the session video approximately 4 minutes and 16 seconds after the Annenberg Media logo.

 Session 1: Index | Notes | Solutions | Video