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How does the number system you’ve been working with in Part A relate to the real number system you usually use? Through analyzing a small, finite system, you’ve gained some understanding of number systems in general. Using units digit arithmetic, you’ve created an independent number system which was relatively easy to manage.

You’ve considered only the units digit for any one answer and in so doing have limited the size of your number system to just 10 numbers — 0 through 9. You’ve seen that the finite system has its own addition and multiplication tables, additive and multiplicative inverse and identity elements, and computational rules. You’ve seen that the finite system does not have unique inverse elements for multiplication, as does the system of real numbers. Furthermore, you’ve seen that the computational rules sometimes coincide with those of the real number system. Let’s explore some of the similarities and differences between the two systems.

**Problem B1
**

**Problem B2
**

**Problem B3**

**a.** Does the distributive law act the same way in the finite system as it does in the infinite, real number system?

**b.** Why do we need a distributive law?

**Problem B4
**In your opinion, what are the most important rules that apply to both this finite system and our infinite system?

**Problem B1**

**a.** The rules are very similar. Addition is closed, commutative, and associative; there is always exactly one answer to an addition problem. One difference is that the result of adding two numbers is not necessarily “larger” than the original numbers in the finite system; for example, 6 + 7 = 3. This means that the use of “larger” is not applicable to this system. Another difference is that this system has only 10 elements, whereas the real number system has an infinite number of elements.

**b.** The rules are very similar; there is always exactly one answer to a subtraction problem. One difference is that there are no “negative” numbers in the finite system. However, each number has an additive inverse, so if we interpret -b as the inverse of b, we can say that (a – b) is the same as (a + (-b)).

**Problem B2
**

**Problem B3
a. **Yes, it acts in the same manner.

b.

**Problem B4
**Answers will vary.