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# What Is a Number System? Part B: Comparing Number Systems (15 minutes)

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
a. In what ways does addition act the same way in the finite system as it does in the infinite, real number system? Which rules are the same, and which are different?

Problem B2
a. In what ways does multiplication act the same way in the finite system as it does in the infinite, real number system? Which rules are the same, and which are different?

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?

### Solutions

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
a. The rules are similar. Multiplication is closed, commutative, and associative. There is always exactly one answer to a multiplication problem. Multiplying by 0 results in 0, and multiplying by 1 results in the original number. One difference is that sometimes, in a finite system, you can multiply the same (non-zero!) number by two different numbers and get the same answer (for example, 4 • 2 and 4 • 7 both equal 8).
b. The rules are not as similar. In the finite system, some division problems have no solution (74, for example), though this is not very different from whole numbers. However, some division problems have more than one solution (84, for example, which in units digit arithmetic can be solved by both 2 and 7). Therefore, you cannot divide by 0, 2, 4, 5, 6, or 8 in this system. In the real number system, we can divide by all non-zero numbers. Also, in the finite system, as in the real number system, you can’t make sense of 0 divided by 0 because there are too many solutions, and you can’t make sense of number a divided by 0 for any other number a, because there are no solutions.

Problem B3
a.
Yes, it acts in the same manner.
b.
The distributive law ties together addition and multiplication: a • (b + c) = (a • b) + (a • c). To consider a system with two operations, we must have a property that tells us how the two operations are related. Otherwise, we would not know how to compute an expression that contains both addition and multiplication.
Similarly, in the real number system, the distributive law allows us to see how these operations relate to each other. It also allows us to use different methods to compute products. For example, if you need to multiply 25 by 99, the distributive law allows you to do the computation as 25 • (100 – 1), which you can do in your head.

Problem B4