Session 9, Part C:
Algebraic Structures

In This Part: Units Digit | A New Algebraic Structure | Properties | More Properties

What we have been exploring is a "units digit arithmetic," an arithmetic whose objects are the digits 0 through 9, and whose operations are "add and take the units digit" and "multiply and take the units digit." The whole system can be captured in two tables (with some of the results left blank for you to complete later):

These tables define an algebraic structure on the set of numbers

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

You can forget that these tables came from "taking the units digit" and concentrate on the system they describe. There are sophisticated words for this process of "ignoring where it came from and concentrating on the behavior" like "abstraction" and "decontextualization." These are useful mathematical habits of mind. In some situations, you want to forget the meanings of these operations and just work with them. In other situations, it's just as important to be able to go back to the source of these things (to "contextualize" them) and to remember that these tables came from looking at units digits or remainders. Note 6

In some of the following problems, you can work with the tables without worrying where the operations come from. In others, especially when you need to think about why a property you see in the table holds, you'll need to go back to the context (taking units digits or taking remainders on division by 10). This fluctuation in and out of contexts is typical of the way mathematics is performed.

Problem C6

Fill in the missing entries in each table. As you complete the tables, look for patterns in the numbers you enter.

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

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

 Session 9: Index | Notes | Solutions | Video