Mathematicians have always been interested in solving equations. Over the past 150 years they have studied techniques for solving equations, properties of operations that allow one to develop strategies for solving equations, and, eventually, entire systems in which one can calculate, and hence solve, equations. Note 4
These algebraic structures have become the primary focus of modern algebra. An algebraic structure is a collection of objects and operations that can be used to calculate and solve equations. The objects can be numbers, polynomials, geometric figures, points in space, card shuffles, or just about any mathematical object you can think of. The operations are usually binary operations, operations that combine two objects and form another of the same type.
Examples of systems include the system of integers and the system of rational (whole and fractional) numbers. Here, the operations are the usual operations of arithmetic -- addition, multiplication, etc. The structural approach to algebra has enormously widened the kinds of systems in which algebraists work, and hence has changed the face of what's considered "algebra."
Algebraic structures come up naturally in mathematical investigations. In Part C, we will investigate units digit arithmetic. Our goal here is to look at the underlying structure of this arithmetic, not just the calculations involved in it. Suppose, for example, that you are looking at the last digit, or units digit, of whole numbers. (Note: In Parts C and D of this session, an "*" is used to represent multiplication.)
Find the units digit of:
(22 * 43 + 59 * 27) * (47 + 1,432 * 268 * 21,343)
One way to do this is to carry out the entire calculation and then to look at the units digit. But there's no need for that much work; you can predict what the units digit will be without making the explicit calculations. For example, the units digit of 22 * 43 will be 2 * 3 = 6. And the units digit of 59 * 27 will be the units digit of 9 * 7 (that is, 3), so (22 * 43 + 59 * 27) will have the same units digit as 6 + 3. In other words, you can replace the numbers in the calculation by their units digits, turning the very large problem into a more manageable one:
(2 * 3 + 9 * 7) * (7 + 2 * 8 * 3)
Then, you can simplify as you go, so that, for example, (2 * 3 + 9 * 7) becomes 6 + 3, which becomes 9. These calculations depend upon order of operations. Look at the tip in Problem C1 below if you are unfamiliar with this concept.