1. The Game of Algebra An Introduction The Game of Algebra An Introduction © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 9

2. In Lesson 1, we discussed how to develop a strategy that would allow us to paraphrase an algebraic equation into the form of a simpler numerical equation. To paraphrase an equation means to change the wording of the equation, without changing the meaning of the equation. To make sure of not changing the meaning, we had to conform to various generally accepted rules of logic (such as If a = b and b = c then a = c). © 2007 Herbert I. Gross next

3. Whenever rules and strategy are involved in a process, we may view the process as being, in a manner of speaking, a game. Therefore, in this Lesson we examine algebra in terms of its being a game. © 2007 Herbert I. Gross next

4. When you first learn a game, the strategies are relatively simple, but as you become a more advanced player, the strategies you need in order to win become more complicated. In this sense, this lesson begins to prepare us to study more complicated algebraic expressions and equations. So with this in mind, Lesson 9 begins with a discussion of what constitutes a game. © 2007 Herbert I. Gross next

5. For example, chess is a game, baseball is a game, and gin rummy is a game. What is it that these three very different games have in common that allows us to call each of them a game? More generally, what is it that all games have in common? Our answer, which will be developed in this Lesson, is… © 2007 Herbert I. Gross next

6. Every game has its own pieces, and we can't even begin to play the game unless we know the definitions (vocabulary) that describe these pieces. © 2007 Herbert I. Gross next The Game Every game has its own rules that tell us how the various pieces that make up the game are related. And every game has winning as its objective; but where winning means that we have to do it in terms of the rules of the game. The process of applying the definitions and the rules to arrive at a winning situation is called strategy. next

7. © 2007 Herbert I. Gross next We define a game to be any system that consists of definitions, rules and the objective. Defining a Game The objective is carried out as an inescapable consequence of the definitions and rules, by means of strategy. next

8. In terms of a diagram… © 2007 Herbert I. Gross next The Rules of the Game (tells us how the terms are related). The Definitions or Vocabulary (tells us what the terminology means). Apply strategy (to the definition and rules). The Objective (to win). next

9. The interesting part about this definition is that it allows us to view almost anything as a game. © 2007 Herbert I. Gross next For example, in any academic subject we have terminology, rules, and an objective. In creative writing, the objective is to use vocabulary and the rules of grammar to communicate in exciting ways with our fellow human beings. next

10. © 2007 Herbert I. Gross next In economics, the objective is to use various accepted principles (which become the accepted rules) to help foster economic growth and stability. In psychology, the objective is to explain and/or account for human behavior in terms of certain observations (rules) that people have developed through the years. next A nd in algebra, the objective is to use various self evident rules of arithmetic in order to paraphrase more complicated expressions into simpler ones. next

11. Even such subjective topics as religion may be viewed as games. © 2007 Herbert I. Gross next For example, in any religion, the objective is to lead what the religion defines as a good life by applying certain rules (often called tenets or dogma) that we accept in that religion.

12. In this sense, life itself may be viewed as a game. That is, each of us accepts certain definitions and rules, and our objective is to lead what we define to be a rewarding life. What makes the game of life even more complicated is that the rules of the game include not only our own but also those of our family, our church, our society and so on. It often requires compromise and sacrifice in order to balance all the sets of rules under which we have to live. © 2007 Herbert I. Gross next

13. All we ask of our rules is that they be consistent. For example, in baseball there is a rule that says 3 strikes is an out. This rule was arbitrary in the sense that 4 strikes is an out could have been chosen just as easily. But what cant be allowed is to have both of these be rules in the same game. Otherwise, the game would be at an impasse the first time a batter had 3 strikes. Note © 2007 Herbert I. Gross next

14. Since we want the results of our game of mathematics to apply to the real world, we must choose our rules of mathematics to be those which we believe are true in the real world. © 2007 Herbert I. Gross next In other words, the rules must be consistent in every game. In addition, when we come to a game that is based on helping to explain the real world, our rules must also be chosen so that they conform to what we believe to be reality. next

15. The problem here is that what one person calls reality may not be what another person calls reality. © 2007 Herbert I. Gross next So whenever possible, what we do is to choose as our rules only those things that everyone is willing to accept. In certain situations, this is not always possible. (Perhaps this is why people often say that its a bad idea to discuss religion or politics.) next

16. © 2007 Herbert I. Gross next The pieces are numbers, and the rules tell us how we may manipulate these numbers. In mathematics, our rules are usually called axioms. A major objective of algebra is to solve equations. next In terms of our general definition of a game; algebra is a game in which… The strategy is usually to paraphrase, if possible, complicated mathematical relationships into simpler, but equivalent ones. next

17. There are times when two relationships may look alike and still be different while at other times, two relationships may look different and yet be equivalent. For example, look at the following three recipes… © 2007 Herbert I. Gross next 1. Start with (x) 2. Add 3 3. Multiply by 2 4. The answer is (y) Program 1 1. Start with (x) 2. Multiply by 2 3. Add 3 4. The answer is (y) Program 2 1. Start with (x) 2 Multiply by 2 3. Add 6 4. The answer is (y) Program 3 next A Major Difficulty

18. At first glance, Programs 1 and 2 may seem to look alike. In fact, the only difference between them is that we have interchanged the order of steps (2) and (3). © 2007 Herbert I. Gross next 1. Start with (x) 2. Add 3 3. Multiply by 2 4. The answer is (y) Program 1 1. Start with (x) 2. Multiply by 2 3. Add 3 4. The answer is (y) Program 2 next

19. On the other hand, Program 3 looks neither like Program 1 nor Program 2. For example, Program 3 contains the command Add 6; a command that is not part of either Program 1 or Program 2. © 2007 Herbert I. Gross next 1. Start with (x) 2. Add 3 3. Multiply by 2 4. The answer is (y) Program 1 1. Start with (x) 2. Multiply by 2 3. Add 3 4. The answer is (y) Program 2 1. Start with (x) 2 Multiply by 2 3. Add 6 4. The answer is (y) Program 3 next Add 6

20. © 2007 Herbert I. Gross next 1. Start with (x) 2. Add 3 3. Multiply by 2 4. The answer is (y) Program 1 1. Start with (x) 2. Multiply by 2 3. Add 3 4. The answer is (y) Program 2 1. Start with (x) 2 Multiply by 2 3. Add 6 4. The answer is (y) Program 3 next To see why they are not, lets see what happens when we replace x by 7 in both programs. next 7 10 20 7 14 17 next Although Programs 1 and 2 may seem to be equivalent, in reality they are not.

21. © 2007 Herbert I. Gross next 1. Start with (x) 7 2. Add 3 10 3. Multiply by 2 20 4. The answer is (y) 20 Program 1 1. Start with (x) 7 2. Multiply by 2 14 3. Add 3 17 4. The answer is (y) 17 Program 2 1. Start with (x) 2 Multiply by 2 3. Add 6 4. The answer is (y) Program 3 next Equivalent next If Programs 1 and 2 were equivalent, we would not have been able to get different outputs (20 and 17) for the same input (7). However, when x is replaced by 7 in Program 3, we obtain the same result as we did in Program 1. 20 17 7 14 20 next

22. © 2007 Herbert I. Gross next However, what we shall eventually show is that by applying the traditional rules of arithmetic, we can prove that Program 1 and Program 3 are equivalent, and therefore it was not a coincidence that we obtained the same output in both these programs when we started with 7 as the input. The fact that we got the same output in Programs 1 and 3 when we started with 7 could have been a coincidence. Caution

23. In order for two programs to be equivalent, they must be two different ways of saying the same thing. Therefore, if even one input gives us a different output in the two programs, the two programs are not equivalent. Summary © 2007 Herbert I. Gross next In other words, in order for two programs to be equivalent, the output we get in one program for each input must always be the same as the output we get in the other program, for the same input.

24. © 2007 Herbert I. Gross next On the other hand, the fact that we got the same output in Programs 1 and 3 when the input was 7 is not sufficient evidence to prove that Programs 1 and 3 are equivalent. In this sense, this lesson and the next will describe how in certain cases (such as with Programs 1 and 3) we can tell for sure whether two programs are equivalent, without our having to resort to either hunches or trial and error.

25. © 2007 Herbert I. Gross next In the meantime, there are some fairly straightforward demonstrations that most of us would accept for showing that Programs 1 and 3 are equivalent. 1. Start with (x) 7 2. Add 3 10 3. Multiply by 2 20 4. The answer is (y) 20 Program 1 1. Start with (x) 7 2. Multiply by 2 14 3. Add 6 20 4. The answer is (y) 20 Program 3

26. © 2007 Herbert I. Gross next One such way is quite visual. Namely, we can use x + 3 = next 2(x + 3) = 2x + 6 = Program 1 Program 3 next to stand for whatever number we want x to be, and to stand for 1. In this way… next

27. © 2007 Herbert I. Gross next we see that for any given input, the output in Programs 1 and 3 is always 3 more than the output in Program 2. This agrees with our earlier result when we got 20 as the output in Programs 1 and 3; but 17 as the output in Program 2 when the input was 7. Comparing Programs 1 and 3 ( ) with Program 2 next

28. © 2007 Herbert I. Gross next However, not all programs are this simple; and as the programs become more complicated, so also would their visual representations. This is one reason why our game of algebra is so important. Important Note

29. © 2007 Herbert I. Gross next By using the rules of the game of algebra, we have a relatively simple, logical way to decide when two relationships are equivalent and when they aren't. Moreover, these same rules often help us reduce a complicated relationship to a simpler relationship that is easier for us to analyze.

30. © 2007 Herbert I. Gross next We shall show what this means in more detail throughout the rest of this course. More specifically, we will begin to define the rules for the game of algebra in this lesson and conclude this discussion in Lesson 10. In Lesson 11, we shall apply the results obtained in Lessons 9 and 10 toward solving more complicated algebraic equations.

31. © 2007 Herbert I. Gross next The first thing we have to notice is that people view numbers in different ways. One person may view a whole number as being a certain number of tally marks (for example, |||). Another person may view it as being a length (for example, ). What we must do is to be sure that any rules that we accept are independent of how we view a number. next

32. © 2007 Herbert I. Gross next For instance, when we say that two numbers are equal, we must make sure that this means the same thing to people whether they use tally marks, lengths or anything else to visualize numbers. So, we define equality by what we believe are its properties. As you look at the rules, ask yourself whether your own definition of equality meets the conditions stated in our rules (axioms).

33. © 2007 Herbert I. Gross next (In these axioms, a, b, and c stand for numbers) E1: a = a (the reflexive property). AXIOMS OF EQUALITY In plain English, every number is equal to itself. However, not every relationship is reflexive. For example, the relationship is older than, is not reflexive because it is false that a person is older than him/ herself. However, is the same age as is reflexive, because a person is the same age as him/herself. next

34. © 2007 Herbert I. Gross next E2: If a = b then b = a (the symmetric property). AXIOMS OF EQUALITY In plain English, if the first number equals the second number, its also true that the second number is equal to the first number. For example, the fact that 3 + 2 = 5 means that 5 = 3 + 2.

35. © 2007 Herbert I. Gross next E2: If a = b then b = a (the symmetric property). AXIOMS OF EQUALITY Not every relationship is symmetric. For example, is the father of is not symmetric because if John is the father of Bill, Bill is not the father of John. But the relationship is the same height as is symmetric, because if John is the same height as Bill, then Bill is the same height as John.

36. © 2007 Herbert I. Gross next E3: If a = b and if b = c, then a = c (the transitive property). AXIOMS OF EQUALITY In plain English, if the first number is equal to the second number and the second number is equal to the third number, then the first number is also equal to the third number.

37. © 2007 Herbert I. Gross next E3: If a = b and if b = c, then a = c (the transitive property). AXIOMS OF EQUALITY The relationship is taller than" is transitive. For example, if John is taller than Bill, and Bill is taller than Mary, then John is also taller than Mary. On the other hand, for example, is the father of is not transitive. That is, if John is the father of Bill, and Bill is the father of Mary, then John is the grandfather (not the father) of Mary.

38. © 2007 Herbert I. Gross next A relationship that has all of the above three properties (that is, it is reflexive, symmetric, and transitive) is a very special relationship, and it is given a special name. Namely, it is called an equivalence relation, and the set of members that obey this relationship is called an equivalence class with respect to this relationship. Thus, for example, equality (more specifically is equal to) is an example of an equivalence relation.

39. © 2007 Herbert I. Gross next In informal terms, if youve seen one member of an equivalence class youve seen them all. That is, with respect to this course, If a = b, then a and b can be used interchangeably in any mathematical expression that involves equality. In effect, this property encompasses the reflexive, symmetric, and transitive properties. So we often use it as a replacement for the other three properties. next

40. © 2007 Herbert I. Gross next E4: (the equivalence property) If a = b, then a and b can be used interchangeably in any mathematical relationship that involves equality. AXIOMS OF EQUALITY More specifically, we use the following property as a shortcut summary of E1, E2, and E3. next

41. When we talk about two things being equivalent, we always mean with respect to a given relationship. © 2007 Herbert I. Gross next An Important Warning Example: when the Declaration of Independence talks about all men are created equal, it does not mean that all men look alike or that all men have the same height or the same amount of money. Rather, its meaning is something like equal in the eyes of God. next

42. Rather, they are equivalent with respect to naming the number that we must multiply by 2 in order to obtain 6 as the product. © 2007 Herbert I. Gross next In a similar way, in arithmetic when we say such things as… 6 ÷ 2 = 12 ÷ 4, we do not mean that these expressions look alike. They dont! next

43. As a non-mathematical example, suppose we write… Mark Twain = Samuel Clemens © 2007 Herbert I. Gross next This does not mean that the two names look alike, but rather that they name the same person; Mark Twain is the pen name of Samuel Clemens! In other words, any statement that is true about the man, Mark Twain, is also true about the man, Samuel Clemens. Thus, the statements Mark Twain wrote Huckleberry Finn and Samuel Clemens wrote Huckleberry Finn are equivalent statements. next

44. © 2007 Herbert I. Gross next E4 is the logical reason behind the earlier rules we accepted, such as equals added to equals are equal". For example, suppose that we know that… Notes x = y …and we decide to add 3 to x. The left hand side of the equation becomes… x + 3 next

45. © 2007 Herbert I. Gross next When x = y, by E4 we may replace x by y in any mathematical relationship. Notes x + 3 The mathematical way of saying that x + 3 and y + 3 are equivalent is to write… y Replacing x by y in x + 3 gives us the equivalent expression… x + 3 = y + 3 next

46. If we now compare… x = y and x + 3 = y + 3, we see that in effect we simply added equals to equals to obtain equal results. © 2007 Herbert I. Gross next x = y + 3 = next

47. Of course, the rule that equals added to equals are equal may have seemed obvious to us without having to talk about E4. What E4 does for us, however, is to demonstrate that this rule is consistent with the rules in what we are calling the game of algebra. © 2007 Herbert I. Gross next Now that weve proved by E4 that equals added to equals are equal, we may use this rule in our new game, just as we did in Lesson 1. next

48. As the algebraic expressions and equations in our course become increasingly more complicated, some of our strategies will become less obvious. If at such a point a strategy is not obvious to all of the game players, it is our obligation to show them that the strategy is indeed a logical (that is, inescapable) consequence of the rules weve accepted. © 2007 Herbert I. Gross next Notes

49. In other words… © 2007 Herbert I. Gross next And it is the obligation of the instructor to ensure that each strategy employed by either the instructor or the students follows inescapably as a consequence of these definitions and the rules. It is the obligation of each student to accept the definitions and rules of arithmetic (and thats why we try to make them as self-evident as possible)… next

50. If we are to play the game of algebra correctly, the next challenge that confronts us is to define the properties (rules) that govern addition, subtraction, multiplication, and division of numbers. © 2007 Herbert I. Gross next More crucially, we have yet to answer the question… What is a number? All of this will be the topic of our next lesson. next