1. Rates to Introduce Signed Numbers © Math As A Second Language All Rights Reserved next #9 Taking the Fear out of Math -3-3 + 3

2. Does it seem strange that the ancient Greeks were able to do extensive work with rational numbers (fractions) but did not know that negative numbers existed? The reason is that they viewed numbers as being lengths and no length could be “less than nothing”. That is, how could a piece of string be so short that even if it was 2 inches longer it would have “zero length”? next © Math As A Second Language All Rights Reserved

3. In text books, signed numbers are often introduced in terms of the number line (a special case of which is a liquid thermometer) or the business model (namely profit/loss). next © Math As A Second Language All Rights Reserved Specifically, a $3 loss is referred to as negative 3 while a $3 profit is referred to as positive 3 1. note 1 However the phrase “a $3 loss” takes away the need to use the word “negative”. In a similar way, saying “3 degrees below zero” takes away the need to talk about “negative 3 degrees”. next

4. In our earlier lessons we stressed the concept of rates. We may also apply the concept of signed numbers to rates. This is illustrated in the following problem. In a certain town the rate of change of the population is 5,000 persons per year. If the population of the town this year is 55,000 people, what was the population of the town last year? next © Math As A Second Language All Rights Reserved

5. The answer is either 50,000 or 60,000. We are not given enough information to arrive at a unique answer. Namely, we have to know whether the change in population represented an increase or whether it represented a decrease. If the population is decreasing at a rate of 5,000 persons per year, then a year ago the population would have been 60,000. However if the population is increasing at a rate of 5,000 persons per year, then a year ago the population would have been 50,000. next © Math As A Second Language All Rights Reserved

6. Thus, in terms of our study of rates, another way to visualize signed numbers is by having positive represent an increase and negative a decrease. 0 would represent the fact that no change took place. This gives us a “real world” reason as to why + 3 and - 3 are connected by the fact that + 3 + - 3 = 0. For example, if you make a $3 profit on one transaction ( + 3) and then you incur a loss of $3 on the second transaction ( - 3), the net result is that you “broke even” (0). next © Math As A Second Language All Rights Reserved

7. Recall that in our treatment of rational numbers, we first established a need for “inventing” them (for example, to be able to perform the division 5 ÷ 3), after which we developed the rules of the game that governed the arithmetic of rational numbers. To apply a similar discussion to the invention of signed numbers, let’s look at the “fill-in-the-blank” problem… 3 +___ = 0 next © Math As A Second Language All Rights Reserved

8. As of now in our course, 0 is the smallest number, and since 3 + 0 = 3, it means that if we add any number to 3 the answer will be greater than 3. So we can either say that… There is no reason for us to have to worry about this since no number can be less than 0. So let’s just leave the number system as is. next © Math As A Second Language All Rights Reserved

9. or… If we have made a $3 profit on one transaction and then a $3 loss on another transaction, we have “broken even” (that is, we made a $0 profit). In other words… $3 profit + $3 loss = $0 profit (or loss). next © Math As A Second Language All Rights Reserved

10. Therefore, if we want a mathematical model that is applicable to the real world, we have to invent a number system in which a fill-in-the-blank problem such as 3 +___ = 0 has a numerical answer. That is why the number - 3 was invented. We generalize this result by adding the property… next © Math As A Second Language All Rights Reserved If a is any number, there exists another number b, such that a + b = 0

11. In this case we refer to a and b as the opposites of one another (or, more formally, the additive inverses of one another). If a and b are opposites of one another, we often rewrite b as - a and/or a as - b. next © Math As A Second Language All Rights Reserved Because 0 + 0 = 0, we see that 0 is its own opposite. So while 0 is neither positive nor negative it does have an opposite (namely, itself). next

12. It is not important which we call positive and which we call negative. What is important is that one is the “opposite” of the other. For example, suppose you go to a candy store and buy a $3 box of candy. When you leave the store you have 1 more box of candy but $3 less than when you came into the store. On the other hand, from the shop keeper’s perspective, when you leave the store he has 1 less box of candy but $3 more than when you came in. © Math As A Second Language All Rights Reserved Notes

13. Notice that the opposite of a number can be positive. For example, the opposite of - 3 is + 3. next © Math As A Second Language All Rights Reserved Notes It is often customary to omit the positive sign when we talk about signed numbers. Namely, when we talk about buying, say, 3 apples, we mean 3 more than 0 (that is, 3 more than 0 apples) next

14. Many textbooks use the notation (-3) rather than - 3. The parentheses are used to indicate that the sign is associated with the number rather than with the operation of subtraction. Using parentheses is cumbersome. Additionally we prefer the notation - 1 to emphasize the difference between subtraction, opposite and the sign of a number. By writing the sign as a “superscript” we are trying to indicate without the use of parentheses that the sign is associated with the number. © Math As A Second Language All Rights Reserved Notes

15. We do not read - 1 as “minus 1”. Rather we read it as “negative 1”. In other words, when we say “minus” it means that we are going to perform subtraction (as in 5 – 3 being read as “5 minus 3”). On the other hand, we say “negative 3” when we are referring to the sign of a signed number, and we say the opposite of - 3 when we are referring to changing the sign of - 3. next © Math As A Second Language All Rights Reserved Notes For example, we would read - 3 – - b as “negative 3 minus the opposite of b”. next

16. To view signed numbers in terms of the “real world”, we often use at least one of the following models… © Math As A Second Language All Rights Reserved In Summary The “Business Model” next The Temperature Model The Directed Distance Model

17. next © Math As A Second Language All Rights Reserved The “Business Model” next In terms of this model 3 +___ = 0 would be the fill in the blank form for answering a question such as “If the first transaction resulted in a $3 profit, what must the second transaction have to be in order for the net result of these two transactions to be a $0 profit?” +3+3 -3-3 Profit/Loss

18. In terms of this model 3 +___ = 0 would be the fill in the blank form for answering a question such as, “If the temperature increases by 3°C in the first hour, what must it do during the second hour in order for the temperature to return to what it was originally?” next © Math As A Second Language All Rights Reserved The Temperature Model 0 +1 +2 +3 +4 +5 +6 +7 -2 -3 -4 -5 next

19. In terms of this model 3 +___ = 0 would be the fill in the blank form for answering a question such as, “If you move 3 units to the right of where you were, what must you do to return to your starting point?” © Math As A Second Language All Rights Reserved The Directed Distance Model next By convention “to the right of 0” is positive and “to the left of 0” is negative. 0 3

20. © Math As A Second Language All Rights Reserved next In general, all of the models are basically rate models (which we may call “increasing/decreasing” models) in which “increasing” means that the sign is positive and “decreasing” means that the sign is negative. In these cases, 0 is used to mean that there was no change.

21. In the next presentation, we will begin a discussion of how we add signed numbers. © Math As A Second Language All Rights Reserved Adding Signed Numbers