1. Multiplying Signed Numbers © Math As A Second Language All Rights Reserved next #9 Taking the Fear out of Math × - 3 + 3 - 6

2. Recall that when we multiply two quantities, we multiply the adjectives and we multiply the nouns. next © Math As A Second Language All Rights Reserved For example… 3 kilowatts × 2 hours = 6 kilowatt hours 3 hundreds × 2 thousand = 6 hundred thousand 3 feet × 2 feet = 6 “feet feet” = 6 feet 2 = 6 square feet next

3. This concept becomes very interesting when we deal with signed numbers because there are only two nouns, “positive”and “negative”. © Math As A Second Language All Rights Reserved Recall that the adjective part of a signed number is the magnitude and the noun part is the sign. If two signed numbers are unequal but have the same magnitude, then they must be opposites of one another. next

4. For example, we know that 3 x 2 = 6 (that is, + 3 x + 2 = + 6). So let’s assume that multiplying a signed number by 2 yields a different product than if we had multiplied it by - 2. © Math As A Second Language All Rights Reserved Since 3 x 2 and 3 x - 2 have the same magnitude (that is, 6), the only way they can be unequal is if they have different signs, and in that case it means that the products are opposites of one another. next

5. Stated more symbolically, if a and b are signed numbers and a ≠ b but |a| = |b| then a = - b or equivalently - a = b © Math As A Second Language All Rights Reserved In terms of a more concrete model, a $3 profit is not the same as a $3 loss, but the size of either transaction is $3. next Let’s now see how this applies to the product of any two signed numbers.

6. However, rather than to be too abstract let’s work with two specific signed numbers. © Math As A Second Language All Rights Reserved So suppose, for example, we multiply two signed numbers whose magnitudes are 3 and 2. next Then the magnitude of their product will be 6 regardless of their signs.

7. That is… © Math As A Second Language All Rights Reserved + 3 × + 2 = 3 pos × 2 pos = 6 “pos pos” next + 3 × - 2 = 3 pos × 2 neg = 6 “pos neg” - 3 × + 2 = 3 neg × 2 pos = 6 “neg pos” - 3 × - 2 = 3 neg × 2 neg = 6 “neg neg” next

8. However, there are only two nouns, positive and negative. © Math As A Second Language All Rights Reserved Therefore, “pos pos” must either be positive or negative. The same holds true for “pos neg”, “neg pos” and “neg neg”. next

9. It’s easy to see that “pos pos” = positive… © Math As A Second Language All Rights Reserved + 3 × + 2 = 3 × 2 = 6 = + 6 = 6 pos next …and at the same time it is equal to 6 “pos pos”. Hence… positive × positive = positive

10. Multiplying by + 2 is not the same as multiplying by - 2. next © Math As A Second Language All Rights Reserved Therefore + 3 × + 2 ≠ + 3 × - 2; and since both numbers have the same magnitude, they must have opposite signs. next Since + 3 × + 2 is positive, + 3 × - 2 must be negative.

11. Thus… © Math As A Second Language All Rights Reserved + 3 × - 2 = - 6 = 6 negative next …but at the same time it is equal to 6 “pos neg”. Hence… positive × negative = negative And since multiplication is commutative… negative × positive = negative next

12. The above results can be visualized rather easily in terms of profit and loss… © Math As A Second Language All Rights Reserved next A $3 profit 2 times is a $6 profit. A $2 loss 3 times is a $6 loss. A $3 loss 2 times is a $6 loss. Physical Models

13. And in terms of the chip model… next © Math As A Second Language All Rights Reserved next 3 positive chips 2 times is 6 positive chips. Physical Models 3 negative chips 2 times is 6 negative chips. 2 negative chips 3 times is 6 negative chips. PPPNNNPPPNNNNNNNNN

14. However, these physical models do not make sense when we talk about negative × negative. next © Math As A Second Language All Rights Reserved For example, we cannot have a loss or a decrease in temperature occur a negative number of times. next However, we do know that - 3 × + 2 cannot be equal to - 3 × - 2 but both numbers have the same magnitude.

15. Hence, they must differ in sign… © Math As A Second Language All Rights Reserved - 3 × + 2 = - 6 = 6 negative next Therefore, since - 3 × + 2 is negative, - 3 × - 2 must be positive. Hence… negative × negative = positive next

16. © Math As A Second Language All Rights Reserved Notes Very often when we make up a rule, we want it to conform to what we feel is reality. One rule of mathematics that we feel conforms to reality is the “cancellation law” which states… If a × b = a × c and a ≠ 0 1, then b = c. next note 1 The assumption that a ≠ 0 is crucial. For example, since the product of 0 and any number is 0, 0 ×10 = 0 × 3, and if we cancel 0 from both sides of the equality we obtain the false result that 10 = 3.

17. next © Math As A Second Language All Rights Reserved With this in mind, let’s see what happens if we were to allow the product of two negative numbers to be negative. We already have accepted the fact that - 3 × + 2 = - 6. Hence, if it was also true that - 3 × - 2 = - 6, it would mean that… next - 3 × + 2 = - 3 × - 2 If we then use the cancellation law by dividing both sides of the equality by - 3, we obtain the false result that + 2 = - 2.

18. next © Math As A Second Language All Rights Reserved In our above “proof” as to why the product of two negative numbers had to be positive, we made the assumption that the cancellation law had to remain in effect. So it might be natural for someone to wonder if it is really a proof if we have to make certain assumptions in order to obtain it. The truth of the matter is that there can never be proof without certain assumptions being made.

19. next © Math As A Second Language All Rights Reserved Very often, there can be different assumptions that lead to the same result. For example, shown on the following slide is a different demonstration of why the product of two negative numbers is positive. In the example, we will look at a chart and them make an assumption that seems obvious to us, and then see what it leads to. next

20. © Math As A Second Language All Rights Reserved next So consider the following pattern… - 3 × + 4= - 12 - 3 × + 3= -9-9 - 3 × + 2= -6-6 - 3 × + 1= -3-3 In the first column every product has - 3 as its first factor. As we read down the rows in the first column we find that the second factor is an integer that decreases by 1 each time. In the last column, we see that each time we go down one row the number increases by 3. next

21. © Math As A Second Language All Rights Reserved Aside We have to be careful when we compare the size of negative numbers. For example, 12 is greater than 9 but - 12 is less than - 9. next In terms of our profit and loss model, the bigger the profit the better it is for us, but the bigger the loss the worse it is for us.

22. © Math As A Second Language All Rights Reserved next So just extending the chart by rote (so to speak) the pattern leads us to… - 3 × + 4= - 12 - 3 × + 3= -9-9 - 3 × + 2= -6-6 - 3 × + 1= -3-3 - 3 × 0=0 - 3 × - 1= +3+3 - 3 × - 2= +6+6 - 3 × - 3= +9+9

23. next © Math As A Second Language All Rights Reserved Notes No one forces us to make sure that the pattern continues or that the cancellation law remains valid. next Thus, we are faced with a choice in the sense that if we want the product of two negative numbers to be negative, we would have to “sacrifice” such things as nice patterns and the “cancellation law”. In short, just as in “real life”, in mathematics there is a price that we sometimes have to pay in order for us for to enjoy the use of “luxuries”

24. © Math As A Second Language All Rights Reserved Of course, once we know that negative × negative = positive it is easy to make up a reason that will explain this physically. next For example, in terms of temperature we may interpret - 3 × - 2 to mean that if the temperature was decreasing by 3° per hour then 2 hours ago it was 6° greater than it is now. In other words, although 2 means the same thing in both quantities, 2 hours after now is the opposite of 2 hours before now.

25. © Math As A Second Language All Rights Reserved Or if you lose $2 on each transaction then 3 transactions ago you had $6 more than you have now. next Multiplying a signed number by either - 1 or + 1 doesn’t change the magnitude of the signed number. Notes

26. next © Math As A Second Language All Rights Reserved However, since the two products cannot be equal it means that when we multiply a signed number by - 1 we do not change its magnitude, but we do change its sign. next In more mathematical terms, for any signed number n, n × - 1 = - n (remember that - n means the opposite of n, not negative n. So, - n will be positive if n is negative). Notes

27. next © Math As A Second Language All Rights Reserved In other words, in terms of the four basic operations of arithmetic, the command “change the sign of a number” means the same thing as “multiply the number by - 1”. next Notes This idea plays a very important role in algebra.

28. next © Math As A Second Language All Rights Reserved Since there are only two signs, when we multiply a signed number twice by - 1, we obtain the original number. next Notes Sometimes this is referred to as “the rule of double negation”. However, we must use this term carefully because while the product of two negative numbers is positive, the sum of two negative numbers is negative.

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