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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 5
Introduction to Signed Numbers: Multiplying and Dividing Signed Numbers Introduction to Signed Numbers: Multiplying and Dividing Signed Numbers × © 2007 Herbert I. Gross next ÷
Recall that when we multiply two quantities, we multiply the adjectives and we also multiply their nouns. For example: 3 kilowatts × 2 hours = 6 kilowatt hours 3 hundred × 2 thousand = 6 hundred thousand 3 feet × 2 feet = 6 “feet feet” = 6 feet 2 = 6 square feet © 2007 Herbert I. Gross next
© 2007 Herbert I. Gross next This concept becomes very interesting when we deal with signed numbers because there are only two nouns, “positive” and “negative”. Recall that the adjective part of a signed number is the magnitude (size) of the number and the noun part is the sign (+or -).
© 2007 Herbert I. Gross next If two signed numbers are unequal but have the same magnitude, then they must be opposites of one another. Stated more symbolically... if a and b are signed numbers and a ≠ b but │a│ = │b│, then a = - b or equivalently - a = b
© 2007 Herbert I. Gross next In terms of a more concrete model, a $3 profit is not the same as a $3 loss, but the size of each transaction is $3. Let's now see how this applies to the product of any two signed numbers. However, rather than be too abstract, let's work with two specific signed numbers and see what happens.
© 2007 Herbert I. Gross next For example, suppose we multiply two signed numbers whose magnitudes are 3 and 2. Then the magnitude of their product will be 6 regardless of the sign. That is… + 3 × + 2 = 3 pos × 2 pos = 6 “pos pos” + 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”
© 2007 Herbert I. Gross next However, there are only two nouns, positive and negative. Therefore, “pos pos” must either be positive or negative. The same holds true for “pos neg”, “neg pos” and “neg neg”. Namely… + 3 × + 2 = 3 × 2 = 6 = + 6 = 6 pos It's easy to see that “pos pos” = positive. And at the same time, + 3 × + 2 = 6 pos pos Therefore… positive × positive = positive.
© 2007 Herbert I. Gross next As for + 3 × - 2, notice that multiplying by + 2 is not the same as multiplying by - 2. Hence, + 3 × + 2 ≠ + 3 × - 2 Since both numbers have the same magnitude but are unequal, they must have opposite signs. Since + 3 × + 2 is positive, + 3 × - 2 must be negative; but at the same time it is equal to 6 “pos neg”. Hence… positive × negative = negative. And since multiplication is commutative… negative × positive = negative.
© 2007 Herbert I. Gross next The above results can be visualized rather easily in terms of our physical models. For example, in terms of profit and loss… A $3 profit 2 times is a $6 profit. Profit + Loss - $3 $6 $6 pos A $2 loss 3 times is a $6 loss. Profit + Loss - $2 $6 $6 neg A $3 loss 2 times is a $6 loss. Profit + Loss - $3 $6 $6 neg
© 2007 Herbert I. Gross next And in terms of temperature change… A 3º increase in temperature 2 times is a net increase of 6º p o s
© 2007 Herbert I. Gross next Temperature Model… A 2º decrease in temperature 3 times is a net decrease of 6º n e g
© 2007 Herbert I. Gross next Temperature Model… A 3º decrease in temperature 2 times is a net decrease of 6º n e g 1 2 3
© 2007 Herbert I. Gross next However, these physical models do not make sense when we talk about negative × negative. For example, with respect to our profit and loss model, we cannot incur a loss a negative number of times. And in our temperature model, we cannot have the temperature decrease a negative number of times.
© 2007 Herbert I. Gross next However, what we do know is that - 3 × - 2 cannot be equal to - 3 × + 2, but both numbers have the same magnitude. Hence, they must differ in sign. Therefore, since - 3 × + 2 is negative; - 3 × - 2 must be positive. In other words… negative × negative = positive
If we want to be more “traditional” in showing why - 3 × + 2 ≠ - 3 × - 2, we can use the so-called cancellation law. Namely… If a × b = a × c and a ≠ 0, then, b = c. Therefore, if we were to assume that - 3 × + 2 = - 3 × - 2, and that the cancellation law was to remain in effect, we could then cancel - 3 from both sides of the equal sign and obtain the false result … + 2 = - 2. © 2007 Herbert I. Gross next Note
© 2007 Herbert I. Gross next Of course once we know that negative × negative = positive, it is easy to interpret this in terms of our physical models. Or in terms of profit and loss, if we lose $2 on each transaction, then 3 transactions ago we had $6 more than we have now. For example, in terms of temperature, we may interpret - 3 × - 2 to mean that if the temperature decreased by 2° per hour, then 3 hours ago it was 6°higher.
© 2007 Herbert I. Gross next Other ways to demonstrate why negative × negative = positive Let’s look at a pattern that we might want to see continued. To begin with, by now most of us accept the fact that negative × positive = negative. (For example a $4 loss 5 times is a net loss of $20.) next
© 2007 Herbert I. Gross next So let's look at the following pattern… - 3 × + 4 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 × × × + 1 = = = =
© 2007 Herbert I. Gross next Thus if we want this pattern to continue as we add on more rows in the top to bottom direction, the second factor in the first column must keep decreasing by 1 and the product in the last column must keep increasing by 3 each time. - 3 × × × × + 1 = = = = next So just extending the chart by rote (so to speak) the pattern leads us to… - 3 × + 0 = × - 1 = × - 2 = × - 3 = +9+9
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. 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. In terms of the number line, greater than means to the right of and the point - 9 is to the right of the point In terms of pure arithmetic, we have to add 3 to - 12 to obtain - 9 as the sum. © 2007 Herbert I. Gross next Note
© 2007 Herbert I. Gross next When we add two numbers there can only be one sum. The same is true for subtraction, multiplication and division. Therefore, when a number is expressed in two different ways, the two expressions must be equivalent. This gives us yet another way to demonstrate why… - 3 × - 2 = + 6 Note
© 2007 Herbert I. Gross next More specifically, let's compute the number named by × ( ) in two different ways = - 6 next On the one hand On the other hand, by the distributive property ( - 3 × - 2) ( - 3 × + 4) + ( - 3 × - 2) = - 3 × ( ) = Since - 3 × ( ) is equal to both - 6 and ( - 3 × - 2), it means that… - 6 = ( - 3 × - 2) ×
© 2007 Herbert I. Gross next Since - 6 = ( - 3 × - 2) - 6 = we may then add + 12 to both sides of the equation to obtain… ( - 3 × - 2) = - 3 × = - 3 × - 2 or…
No one forces us to make sure that the pattern continues or that the distributive property remains valid. The point is, the pattern can only continue if we define negative × negative to be positive. Thus we are faced with a choice in the sense that if we wanted the product of two negative numbers to be negative, we would have to “sacrifice” such things as nice patterns and the cancellation law, etc. © 2007 Herbert I. Gross next Note
In short, just as in “real life”; in mathematics there is a price that we sometimes have to pay in order for us to enjoy the use of “luxuries”. © 2007 Herbert I. Gross next Note
© 2007 Herbert I. Gross next Reminder Multiplying a signed number by either + 1 or - 1 doesn't change the magnitude of the signed number.
© 2007 Herbert I. Gross next Discussion When we multiply a signed number by - 1, we do not change its magnitude, but we do change its sign. In more mathematical terms, for any signed number, n, n × - 1 = - n Remember: - n means the opposite of n, not negative n, - n will be positive if n is negative.
© 2007 Herbert I. Gross next Note So 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”. As we shall see later, this idea plays a very important role in algebra.
© 2007 Herbert I. Gross next Note Since there are only two signs when we multiply a signed number twice by - 1, we obtain the original number. In particular, - 1 × - 1 = + 1
© 2007 Herbert I. Gross next Note Combined with the associative and commutative properties, this gives us yet another way to demonstrate why - n × - m = n × m. n × m 1 × (n × m) = ( - 1 × - 1) × (n × m) = ( - 1 × n) × ( - 1 × m) = - n × - m =
© 2007 Herbert I. Gross next Note Sometimes neg × neg = pos is referred to as “the rule of double negation”. We should use this term carefully because, while the product of two negative numbers is positive, the sum of two negative numbers is negative. next
© 2007 Herbert I. Gross next If we use our “unmultiplying” model for division, the division of signed numbers almost becomes an anecdotal footnote to our discussion of multiplication of signed numbers. For example, consider a problem like - 12 ÷ - 4, which means we want to find the number that, when multiplied by - 4, yields - 12 as the answer. In terms of a “fill in the blank” problem, we are saying that the problem - 12 ÷ - 4 = __ is equivalent to the problem - 4 × __ = - 12.
© 2007 Herbert I. Gross next Recall that when we multiply two signed numbers, we multiply the magnitudes (sizes) to get the magnitude of the product, and we multiply the signs to get the sign of the product. So we see that we have to multiply 4 by 3 to obtain 12, and we have to multiply negative by positive, in order to obtain negative. Hence we conclude that we must multiply - 4 by + 3 to obtain - 12 as the product. Since - 4 × __ = - 12 is simply a restatement of - 12 ÷ - 4 = __, we see that… - 12 ÷ - 4 = + 3.
© 2007 Herbert I. Gross next The fact that we were dealing with the specific numbers - 12 and - 4 is just a special case of what happens when we divide one negative number by another. More specifically, if we concentrate just on the signs, the fact that… positive × negative = negative means positive negative negative = × ÷
© 2007 Herbert I. Gross next In summary, when we divide two negative numbers the sign of the quotient is positive and the magnitude is the quotient of the two magnitudes. In a similar way the fact that… means that… negative × negative = positive negative negative positive ×= ÷
© 2007 Herbert I. Gross next More concretely… + 6 ÷ - 2 = __ means the same thing as… - 2 × __ = + 6 We have to multiply 2 by 3 to get 6 as the magnitude, and we have to multiply negative by negative to get positive. That is… - 2 × - 3 = + 6 or + 6 ÷ - 2 = - 3
The magnitude of the product is the product of the two magnitudes. For example, the magnitude of each of + 4 × + 3‚ + 4 × - 3, - 4 × + 3, - 4 × - 3 are 12 because, in each case, the magnitudes of the factors are 4 and 3. In short, notice that the magnitude of the product does not depend on the signs of the factors. © 2007 Herbert I. Gross next Summary To Multiply Two Signed Numbers
The sign of the product is positive if the two factors have the same sign… © 2007 Herbert I. Gross next To Multiply Two Signed Numbers For example, + 4 × + 3 = + 12, and - 4 × - 3 = + 12 and negative if the two factors have different signs. For example, + 4 × - 3 = - 12, and - 4 × + 3 = - 12 next
The magnitude of the quotient is the quotient of the two magnitudes. For example, the magnitude of each of the numbers… + 12 ÷ + 3‚ + 12 ÷ - 3, - 12 ÷ + 3, and - 12 ÷ - 3 is 4 because, in each case, the magnitudes of the two numbers are 12 and 3 respectively. In short, notice that the magnitude of the quotient does not depend on the signs of the numbers. © 2007 Herbert I. Gross next To Divide Two Signed Numbers
The sign of the quotient is positive if the two factors have the same sign… © 2007 Herbert I. Gross next To Divide Two Signed Numbers For example, + 12 ÷ + 3 = + 4, and - 12 ÷ - 3 = + 4 and negative if the two numbers have different signs. For example, + 12 ÷ - 3 = - 4, and - 12 ÷ + 3 = - 4 next
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