multiplying signed numbers © math as a second language all rights reserved next #9 taking the fear...

MultiplyingSigned Numbers

© Math As A Second Language All Rights Reserved

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#9

Taking the Fearout of Math

× -3

+3

-6

Recall that when we multiply two quantities, we multiply the adjectives

and we multiply the nouns.

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For example…

3 kilowatts × 2 hours = 6 kilowatt hours

3 hundreds × 2 thousand =

6 hundred thousand

3 feet × 2 feet = 6 “feet feet” = 6 feet2 =

6 square feet

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This concept becomes very interesting when we deal with signed numbers because there are only two nouns,

“positive”and “negative”.

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Recall that the adjective part of a signednumber 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.

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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.

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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 .

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Stated more symbolically, if a and b are signed numbers and a ≠ b but

|a| = |b| then a = -b or equivalently -a = b

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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.

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Let’s now see how this applies to the product of any two signed numbers.

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However, rather than to be too abstract let’s work with two specific

signed numbers.

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So suppose, for example, we multiply two signed numbers whose magnitudes are

3 and 2.

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Then the magnitude of their product will be 6 regardless of their signs.

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That is…

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+3 × +2 = 3 pos × 2 pos = 6 “pos pos”

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+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”

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However, there are only two nouns, positive and negative.

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Therefore, “pos pos” must either be positive or negative. The same

holds true for “pos neg”, “neg pos” and “neg neg”.

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It’s easy to see that “pos pos” = positive…

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+3 × +2 = 3 × 2 = 6 = +6 = 6 pos

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…and at the same time it is equal to 6 “pos pos”.

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Hence…

positive × positive = positive

Multiplying by +2 is not the same as multiplying by -2.

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Therefore +3 × +2 ≠ +3 × -2; and since both numbers have the same magnitude, they

must have opposite signs.

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Since +3 × +2 is positive, +3 × -2 must be negative.

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Thus…next


+3 × -2 = -6 = 6 negative

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…but at the same time it is equal to 6 “pos neg”.

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Hence…

positive × negative = negative

And since multiplication is commutative…

negative × positive = negative

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The above results can be visualized rather easily in terms of profit and loss…

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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.

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Physical Models

And in terms of the chip model…

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3 positive chips 2 times is 6 positive chips.

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Physical Models

3 negative chips 2 times is 6 negative chips.

2 negative chips 3 times is 6 negative chips.

P P P

N N N

P P P

N N

N N N

N NN N

However, these physical models do not make sense when we talk about

negative × negative.

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For example, we cannot have a loss or a decrease in temperature occur a negative

number of times.

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However, we do know that -3 × +2 cannot be equal to -3 × -2 but both numbers have

the same magnitude.

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Hence, they must differ in sign…next


-3 × +2 = -6 = 6 negative

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Therefore, since -3 × +2 is negative, -3 × -2 must be positive.

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Hence…

negative × negative = positive

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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 feelconforms to reality is the

“cancellation law” which states…

If a × b = a × c and a ≠ 01, then b = c.

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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.

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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…

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-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.

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In our above “proof” as to why the product of two negative numbershad to be positive, we made the

assumption that the cancellation lawhad 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.

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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.

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next So consider the following pattern…

-3 × +4 = -12-3 × +3 = -9-3 × +2 = -6-3 × +1 = -3

In the first column every product has -3 as its first factor. As we read down the rows

inthe first column we find that the second factor is an integer that decreases by 1

each time. In the last column, we see thateach time we go down one row the number

increases by 3.

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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.

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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.

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So just extending the chart by rote (so to speak) the pattern leads us to…

-3 × +4 = -12-3 × +3 = -9-3 × +2 = -6-3 × +1 = -3-3 × 0 = 0-3 × -1 = +3-3 × -2 = +6-3 × -3 = +9

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Notes

No one forces us to make sure that the pattern continues or that the cancellation

law remains valid.

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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”

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Of course, once we know that negative × negative = positive

it is easy to make up a reason that will explain this physically.

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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.

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Or if you lose $2 on each transaction then 3 transactions ago you had $6 more

than you have now.

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Multiplying a signed number by either -1 or +1 doesn’t change the magnitude of

the signed number.

Notes

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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.

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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

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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”.

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Notes

This idea plays a very important role in algebra.

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Since there are only two signs, when we multiply a signed number twice by -1, we

obtain the original number.

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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.

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In the next presentation, we will begin a

discussion of how we divide signed numbers.


Dividing Signed Numbers

multiplying signed numbers © math as a second language all rights reserved next #9 taking the fear...

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