Transcript 4. Multiplying Signed Numbers

Taking the Fear
out of Math
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#9
+3
× -3
-6
Multiplying
Signed Numbers
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Recall that when we multiply two
quantities, we multiply the adjectives
and we multiply the nouns.
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”.
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.
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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.
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
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.
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.
So suppose, for example, we multiply two
signed numbers whose magnitudes are
3 and 2.
Then the magnitude of their product will
be 6 regardless of their signs.
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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”
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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”.
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It’s easy to see that “pos pos” = positive…
+3
× +2 = 3 × 2 = 6 = +6 = 6 pos
…and at the same time it is equal to
6 “pos pos”.
Hence…
positive × positive = positive
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Multiplying by +2 is not the same as
multiplying by -2.
Therefore +3 × +2 ≠ +3 × -2; and since both
numbers have the same magnitude, they
must have opposite signs.
Since +3 × +2 is positive,
+3 × -2 must be negative.
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Thus…
+3
× -2 = -6 = 6 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
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Physical Models
The above results can be visualized rather
easily in terms of profit and loss…
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…
3 positive chips 2 times is
6 positive chips.
P P P
P P P
3 negative chips 2 times is
6 negative chips.
N N N
N N N
2 negative chips 3 times is
6 negative chips.
N N
N N
N N
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However, these physical models do not
make sense when we talk about
negative × negative.
For example, we cannot have a loss or a
decrease in temperature occur a negative
number of times.
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…
-3
× +2 = -6 = 6 negative
Therefore, since -3 × +2 is negative, -3 × -2
must be positive.
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 feel
conforms to reality is the
“cancellation law” which states…
If a × b = a × c and a ≠ 01, then b = c.
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.
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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.
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We already have accepted the fact that
-3 × +2 = -6.
Hence, if it was also true that -3 × -2 = -6, it
would mean that…
-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 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.
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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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So consider the following pattern…
× +4
-3 × +3
-3 × +2
-3 × +1
-3
=
=
=
=
-12
-9
-6
-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.
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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.
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…
× +4
-3 × +3
-3 × +2
-3 × +1
-3 × 0
-3 × -1
-3 × -2
-3 × -3
-3
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=
=
=
=
=
=
=
=
-12
-9
-6
-3
0
+3
+6
+9
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Notes
No one forces us to make sure that the
pattern continues or that the cancellation
law remains valid.
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.
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.
Notes
Multiplying a signed number by either
-1 or +1 doesn’t change the magnitude of
the signed number.
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Notes
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.
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).
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Notes
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”.
This idea plays a very important
role in algebra.
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Notes
Since there are only two signs, when we
multiply a signed number twice by -1, we
obtain the original number.
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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Dividing Signed Numbers
In the next presentation,
we will begin a
discussion of how we
divide signed numbers.
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