Download Signed Numbers: Mulitplication, Divsion

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 5
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
next
Introduction to
Signed Numbers:
× Multiplying and
Dividing ÷
Signed Numbers
next
© 2007 Herbert I. Gross
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 feet2 =
6 square feet
next
© 2007 Herbert I. Gross
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 -).
next
© 2007 Herbert I. Gross
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
next
© 2007 Herbert I. Gross
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.
next
© 2007 Herbert I. Gross
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”
next
© 2007 Herbert I. Gross
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”.
It's easy to see that “pos pos” = positive.
Namely… +3 × +2 = 3 × 2 = 6 = +6 = 6 pos
And at the same time, +3 × +2 = 6 pos pos
Therefore…
positive × positive = positive.
next
© 2007 Herbert I. Gross
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
$3
$6
$6 pos
© 2007 Herbert I. Gross
A $2 loss 3
times is a
$6 loss.
A $3 loss 2
times is a
$6 loss.
Profit + Loss $2
$2
Profit + Loss $3
$3
$2
$6
$6 neg
$6
$6 neg
next
And in terms of temperature change…
+8
6
p
o
s
A 3º increase in temperature 2
times is a net increase of 6º.
3
2
1
3
2
1
+7
+6
+5
+4
+3
+2
+1
0
-1
-2
-3
-4
-5
-6
-7
-8
next
© 2007 Herbert I. Gross
Temperature Model…
+8
+7
+6
+5
+4
+3
+2
6
n
e
g
A 2º decrease in temperature 3
times is a net decrease of 6º.
1
2
1
2
1
2
+1
0
-1
-2
-3
-4
-5
-6
-7
-8
next
© 2007 Herbert I. Gross
Temperature Model…
+8
+7
+6
+5
+4
+3
+2
6
n
e
g
A 3º decrease in temperature 2
times is a net decrease of 6º.
1
2
3
1
2
3
+1
0
-1
-2
-3
-4
-5
-6
-7
-8
next
© 2007 Herbert I. Gross
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.
next
© 2007 Herbert I. Gross
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
next
© 2007 Herbert I. Gross
Note
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. next
© 2007 Herbert I. Gross
Of course once we know that
negative × negative = positive, it is easy to
interpret this in terms of our physical
models.
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.
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.
next
© 2007 Herbert I. Gross
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
So let's look at
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.
next
© 2007 Herbert I. Gross
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.
+
So just extending
the chart by rote
(so to speak) the
pattern leads us
to…
© 2007 Herbert I. Gross
3× 4
-3 × +3
-3 × +2
-3 × +1
-3 × +0
-3 × -1
-3 × -2
-3 × -3
= 12
= -9
= -6
= -3
= 0
= +3
= +6
= +9
next
Note
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 -12.
In terms of pure arithmetic, we have to
add 3 to -12 to obtain -9 as the sum.
© 2007 Herbert I. Gross
next
Note
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
next
© 2007 Herbert I. Gross
More specifically, let's compute the number
named by... -3 × (+4 + -2) in two different ways.
On the one hand....
-3
× (+4 ++2-2) =
×
On the other hand,
by the distributive
property....
= -6
(-3-3××+4)
(+4++(--32)×=-2) =
-12
+ (-3 × -2)
Since -3 × (+4 + -2) is equal to both -6 and
-12 + (-3 × -2), it means that…
-6 = -12 + (-3 × -2)
© 2007 Herbert I. Gross
next
Since -6 = -12 + (-3 × -2)
we may then add +12 to both sides of the
equation to obtain…
= -12 + (-3 × -2)
+12 +12
-6 + +12
+6 = - 3 × - 2
or…
-6
next
© 2007 Herbert I. Gross
Note
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
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 ×
(-1 × n)
= m) =
(-1 × -1) × (n × m) =
1 × (n × 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.
© 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.
next
© 2007 Herbert I. Gross
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.
next
© 2007 Herbert I. Gross
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
next
© 2007 Herbert I. Gross
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…
negative × negative = positive
means that…
negative
negative ÷ positive
next
© 2007 Herbert I. Gross
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
© 2007 Herbert I. Gross
× -3 = +6 or
+6
÷ -2 = -3
next
Summary
To Multiply Two Signed Numbers
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. next
© 2007 Herbert I. Gross
To Multiply Two Signed Numbers
The sign of the product is positive if the
two factors have the same sign…
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
© 2007 Herbert I. Gross
To Divide Two Signed Numbers
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
next
the numbers.
© 2007 Herbert I. Gross
To Divide Two Signed Numbers
The sign of the quotient is positive if the
two factors have the same sign…
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
© 2007 Herbert I. Gross