Download Exponents that are Not Whole Numbers

The Game of Algebra
or
The Other Side of
Arithmetic
Lesson 7
Part 2
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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The
Arithmetic of
Exponents
When the exponents are not whole
numbers!
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© 2007 Herbert I. Gross
There are times when exponents must be
whole numbers. For example, we cannot
flip a coin a fractional or a negative
number of times. However suppose you
have a long term investment in which
the interest rate is 7% compounded
annually. Knowing the present value of
the investment, it makes sense to ask
what the value of the investment was,
say, 3 years ago.
© 2007 Herbert I. Gross
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Moreover we might even want to invent
exponents that are not integers. For
example suppose the cost of living
increases by 6% annually. We might want
to know how much it increases by every
6 months (that is, in 1/2 of a year). It
might come as a bit of a surprise, but as
we shall see later in this presentation the
answer is not 3%)
© 2007 Herbert I. Gross
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A device that is often used in mathematics is
that when we extend a definition we do it in a
way that preserves the original definition. In
the case of exponents we like the properties
that were discussed in Part 1 of this
presentation, namely… bm × bn = bm+n
bm ÷ bn = bm-n
(At this point we have not yet defined negative
exponents so we have to remember that so far
this property assumes that m is greater than n;
that is, m – n cannot be negative.)
(bm)n
© 2007 Herbert I. Gross
=
bmn
(bn × cn) = (b ×c)n
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With this in mind, let's look at an
expression such as 20. Notice that so
far we have only defined 2n in the case
for which n is a positive integer. 0 is
considered to be neither positive nor
negative. Thus, we are free to define 20
in any way that we wish.
© 2007 Herbert I. Gross
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Thinking in terms of flipping a coin,
it seems that 20 should represent the
number of possible outcomes if a coin is
never flipped. So we might be tempted to
say that 20 = 0 because there are no
outcomes. On the other hand, the fact
that nothing happens is itself an
outcome, so perhaps we should define
20 to be 1.
© 2007 Herbert I. Gross
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However, how we choose to
0
define 2 will be based on
the decision that we would
like…
m
n
m+n
b ×b =b
to still be correct even
when one or both of the
exponents are 0.
© 2007 Herbert I. Gross
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So suppose for example that we insist
that 23 × 20 = 23+0. Since 3 + 0 = 3, this
would mean that… This tells us that 20 is
that number which when multiplied by 23
yields 23 as the product, and this is
precisely what it means to multiply a
number by 1. That is 20 must equal 1.
3
21 ×
3
2
0
2
=
3
21
3
2
Another way to obtain this result is to divide
both sides of the equation 23 × 20 = 23 by 23
to obtain… we see that 20 = 1.
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© 2007 Herbert I. Gross
This same result can be obtained
algebraically without the use of exponents
by replacing 23 by 8 and 20 by x …
3
28 ×
0
2x
=
3
28
x == 1
And dividing both sides by 8, we obtain…
Since x = 20 we see that 20 = 1.
© 2007 Herbert I. Gross
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Key Point
More generally, if we replace 2 by b and 3
by n, and cancel bn from both sides of
the equation, we see that for any non-zero
number b, b0 = 1
n
3
b2
n
b
© 2007 Herbert I. Gross
×
0
b2
n
3
+
0
=b
12
n
b
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Note
• The reason we must specify that b ≠ 0 is
based on the fact that any number
multiplied by 0 is 0. More specifically, if we
replace b by 0 in the equation
b3 × b0 = b3, we obtain the result that
03 × 00 = 0. Since 03 = 0, this says that
0 × 00 = 0; and since any number times 0 is
0, we see that 00 is indeterminate, where by
indeterminate, we mean that it can be any
number.
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© 2007 Herbert I. Gross
Note
• This is similar to why we call 0 ÷ 0
indeterminate. Namely 0 ÷ 0 means the set of
numbers which when multiplied by 0 yield 0
as the product; and any number times 0 is
equal to 0.
We say such things as 6 ÷ 3 = 2 when, in
reality, we should say that 6 ÷ 2 denotes the
set of all numbers which when multiplied by 2
yield 6 as the product. However since there
is only one such number (namely, 3) there is
no harm in leaving out the phrase the “set of
next
numbers”.
•
© 2007 Herbert I. Gross
Note
• The time that it is important to refer to
the “set of numbers” is when we talk about
dividing by 0.
For example, b ÷ 0 denotes the set of all
numbers which when multiplied by 0 yield b
as the product. Since any number
multiplied by 0 is 0, if b is not 0, then there
are no such numbers, and if b is 0 the set
includes every number.
© 2007 Herbert I. Gross
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Note
• For example, suppose that for some
“strange” reason we wanted to define 00 to
be 7. If we replace 00 by 7, it becomes …
0×
0
07
=0
which is a true statement.
© 2007 Herbert I. Gross
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Key Point
We are not saying if b ≠ 0 that b0 has to be 1.
Rather what we are saying is that if we don't
define b0 to equal 1, then we cannot use the
rule, bm × bn = bm+n if either m or n is equal to
0. In other words by electing to let b0 = 1,
we are still allowed to use this rule.
Note
For example if we were to let
20 = 9, the equation 23 x 20 = 20 would lead
to the false statement 23 x 20 = 23; that is it
would imply that 8 x 9 = 8.
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© 2007 Herbert I. Gross
Note
• There are other motivations for defining
b0 to be 1.
For Example
If we write bn as 1 x bn, then n tells us the
number of times we multiply 1 by b. If we
don't multiply 1 by b, then we still have 1.
This is especially easy to visualize when b =
10. Namely, in this case 10n is a 1 followed
by n zeros. Thus 100 would mean a 1
followed by no 0’s which is simply 1.
© 2007 Herbert I. Gross
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Note
 In talking about an interest rate of 7%
compounded annually. Namely, if
(1.07)n denotes the value of $1 at the end of
n years, n = 0 represents the value of the
dollar when it is first invested; which at
that time is still $1. In this context
($1.07)0 = $1.
© 2007 Herbert I. Gross
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Using the same approach as above a clue
to one way in which we can define bn in
the case that n is a negative integer can
be seen in the answer to the following
question.
Practice Question 1
How should we define 2-3 if we want Rule
#1
to still be correct even in the case of
negative exponents?
Hint: 3 + -3 = 0
© 2007 Herbert I. Gross
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Solution
Answer: 2-3 = 1 ÷ 23 (= 1/23)
In order for it to still be true that bm × bn =
bm + n, it must be that 23 × 2-3 = 23 + -3 = 20 = 1.
The fact that 23 × 2-3 = 1 means that 23 and
2-3 are reciprocals; that is, 2-3 = 1 ÷ 23.
Another way to see this is to start with
23 × 2-3 = 1 and then divide both sides by 23
to obtain that 2-3 = 1/23. If we now replace 2
by b and 3 by n, we obtain the more general
result that if b ≠ 0 and if n is any integer
(positive or negative), b-n = 1 ÷ bn
© 2007 Herbert I. Gross
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Important Note
Even though the exponent is negative, 2-3
is positive. It is the reciprocal of 23. What
is true is that as the magnitude of the
negative exponent increases, the number
gets closer and closer to zero. For
example, since 210 = 1,024; 2-10 = 1/1024 or
in decimal form approximately 0.000977,
and since 220 equals 1,048,576, 2-20 is
approximately 0.0000000954
© 2007 Herbert I. Gross
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Note
In fact the property bm ÷ bn = bm-n gives us
yet another way to visualize why we define
b0 to equal 1. Namely if we choose m and
n to be equal, we may replace m by n in the
above property to obtain…
bn ÷ bn = bn-n = b0
And since bn ÷ bn = 1 (unless b = 0 in which
case we get the indeterminate form 0 ÷ 0)
we may rewrite the above equality as
1 = b0.
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© 2007 Herbert I. Gross
Summary
If we define b0 to equal 1 (unless b = 0, in
which case 00 is undefined), and if we
also define b-n = 1 ÷ bn then the rules that
apply in the case of positive whole
number exponents also apply in the case
where the exponents are any integers.
© 2007 Herbert I. Gross
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Practice Question 2
For what value of n is it true that
42 ÷ 45 = 4n?
Solution
Answer: n = -3
If we still want bm ÷ bn = bm – n to apply, it
means that 42 ÷ 45 = 42-5 = 4-3.
© 2007 Herbert I. Gross
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Solution
Answer: n = -3
If you prefer, instead, to “return to basics”
use the definition for whole number
exponents to obtain…
2
4
1 4×4 1
2
5
4 ÷4 = 5 =
= 3
4
4 × 4 × 4 × 4 4× 4
and since by definition 1/43 means the
same thing as 4-3, the result follows.
© 2007 Herbert I. Gross
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The calculator can also be used as a
“laboratory” to verify some of the
properties about exponents.
For example
To verify
that 150 = 1,
simply enter
“15”,
press the
xy key;
0
15 1
9
8
7
+
6
5
4
-
3
2
1
×
% 0
.
÷
On/off
© 2007 Herbert I. Gross
1/x xy
=
enter“0”;
press
the = key
The display
window of
the calculator
displays 1 as
the answer.
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This is not really a proof of the formula
b0 = 1.
Rather it is a demonstration that the formula
is plausible. In other words, all we've
actually done is verified that 150 = 1. It tells
us nothing about the value of b0 when b ≠ 15.
© 2007 Herbert I. Gross
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Of course, we can repeat the above
procedure for as many values of b as we
wish, and each time we will find that b0 = 1.
However, until we test the next value of b,
all we have is an educated guess that the
result will again be 1. Yet seeing that the
formula is correct in every case we
look at tends to give us a better feeling
about the validity of the formula.
© 2007 Herbert I. Gross
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Note
• No matter how small b is, as long as it
isn’t 0, b0 will equal 1.
For example, (0.000000001)0 = 1.
Sometimes even a calculator cannot
distinguish between 0.000000001 and 0;
hence it might give 1 as the value of 00
© 2007 Herbert I. Gross
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If your calculator has a +/ – key (the “sign
changing” key), you can even verify results
that involve negative exponents.
For example
Suppose you
weren’t
certain that
10-3 = 1/103 =
“0.001”. You
could enter
“10”.
© 2007 Herbert I. Gross
-3
0.001
10
9
8
7
+
6
5
4
-
3
2
1
×
% 0
.
÷
On/off
+/- xy =
press the xy
key;
Enter “3”;
press the +/key
press the = key
“0.001” now
appears in the
display window.
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Scientific calculators also have a reciprocal
key. It looks like 1/x.
For example
If you enter
“2” and press
the 1/x key,
“0.5” appears
in the
calculator's
display
window.
© 2007 Herbert I. Gross
0.5
2
9
8
7
+
6
5
4
-
3
2
1
×
% 0
.
÷
On/off
1/x xy =
(Notice that
0.5 is the
decimal
equivalent
of the
reciprocal of
2. That is,
1 ÷ 2 = 0.5).
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Since 10-3 is the reciprocal of 103, another
way to compute the value of 10-3 is to
compute 103 and then press the 1/x key.
Caution
Not all fractions are represented by
terminating decimals. For example, if you
use the calculator to compute the value of
3-2 , the answer will appear as “0.111111111”
which is a rounded off value for 1/9 or 3-2.
In this sense, using the calculator will give
you an excellent approximation to the exact
answer, but it might tend to hide the
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structure of what is happening.
© 2007 Herbert I. Gross
As a practical application of negative
exponents let's return to an earlier
discussion where we talked about the
growth of, say, $10,000 if it was invested at
an interest rate of 7% compounded
annually for 10 years. We saw that after ten
years, the amount of the investment, (A),
would be given by…
A=
© 2007 Herbert I. Gross
10
$10,000(1.07)
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A companion question might have been to
find the amount of money that would have
had to have been invested 10 years ago at
an interest rate of 7% compounded
annually in order for the investment to be
worth $10,000 today. In that case, the
amount (A) would be given by…
A=
© 2007 Herbert I. Gross
-10
$10,000(1.07)
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That is, if we knew how much money there
was at the end of a year, we would simply
divide that amount by 1.07 in order to find
what the amount was at the start of that
year. Dividing by 1.07 ten times is the
same as dividing by (1.07)10 which is the
same as multiplying by (1.07)-10.
© 2007 Herbert I. Gross
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To compute the value of 10,000(1.07)-10 using
a calculator, try the following sequence of
key strokes…
1.07 xy
10 +/-
=
×
10000
=
5,083.49
The display will now show that rounded
off to the nearest cent, $5,083.49 would
have had to have been invested 10 years
ago in order for the investment to be
worth $10,000 today.
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© 2007 Herbert I. Gross
In fact on a year by year basis, the growth
of $5,083.49 would have looked like…
10 years ago
9 years ago
8 years ago
7 years ago
6 years ago
5 years ago
4 years ago
3 years ago
2 years ago
1 year ago
Now
© 2007 Herbert I. Gross
$5,083.49
$5,439.33
$5,820.09
$6,227.49
$6,663.42
$7,129.86
$7,628.95
$8,162.97
$8,734.38
$9,345.79
$9,999.99
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Although fractional exponents are beyond our
scope at this point, notice that they can be
motivated by asking such questions as “How
much will the above investment be worth 6
months (that is, 1/2 year) from now?”. In that
situation it makes sense to assume that if we
want our definitions and rules to still be
obeyed, the formula should be…
A = $10,000(1.07)1/2 = A = $10,000(1.07)0.5
So to give a bit of the flavor of fractional
exponents let's close this presentation with
what to do when the exponent is 1/2 .
© 2007 Herbert I. Gross
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As an example, suppose we were given an
expression such as 91/2. We know that
1/2 + 1/2 = 1. Therefore, if we want bm × b n
to still be equal to bm+n, then we would have
to agree that …
91/2 × 91/2 = 91/2 + 1/2 = 91 = 9
In other words 91/2 is the number which
when multiplied by itself is equal to 9. This
is precisely what is meant by the (positive)
square root of 9 (that is, √9 ). In other
words 91/2 = 3.
© 2007 Herbert I. Gross
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The same result can be obtained
algebraically by letting x = 91/2
1/2
1/2
x
x
9 ×9 =9
x × x =9
2
© 2007 Herbert I. Gross
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Note
• Based on our knowledge of signed
numbers there are two square roots of 9,
namely 3 and -3. However since 91/2 is
between 90 (=1) and 91 (=9), 91/2 has to be 3.
• Notice that although 1/2 is midway
between 0 and 1, 91/2 is not halfway
between 90 and 91.
90 = 1
© 2007 Herbert I. Gross
91/2 = 3
91 =
9
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Applying our above discussion to
(1.07)0.5, we may use the xy key on our
calculator to see that (1.07)0.5 =
1.034408…
(i.e., (1.034408…)2 = 1.07) In other words
at the end of a half year the value has
increased by a little less than half of 7%.
That is, each dollar is then worth
$1.034408...
This concludes
our discussion of the
arithmetic of exponents. In the next
lesson, we will apply this knowledge to
the topic known as scientific notation.
© 2007 Herbert I. Gross
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