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The Game of Algebraor
The Other Side of Arithmetic
The Game of Algebraor
The Other Side of Arithmetic
© 2007 Herbert I. Gross
byHerbert I. Gross & Richard A. Medeiros
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Lesson 7Part 2
The Arithmetic of
Exponents
The Arithmetic of
ExponentsWhen the exponents are not whole
numbers!© 2007 Herbert I. Gross
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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 whichthe 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. Grossnext
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. Grossnext
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…
© 2007 Herbert I. Gross
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bm × bn = bm+n
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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 = bmn (bn × cn) = (b ×c)nnext
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. Grossnext
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. Grossnext
However, how we choose to define 20 will be based on
the decision that we would like…
bm × bn = bm+n to still be correct even
when one or both of the exponents are 0.
© 2007 Herbert I. Grossnext
So suppose for example that we insist that 23 × 20 = 23+0. Since 3 + 0 = 3, this would mean that…
© 2007 Herbert I. Grossnext
Another way to obtain this result is to divide both sides of the equation 23 × 20 = 23 by 23 to obtain…
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.
we see that 20 = 1. nextnextnext
23 × 20 = 23
23 231 1
This same result can be obtained algebraically without the use of exponents
by replacing 23 by 8 and 20 by x …
© 2007 Herbert I. Grossnext
And dividing both sides by 8, we obtain…
Since x = 20 we see that 20 = 1.nextnextnext
x = 1
23 × 20 = 238x8
=
More generally, if we replace 2 by b
© 2007 Herbert I. Grossnext
Key Point
23 × 20 = 23 + 0
and cancel bn from both sides of the equation,
b
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nb
and 3 by n,
nb
we see that for any non-zero number b, b0 = 1
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bn bn1
• 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. © 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Grossnext
Note
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• 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 numbers”.
• 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 …
© 2007 Herbert I. Grossnext
Note
0 × 00 = 07
which is a true statement.
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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.
© 2007 Herbert I. Grossnext
Key Point
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.
Note
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• There are other motivations for defining b0 to be 1.
© 2007 Herbert I. Gross
Note
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.
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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. Grossnext
Note
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.
© 2007 Herbert I. Grossnext
How should we define 2-3 if we want Rule #1 to still be correct even in the case of
negative exponents?
Practice Question 1
Hint: 3 + -3 = 0next
© 2007 Herbert I. Gross next
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
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. Grossnext
Important 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.
© 2007 Herbert I. Grossnext
Note
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. Grossnext
Summary
© 2007 Herbert I. Grossnext
For what value of n is it true that 42 ÷ 45 = 4n?
Practice Question 2
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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. Grossnextnext
Solution Answer: n = -3
If you prefer, instead, to “return to basics” use the definition for whole number
exponents to obtain…
42 ÷ 45 =42 45
= 4 × 4
4 × 4 × 4 × 4 × 4 1 1
43=
and since by definition 1/43 means the same thing as 4-3, the result follows.
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The display window of
the calculator displays 1 as the answer.
© 2007 Herbert I. Gross nextnextnext
To verify that 150 = 1, simply enter
“15”,
press the xy key;
enter“0”; press
the = key
=
0
9 8 7 +
6 5 4 -
3 2 1 ×
%
xy =
÷
On/off 1/x
0 .
1
1
5
5
xy
0
0
=
1
The calculator can also be used as a “laboratory” to verify some of the
properties about exponents.
For example
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. Grossnext
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 welook at tends to give us a better feeling
about the validity of the formula.
© 2007 Herbert I. Grossnext
• 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. Grossnext
Note
“0.001” now appears in the
display window.© 2007 Herbert I. Gross nextnextnext
Suppose you weren’t
certain that 10-3 = 1/103 = “0.001”. You could enter
“10”.
press the xy
key;
Enter “3”;
press the +/- key
=
0
9 8 7 +
6 5 4 -
3 2 1 ×
%
xy =
÷
On/off +/-
0 .
1
1
0
0
xy
3
-
=
If your calculator has a +/ – key (the “sign changing” key), you can even verify results
that involve negative exponents.
For example
press the = key
+/-
30.001
(Notice that 0.5 is the decimal
equivalentof the
reciprocal of 2. That is, 1 ÷ 2 = 0.5).
© 2007 Herbert I. Gross nextnextnext
If you enter “2” and press
the 1/x key, “0.5” appears
in thecalculator's
display window. =
0
9 8 7 +
6 5 4 -
3 2 1 ×
%
xy =
÷
On/off 1/x
0 .
2
2
=
Scientific calculators also have a reciprocal key. It looks like 1/x.
For example
1/x
0.5
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.
© 2007 Herbert I. Grossnext
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
structure of what is happening.
Caution
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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 = $10,000(1.07)10
© 2007 Herbert I. Grossnext
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 = $10,000(1.07)-10
© 2007 Herbert I. Grossnext
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. Grossnext
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.© 2007 Herbert I. Gross
nextnextnext
1.07 xy 10 +/- =
To compute the value of 10,000(1.07)-10 using a calculator, try the following sequence of
key strokes…
× = 5,083.4910000
In fact on a year by year basis, the growth of $5,083.49 would have looked like…
© 2007 Herbert I. Grossnext
10 years ago $5,083.499 years ago8 years ago7 years ago6 years ago5 years ago4 years ago3 years ago2 years ago1 year ago
Now
$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
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
© 2007 Herbert I. Grossnext
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 .next
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
© 2007 Herbert I. Grossnext
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.
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The same result can be obtained algebraically by letting x = 91/2
© 2007 Herbert I. Grossnextnext
x = 9x
× x = 9x2
91/2 × 91/2 = 9
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• 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.
© 2007 Herbert I. Grossnext
Note
• Notice that although 1/2 is midway between 0 and 1, 91/2 is not halfway
between 90 and 91.
90 = 1 91/2 = 3 91 = 9next
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...
© 2007 Herbert I. Grossnext
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.
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