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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 7 Part 2
The Arithmetic of Exponents The Arithmetic of Exponents When the exponents are not whole numbers! © 2007 Herbert I. Gross next
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 next
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 next
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 next b m × b n = b m+n b m ÷ b n = b m-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.) (b m ) n = b mn (b n × c n ) = (b ×c) n
With this in mind, let's look at an expression such as 2 0. Notice that so far we have only defined 2 n 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 2 0 in any way that we wish. © 2007 Herbert I. Gross next
Thinking in terms of flipping a coin, it seems that 2 0 should represent the number of possible outcomes if a coin is never flipped. So we might be tempted to say that 2 0 = 0 because there are no outcomes. On the other hand, the fact that nothing happens is itself an outcome, so perhaps we should define 2 0 to be 1. © 2007 Herbert I. Gross next
However, how we choose to define 2 0 will be based on the decision that we would like… b m × b n = b m+n to still be correct even when one or both of the exponents are 0. © 2007 Herbert I. Gross next
So suppose for example that we insist that 2 3 × 2 0 = Since = 3, this would mean that… © 2007 Herbert I. Gross next Another way to obtain this result is to divide both sides of the equation 2 3 × 2 0 = 2 3 by 2 3 to obtain… This tells us that 2 0 is that number which when multiplied by 2 3 yields 2 3 as the product, and this is precisely what it means to multiply a number by 1. That is 2 0 must equal 1. we see that 2 0 = × 2 0 =
This same result can be obtained algebraically without the use of exponents by replacing 2 3 by 8 and 2 0 by x … © 2007 Herbert I. Gross next And dividing both sides by 8, we obtain… Since x = 2 0 we see that 2 0 = 1. x = × 2 0 = 2 3 8x8 =
More generally, if we replace 2 by b © 2007 Herbert I. Gross next Key Point 2 3 × 2 0 = and cancel b n from both sides of the equation, b next n b and 3 by n, n b we see that for any non-zero number b, b 0 = 1 bnbn bnbn 1
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 b 3 × b 0 = b 3, we obtain the result that 0 3 × 0 0 = 0. Since 0 3 = 0, this says that 0 × 0 0 = 0; and since any number times 0 is 0, we see that 0 0 is indeterminate, where by indeterminate, we mean that it can be any number. © 2007 Herbert I. Gross next 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. Gross next Note next 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 next Note
For example, suppose that for some strange reason we wanted to define 0 0 to be 7. If we replace 0 0 by 7, it becomes … © 2007 Herbert I. Gross next Note 0 × 0 0 = 0 7 which is a true statement. next
We are not saying if b 0 that b 0 has to be 1. Rather what we are saying is that if we don't define b 0 to equal 1, then we cannot use the rule, b m × b n = b m+n if either m or n is equal to 0. In other words by electing to let b 0 = 1, we are still allowed to use this rule. © 2007 Herbert I. Gross next Key Point For example if we were to let 2 0 = 9, the equation 2 3 x 2 0 = 2 0 would lead to the false statement 2 3 x 2 0 = 2 3 ; that is it would imply that 8 x 9 = 8. Note next
There are other motivations for defining b 0 to be 1. © 2007 Herbert I. Gross Note For Example If we write b n as 1 x b n, 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 10 n is a 1 followed by n zeros. Thus 10 0 would mean a 1 followed by no 0 s which is simply 1. next
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 next Note
Using the same approach as above a clue to one way in which we can define b n in the case that n is a negative integer can be seen in the answer to the following question. © 2007 Herbert I. Gross next 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: = 0 next
© 2007 Herbert I. Gross next Solution Answer: 2 -3 = 1 ÷ 2 3 (= 1/2 3 ) In order for it to still be true that b m × b n = b m + n, it must be that 2 3 × 2 -3 = = 2 0 = 1. The fact that 2 3 × 2 -3 = 1 means that 2 3 and 2 -3 are reciprocals; that is, 2 -3 = 1 ÷ 2 3. Another way to see this is to start with 2 3 × 2 -3 = 1 and then divide both sides by 2 3 to obtain that 2 -3 = 1/2 3. 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 ÷ b n
Even though the exponent is negative, 2 -3 is positive. It is the reciprocal of 2 3. What is true is that as the magnitude of the negative exponent increases, the number gets closer and closer to zero. For example, since 2 10 = 1,024; = 1/1024 or in decimal form approximately , and since 2 20 equals 1,048,576, is approximately © 2007 Herbert I. Gross next Important Note
In fact the property b m ÷ b n = b m-n gives us yet another way to visualize why we define b 0 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… b n ÷ b n = b n-n = b 0 And since b n ÷ b n = 1 (unless b = 0 in which case we get the indeterminate form 0 ÷ 0) we may rewrite the above equality as 1 = b 0. © 2007 Herbert I. Gross next Note
If we define b 0 to equal 1 (unless b = 0, in which case 0 0 is undefined), and if we also define b -n = 1 ÷ b n 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 next Summary
© 2007 Herbert I. Gross next For what value of n is it true that 4 2 ÷ 4 5 = 4 n ? Practice Question 2 next Solution Answer: n = - 3 If we still want b m ÷ b n = b m – n to apply, it means that 4 2 ÷ 4 5 = = 4 -3.
© 2007 Herbert I. Gross next Solution Answer: n = - 3 If you prefer, instead, to return to basics use the definition for whole number exponents to obtain… 4 2 ÷ 4 5 = = 4 × 4 4 × 4 × 4 × 4 × = and since by definition 1/4 3 means the same thing as 4 -3, the result follows. next
The display window of the calculator displays 1 as the answer. © 2007 Herbert I. Gross next To verify that 15 0 = 1, simply enter 15, press the x y key; enter0; press the = key = × % xyxy = ÷ On/off 1/x xyxy 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 b 0 = 1. Rather it is a demonstration that the formula is plausible. In other words, all we've actually done is verified that 15 0 = 1. It tells us nothing about the value of b 0 when b 15. © 2007 Herbert I. Gross next
Of course, we can repeat the above procedure for as many values of b as we wish, and each time we will find that b 0 = 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 next
No matter how small b is, as long as it isn t 0, b 0 will equal 1. For example, ( ) 0 = 1. Sometimes even a calculator cannot distinguish between and 0; hence it might give 1 as the value of 0 0 © 2007 Herbert I. Gross next Note
0.001 now appears in the display window. © 2007 Herbert I. Gross next Suppose you werent certain that = 1/10 3 = You could enter 10. press the x y key; Enter 3; press the +/- key = × % xyxy = ÷ On/off +/ xyxy 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 +/
(Notice that 0.5 is the decimal equivalent of the reciprocal of 2. That is, 1 ÷ 2 = 0.5). © 2007 Herbert I. Gross next If you enter 2 and press the 1/x key, 0.5 appears in the calculator's display window. = × % xyxy = ÷ On/off 1/x = Scientific calculators also have a reciprocal key. It looks like 1/x. For example 1/x 0.5
Since is the reciprocal of 10 3, another way to compute the value of is to compute 10 3 and then press the 1/x key. © 2007 Herbert I. Gross next 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 which is a rounded off value for 1/9 or 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 next
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. Gross next
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. Gross next
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 next
The display will now show that rounded off to the nearest cent, $5, 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 next 1.07xyxy 10+/-= To compute the value of 10,000(1.07) -10 using a calculator, try the following sequence of key strokes… × = 5,
In fact on a year by year basis, the growth of $5, would have looked like… © 2007 Herbert I. Gross next 10 years ago$5, 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 $5, $5, $6, $6, $7, $7, $8, $8, $9, $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. Gross next 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.
As an example, suppose we were given an expression such as 9 1/2. We know that 1/2 + 1/2 = 1. Therefore, if we want b m × b n to still be equal to b m+n, then we would have to agree that … 9 1/2 × 9 1/2 = 9 1/2 + 1/2 = 9 1 = 9 © 2007 Herbert I. Gross next In other words 9 1/2 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 9 1/2 = 3. next
The same result can be obtained algebraically by letting x = 9 1/2 © 2007 Herbert I. Gross next x = 9 x × x x 2 9 1/2 × 9 1/2 = 9 next
Based on our knowledge of signed numbers there are two square roots of 9, namely 3 and - 3. However since 9 1/2 is between 9 0 (=1) and 9 1 (=9), 9 1/2 has to be 3. © 2007 Herbert I. Gross next Note Notice that although 1/2 is midway between 0 and 1, 9 1/2 is not halfway between 9 0 and = 19 1/2 = 39 1 = 9 next
Applying our above discussion to (1.07) 0.5, we may use the x y key on our calculator to see that (1.07) 0.5 = … (i.e., ( …) 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 $ © 2007 Herbert I. Gross next 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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