1. Extending the Definition of Exponents © Math As A Second Language All Rights Reserved next #10 Taking the Fear out of Math 2 -8

2. In illustrating the use of positive integer exponents, we have chosen several different applications including how to compute the number of outcomes when a coin is flipped a certain number of times. next © Math As A Second Language All Rights Reserved This was a good application to use because a coin can only be flipped a whole number of times. It makes no sense to talk about what happens, for example, when a coin is flipped “negative 2” times.

3. next © Math As A Second Language All Rights Reserved next However, there are other applications in which we have occasion to use exponents that include other integers (that is, 0 and the negative integers). One such application is when we want to extend the use of exponential notation to represent the powers of 10 when we deal with decimals numbers.

4. When all we had were the positive integers, the denominations were represented as shown below… next © Math As A Second Language All Rights Reserved next Place Value Notation Exponential Notation 10,00010 4 1,00010 3 10010 2 1010 1

5. As we have mentioned in previous lessons, definitions and rules are often based on things we would like to be true or patterns we would like to continue. next © Math As A Second Language All Rights Reserved next Notice that in the table, each number is divided by 10 to get to the number below it.” Place Value Notation Exponential Notation 10,00010 4 1,00010 3 10010 2 1010 1

6. next © Math As A Second Language All Rights Reserved next Therefore, if we want this pattern to continue, the next entries would have to be based on the facts that 10 ÷ 10 = 1, 1 ÷ 10 = 1 / 10 ; 1 / 10 ÷ 10 = 1 / 100 = 1 / 10 2 etc. Thus, the extension of the first column would have to look like. Place Value Notation Exponential Notation 10,00010 4 1,00010 3 10010 2 1010 1 1 1 / 10 1 / 100 1 / 1000 next

7. © Math As A Second Language All Rights Reserved next Again, assuming that we want the same pattern to continue, we notice that as we read down the second column, the exponent decreases by 1 from its value in the row above. In other words, the sequence of exponents would look like… 10 4, 10 3, 10 2, 10 1, 10 0, 10 -1, 10 -2, 10 -3, etc…

8. © Math As A Second Language All Rights Reserved next Our chart would then look like… Place Value Notation Exponential Notation 10,00010 4 1,00010 3 10010 2 1010 1 10 0 10 -1 10 -2 1 1 / 10 1 / 100 1 / 1000 10 -3

9. next © Math As A Second Language All Rights Reserved So if we want the chart to continue in the same way, what must be true is that… next 10 0 = 1 10 -1 = 1 / 10 1 10 -2 = 1 / 100 = 1 / 10 2 10 -3 = 1 / 1000 = 1 / 10 3 In other words, 10 n and 10 -n are reciprocals of one another. That is, if n is any positive integer 10 -n = 1 ÷ 10 n next

10. © Math As A Second Language All Rights Reserved Notice that defining 10 0 to be 1 is reasonable in the sense that if n is any positive integer, 10 n is a 1 followed by n zeroes. So it seems natural that if the exponent was 0, the number would be a 1 followed by no 0’s (that is, 1). Notes

11. next © Math As A Second Language All Rights Reserved We know that to multiply a decimal number by 1,000, we move the decimal point 3 places to the right, and to divide a decimal number by 1,000 we move the decimal point 3 places to the left. In that context, it seems natural that if 10 +3 tells us to move the decimal point 3 places to the right, that 10 -3 should tell us to move the decimal point 3 places to the left. Notes

12. next © Math As A Second Language All Rights Reserved However, mathematicians have a better reason for defining integer exponents the way we do. Without going into the reasons behind the decision, it turns out that there are good reasons to make sure that the rules that govern the arithmetic of non-zero whole number exponents should also apply to any other exponents as well. Notes

13. next © Math As A Second Language All Rights Reserved With this in mind, let’s see how we would have to define b 0. We already know that for positive integer exponents b m × b n = b m+n. next So if we were to let n = 0, the rule would become… b m × b 0 = b m+0 next Since m + 0 = m, this would mean that… b m × b 0 = b m If we now divide both sides of the above equation by b m, we see that b 0 = 1. 1 next note 1 If b = 0 then b m is also equal to 0. However, we are not allowed to divide by 0. Therefore, we have to add the restriction that b ≠ 0. next

14. © Math As A Second Language All Rights Reserved Another way of saying this is to observe that since b m × b 0 = b m, this equation tells us that b 0 is that number which when multiplied by b m yields b m as the product, and this is precisely what it means to multiply a number by 1. Note

15. next © Math As A Second Language All Rights Reserved Note By way of review, the reason we had to add the restriction that b ≠ 0 when defining b 0 is based on the fact that any number multiplied by 0 is 0. Notice that if we replace b by 0 in the equation b m × b 0 = b m, we obtain the result that 0 3 × 0 0 = 0 3. Since we know that 0 3 = 0, this says that 0 × 0 0 = 0. But, any number times 0 is 0! Therefore 0 0 can equal any number. When this happens, we say that the value of 0 0 is indeterminate; meaning that it can be any number. next

16. © Math As A Second Language All Rights Reserved next For example, suppose that for some “strange” reason we wanted to define 0 0 to be 7. If we replace 0 0 by 7, 0 × 0 0 = 0 becomes 0 × 7 = 0, which is a true statement. For b ≠ 0, we have defined b 0 to be 1. It does not have to be defined to equal 1, but 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 the rule for multiplying like bases.

17. © Math As A Second Language All Rights Reserved Students often feel that b 0 should equal 0 because there are no factors of b. What we can tell students when this happens is that there is nothing wrong with thinking that it should be 0, but if we let it equal 0, we lose the use of the rules that make computing so convenient. Key Point

18. next © Math As A Second Language All Rights Reserved Perhaps it will make it easier for students to feel comfortable with our defining b 0 to be equal 1 if we point out that since b n × 1 = b n to the exponent, 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. As we have already discussed, this is especially easy to visualize when b = 10. Namely, in this case 10 n is a 1 followed by n zeroes. Thus, 10 0 would mean a 1 followed by no 0’s, which is simply 1.

19. next © Math As A Second Language All Rights Reserved In the case where n = 0, 2 0 would be the number of outcomes that can occur if we don’t flip the coin at all. There is ONE outcome - the current state of the coin.

20. next © Math As A Second Language All Rights Reserved One reason that mathematicians prefer to work abstractly is that physical models do not always exist, and even when they exist, they might not make sense in some instances. For example, we cannot use the “flipping a coin” model to explain the use of negative exponents. However, as we have explained earlier, negative exponents make sense when we are dealing with powers of 10. next

21. © Math As A Second Language All Rights Reserved The “powers of 10” model gives us a clue as to why b -n = 1 / b n might still be a correct rule to use even if b ≠ 10. More specifically, we have already given a plausible explanation as to why this is true in the case where b = 10. So based on the fact that 10 -n = 1 / 10 n, we might want to conjecture that for any base b, b -n = 1 / b n. A more consistent reason might be that we still want b m × b n = b m+n to be true even when m and/or n is a negative integer. next

22. © Math As A Second Language All Rights Reserved To this end we already know that when we multiply like bases we add the exponents, and we also know that for any number, n, n + -n = 0. next Therefore… b n × b -n = b n + -n = b 0 next And since we have already accepted the definition that b 0 = 1, it follows that… So if now we divide both sides of the equality b n × b -n = 1 by b n, it follows that b -n = 1 ÷ b n. b n × b -n = b n + -n = b 0 = 1

23. next © Math As A Second Language All Rights Reserved More generally, if m and n are any integers and b and c are any numbers, then it is still true that… (6) (b × c) n = b n × c n (2) b 0 = 1 provide that b ≠ 0; 0 0 is indeterminate 2 (3) b n = 1 ÷ b -n (4) b m ÷ b n = b m-n (5) (b m ) n = b m×n (1) b m × b n = b m+n next note 2 Whenever an exponent is 0 or negative the base b cannot equal 0. next

24. © Math As A Second Language All Rights Reserved next To help you internalize the rules let’s suppose that you wanted to rewrite the expression 10 2 × 10 -5 as a decimal number. By Rule (1) (with b = 10, m = 2 and n = - 5) 10 2 × 10 -5 = 10 -3 = 0.001. As a check, notice that… 10 2 × 10 -5 = 10 2 × 1 / 10 - 5 = 100 × 1 / 100,000 = 100 / 100,000 = 1 / 1000 = 0.001

25. next © Math As A Second Language All Rights Reserved In the same way that something may increase exponentially, another thing might decrease exponentially. One such example is in terms of radioactive decay where we talk about the half-life of a radioactive substance. The half life is the amount of time it takes for the substance to “shrink” to half its present weight (mass). An Enrichment Note on Integer Exponents

26. next © Math As A Second Language All Rights Reserved next Rather than talk about radioactivity, let’s suppose instead that you have received a gift of $128 and you don’t want to spend it all at once. Instead you decide to spend half of it now and then each of the following weeks, you will spend half of what is left. So the first week you spend half of the $128, leaving you with $64; the next week you spend half of $64, leaving you with $32; the next week you would spend half of the $32, leaving you with $16, etc.

27. © Math As A Second Language All Rights Reserved next Generally, if P denotes the present amount, a week later the amount left is 1 / 2 × P, after the second week the amount left is 1 / 2 × ( 1 / 2 × P) or ( 1 / 2 ) 2 × P, and after the third week the amount left is ( 1 / 2 ) 3 × P., etc. However, in order not to have to use fractional notation we may rewrite an expression such as ( 1 / 2 ) 3 × P in the form 2 -3 × P. After n weeks the amount left could then be expressed as 2 -n × P.

28. © Math As A Second Language All Rights Reserved Until now there seems to be no pressing reason to define fractional exponents. However, let’s suppose that we knew that the cost of living was increasing at a rate of 4% per year. That means what costs $1.00 this year will cost $1.04 next year. So if C represents the cost of living this year, the cost of living next year will be 1.04 × C An Enrichment Note on Fractional Exponents

29. next © Math As A Second Language All Rights Reserved Thus, the next year it will be 1.04 × C, not C, that is increasing by 4% a year. So at the end of the second year the cost of living is 1.04 × (1.04 × C), or 1.04 2 × C. And in a similar way at the end of the third year the cost of living is 1.04 3 × C. More generally, at this rate at the end of n years the cost of living would be 1.04 n × C. However, if we wanted to know the cost of living 6 months (i.e., 1 / 2 year) from now, using the above formula, it would be given by 1.04 1/2 × C. next

30. © Math As A Second Language All Rights Reserved This motivates us to try to define 1.04 1/2 More specifically, if we still want it to be true that b m × b n = b m+n, we may replace m and n by 1/2 to obtain… b 1/2 × b 1/2 = b 1/2 + 1/2 = b 1 = b. In other words, b 1/2 is that number which when multiplied by itself is equal to b. By definition of the square root (that is, the square root of a given number is the positive number which when multiplied by itself is equal to the given number), it means that b 1/2 = √b. next

31. © Math As A Second Language All Rights Reserved As a check that this definition is plausible, we can use our calculator and representing 1 / 2 as 0.5, we see, for example, that 9. 0.5 = 3, and as a check, we know that 3 x 3 = 9. Hence, 9. 0.5 is the square root of 9. In a similar way, we see that 1.04 1/2 = 1.01980…, and this, in turn tells us that if something costs $100 today, 6 months from now it will cost 1.01980… × $100 or, to the nearest cent, $101.98. next

32. This discussion completes our arithmetic course. © Math As A Second Language All Rights Reserved Algebra is next. We hope you will join us for our algebra course.