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

2. In our discussion of the development of our number system we talked about how even using place value notation it became cumbersome to denote large numbers. next © Math As A Second Language All Rights Reserved For example, an important constant in the study of chemistry is known as Avogadro's Number, which is the number of atoms in a mole of water (a mole of water weighs a little bit more than a half ounce). In place value notation it is approximately… 600,000,000,000,000,000,000,000.

3. So we invented a new notation for expressing powers of 10. Specifically, we observed that every time we multiply a whole number by 10 we annex a 0 in the ones place. next © Math As A Second Language All Rights Reserved next And since… 600,000,000,000,000,000,000,000 = 6 × 100,000,000,000,000,000,000,000 …we see that to obtain Avogadro’s Number in place value notation, we multiply 6 by 23 factors of 10.

4. To abbreviate writing the product of 23 factors of 10, we invented the notation 10 23. next © Math As A Second Language All Rights Reserved We referred to 10 as the base and to 23 as the exponent and, as we just saw, in place value notation it is a 1 followed by 23 zeroes. Hence, in exponential notation, Avogadro’s Number is written as 6 × 10 23. next 600,000,000,000,000,000,000,000.

5. Think for a moment about how you would answer the question… © Math As A Second Language All Rights Reserved “How much is twenty 2’s?” next Cautionary Note Although “40” is the answer that most people would give (and it’s also the answer that is usually accepted as being correct), the wording of the question leaves much to be desired!

6. Most people tend to hear the question as if it were… next © Math As A Second Language All Rights Reserved “How much is the sum of twenty 2’s?” next Cautionary Note However, while it might not seem obvious at first, there are times when we are interested in answering the question… “How much is the product of twenty 2’s?”

7. next © Math As A Second Language All Rights Reserved next Cautionary Note There is a huge difference between the product of a given number of 2’s and the sum of the same number of 2's. For example, the sum of twenty 2’s is 40, and granted that the computation is tedious we can show that the product of twenty 2’s is 1,048,576.

8. next © Math As A Second Language All Rights Reserved next Definition In the same way that we wrote 10 20 to indicate the product of 20 ten’s, we use the notation 2 20 to indicate the product of 20 two’s. If b is any number and n is any positive integer, we use the notation b n to denote the product of n factors of b. We refer to b as the base and to n as the exponent and to b n as the n th power of b.

9. next © Math As A Second Language All Rights Reserved next So, for example, 4 3 means 4 × 4 × 4 or 64. In this case 4 is the base, 3 is the exponent and 64 is the 3rd power of 4. Be careful not to confuse the 3 rd power of 4 with the 4 th power of 3. The two numbers are not the same. Notes The 3 rd power of 4 = 4 3 = 4 × 4 × 4 = 64 The 4 th power of 3 = 3 4 = 3 × 3 × 3 × 3 = 81 while… next

10. © Math As A Second Language All Rights Reserved Notice that as the exponent increases it becomes increasingly more cumbersome to compute the power of the base. next In essence, to compute the value of 4 4, we first have to know the value of 4 3 ; to compute the value of 4 5, we first have to know the value of 4 4 ; and to compute the value of 4 6, we first have to know the value of 4 5. Notes

11. next © Math As A Second Language All Rights Reserved So, as you can see, if we use pencil and paper computation, it would be exceedingly cumbersome and time-consuming to compute, say, 4 20. next For that reason, it is helpful to have a scientific calculator - that is, a calculator that has a key that looks like… Notes xyxy xyxy

12. next © Math As A Second Language All Rights Reserved In that case, we could quickly compute the value of 4 20 by the following sequence of keystrokes. next Step 1: Enter 4 Notes 4 xyxy Step 2: Enter x y Step 3: Enter 20 Step 4: Enter = 20 1,099,511,627,776 next

13. © Math As A Second Language All Rights Reserved The calculator display tells us that 4 20 = 1,099,511,627,776 next Students might appreciate the power of the calculator by seeing how computing the answer “long hand” quickly becomes tedious and time consuming (not to mention, the chances of making a computational error during the process). Notes

14. next © Math As A Second Language All Rights Reserved However, the calculator is not a cure-all in the sense that while it can carry out the instructions you give it, it cannot tell you whether the instructions you gave it were correct. next For example, suppose we flip a penny, a nickel and a dime, and that we want to know the number of different ways in which the coins can turn up “heads” or “tails”. As we shall see below, the correct answer is given by 2 × 2 × 2 and not by 2 + 2 + 2.

15. © Math As A Second Language All Rights Reserved More specifically, looking at only the penny, we realize that it can turn up either “heads” or “tails”. There are just two possible outcomes. In terms of a chart… next Penny Outcome #1headsOutcome #2tails

16. next © Math As A Second Language All Rights Reserved Notice that the labels “Outcome 1” and “Outcome 2” are used simply to count the possible outcomes. next It doesn’t mean, for example, that Outcome 1 is more likely or more desirable than Outcome 2. Note

17. next © Math As A Second Language All Rights Reserved Regardless of which of the two outcomes occurs when the penny is tossed, the nickel can turn up in two possible ways, namely, either “heads” or “tails”. This doubles the number of possible outcomes. That is, whichever of the two ways the penny turns up, the nickel can be either “heads” or “tails”. Again, in terms of a chart… next Outcome #1heads Outcome #2tailsheadsPennyNickel OutcomesOutcome #3headstailsOutcome #4tails

18. next © Math As A Second Language All Rights Reserved Continuing in this way, we see that the same process applies with respect to the dime. No matter which of the above four possible outcomes occurs, the dime can turn up either “heads” or “tails”. next This again doubles the number of possible outcomes.

19. © Math As A Second Language All Rights Reserved In terms of a chart… next PennyNickel OutcomesDime Outcome #1heads Outcome #2tailsheads Outcome #3headstailsheadsOutcome #4tails headsOutcome #5heads tailsOutcome #6tailsheadstailsOutcome #7headstails Outcome #8tails

20. next © Math As A Second Language All Rights Reserved In the above case, listing the possible outcomes and then counting them is a simple process, but one that can quickly become tedious. Namely, with each additional coin, we double the number of possible outcomes. next In terms of actually listing the possible outcomes, it means that with each additional coin, we would double the number of rows in our chart.

21. © Math As A Second Language All Rights Reserved In particular, if we want to find the number of outcomes we could obtain if a fair coin was tossed 20 times (or if 20 coins were tossed once), we would have to compute the value of 2 20. next So while it is potentially “dangerous” to use the calculator for solving problems you don’t understand, the calculator is a great device for relieving you of the need to do tedious computations.

22. © Math As A Second Language All Rights Reserved In other words the calculator will not tell you that the correct instruction is to compute 2 20 (as opposed to, say, 2 × 20), but it will quickly show you that 2 20 = 1,048,576 next There is a big difference between knowing how to count and knowing what to count.. Note

23. next © Math As A Second Language All Rights Reserved In our previous discussion we saw that for a small number of flips of a coin, it was easy to count the number of possible outcomes simply by listing them. next However, while it might have been tedious to compute the product of twenty 2’s, it’s a lot more tedious to list the 1,048,576 possible outcomes that can occur if 20 coins are flipped! Notes

24. next © Math As A Second Language All Rights Reserved With respect to our present discussion, it wasn’t important from a mathematical point of view that we were flipping coins. next Notes The more important thing is that we were dealing with an event that had only two possible outcomes.

25. next © Math As A Second Language All Rights Reserved The same mathematical reasoning can be applied to guessing answers on a true-false type of quiz (each statement must be either true or false but not both) to constructing sets (each object can either belong to a set or not; one or the other but not both); to betting on athletic events where there are no ties (that is, a team can either win or lose; one or the other, but not both). Notes

26. next © Math As A Second Language All Rights Reserved To list the possible outcomes of guessing on a test that has three “true-false” questions, all we have to do is take the chart we made for “flipping a coin 3 times” and (1) replace the words “heads” and “tails” by “true” and “false”, respectively, (2) replace the “coin” in the column heading by “Question”… next Question 1Question 2 OutcomesQuestion 3 Outcome #1true Outcome #2falsetrue Outcome #3truefalsetrue Outcome #4false true Outcome #5true false Outcome #6falsetruefalse Outcome #7truefalse Outcome #8false

27. next © Math As A Second Language All Rights Reserved Because virtually everyone has experienced such events, we have so far limited our discussion of exponential growth to flipping coins and looking at true/false questions. next However, exponential growth plays an important role in many aspects of life, including economics, science and financial planning.

28. © Math As A Second Language All Rights Reserved For our purposes at the elementary school level, it’s probably best to illustrate this with an application to money, which is something all students can relate to. next So, for example, suppose you are 25 years old and that you want to retire at age 60. Investments

29. next © Math As A Second Language All Rights Reserved You have been lucky enough to find an investment fund that will double the amount of money you have in the account every 7 years. next You have $10,000 that you can invest now, and you want to know how much your investment will be worth when you retire (that is, 35 years from now). Investments

30. next © Math As A Second Language All Rights Reserved The chart below gives us the answer… next Investments Time Amount in Your Retirement Fund Now$10,000 After 7 years $20,000 After 14 years $40,000 After 21 years $80,000 After 28 years $160,000 After 35 years $320,000

31. next © Math As A Second Language All Rights Reserved Notice that the doubling process doesn’t depend on how much you have invested in the fund. next Since every 7 years your investment doubles and since 35 is the 5th multiple of 7, in 35 years your investment will double five times. That is, your original investment is 2 5 or 32 times what you have now. Notes

32. next © Math As A Second Language All Rights Reserved In the present illustration, the original investment happened to be $10,000; and in this case 32 × $10,000 = $320,000. next Notice the special property of exponential growth. Notes That is, every 7 years the investment becomes double what it was 7 years earlier; not from what it was at the start of the original investment. next

33. © Math As A Second Language All Rights Reserved After 7 years it is the $10,000 that doubles but after the 14th year it is the $20,000 that doubles. next Notice, then, that if at the end of the 35th year you decided to renew the investment for another 7 years, it is the $320,000 that will double! Notes

34. next © Math As A Second Language All Rights Reserved Science and mathematics are in themselves, neither good nor bad. The principles are the same whether they are applied to “good” things or to “bad” things. next Enrichment

35. next © Math As A Second Language All Rights Reserved For example, the fact that 2 20 is greater than 1,000,000 tells us that if a person went out and helped 2 people, and then each of these 2 people went out and helped 2 other people, etc.; by the 20th link in this “chain” over one million people would have been helped with nobody having to help more than 2 people! Enrichment

36. next © Math As A Second Language All Rights Reserved Yet the same mathematics tells us that if a person infected 2 people with AIDS; and then each of these 2 people infected 2 other people with AIDS, etc.; by the 20th link in the chain over one million people would have been infected by AIDS, with no one having to infect more than 2 people. Enrichment

37. next © Math As A Second Language All Rights Reserved Which way the mathematical result is used depends on society as a whole; not just on the scientist and mathematician. This is why the humanities, the social sciences and the physical sciences should be studied as a unified whole rather than in fragmented form. Enrichment

38. next In the next presentation, we will begin a discussion of the arithmetic of whole number exponents. © Math As A Second Language All Rights Reserved Arithmetic of Whole Number Exponents