1. 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 8

2. Scientific Notation and Significant Figures Scientific Notation and Significant Figures © 2007 Herbert I. Gross next

3. As a lead-in to scientific notation let's revisit mixed numbers. Recall that a mixed number consists of two parts; a whole number plus a fraction that is less than 1. Thus, whenever we see a mixed number as an answer to a question, we know immediately between what two consecutive whole numbers the answer lies. © 2007 Herbert I. Gross 6 11 / 25 6 7 next

4. For example, when we say 4 hours and 29 minutes (that is, 4 29 / 60 hours), we’d know immediately that the time was more than 4 hours but less than 5 hours. We could have described the time by saying such things as 269 minutes or 2 hours and 149 minutes etc. However, the mixed number format gives us a much better idea of what the number of hours was. © 2007 Herbert I. Gross next

5. Scientific notation is based on somewhat the same idea. Namely, in place value the powers of ten are our nouns and whenever we get to 10 of any power of ten, we can exchange them for the next higher power of ten. Thus, just as we can exchange ten 1's for one ten or ten 10's for one 100, we can continue this process for any integral power of 10. For example, we can exchange ten 10 11 's for one 10 12. © 2007 Herbert I. Gross next

6. Notice that 10 11 may be viewed as 1 × 10 11 and 10 12 may be viewed as 10 × 10 11. Hence if c is any number such that 1 ≤ c < 10, then c × 10 11 is at least as great as 10 11 but less than 10 12. © 2007 Herbert I. Gross next More generally if 1 ≤ c < 10, then c times any power of 10 is a number between that power of 10 and the next greater power of 10.

7. For example, let's look at 6.34 × 10 11. Every time we multiply by 10 we move the decimal point one place to the right. Since multiplying by 10 11 is the same as multiplying by 10 eleven times, we move the decimal point in 6.34 eleven places to the right (and since there are only two digits to the right of the decimal point, we have to annex nine 0's as place holders). © 2007 Herbert I. Gross next 6.34 × 10 11 =6 3 4.000000000 1234567891011 places which is greater than 10 11 (100,000,000,000) but less than 10 12 (1,000,000,000,000).,,, next

8. © 2007 Herbert I. Gross next 10 11 next = 1 × 10 11 = 100 billion 10 12 = 10 × 10 11 = 1 trillion 6.34 × 10 11 = 634 billion = 100,000,000,000 = 1,000 billion = 634,000,000,000 = 1,000,000,000,000

9. A number is said to be in scientific notation if it is written in the form c × 10 n where n is an integer and c is a constant such that 1 ≤ c < 10. The value of n is called the order of magnitude of the number. © 2007 Herbert I. Gross next With this as motivation, we define scientific notation in the following way. Definition next

10. Write 67,897 in scientific notation. Practice Problem #1 © 2007 Herbert I. Gross Answer: 6.7897 × 10 4 Solution: To be in scientific notation, the number that modifies the power of 10 must be at least 1 but less than 10. Hence given the sequence of digits 67,897 we want to write it in the form 6.7897×10 n next

11. Practice Problem #1 © 2007 Herbert I. Gross Solution: To convert 67,897 into 6.7897 we have to move the decimal point four places to the left. This is the same as dividing the number by 10,000, or, equivalently, multiplying it by 10 -4. next To undo dividing by 10,000 we have to multiply by 10,000, and this is the same as multiplying by 10 4. next

12. Practice Problem #1 © 2007 Herbert I. Gross Solution: Hence a “cute” way to multiply 67,897 by 1 is to rewrite it as… next 67,897 × 10 -4 × 10 4 ( ) next (6 7 8 9 7 × 10 -4 ) ×,...,.. next -3-3 -2 0 10 4 ( 6.7 8 9 7 × 1 ) ) × 10 4

13. Practice Problem #1 © 2007 Herbert I. Gross Solution: Hence a “cute” way to multiply 67,897 by 1 is to rewrite it as… next 67,897 × 10 -4 × 10 4 ( ) next (6 7 8 9 7 × 10 -4 ) ×.... next -3-3 -2 0 10 4 ( 6.7 8 9 7 × 1 ) ) × 10 4.

14. As a check, notice that multiplying by 10 4 means that we move the decimal point 4 places to the right. Hence 6.7897 × 10 4 = 67,897. Note © 2007 Herbert I. Gross next Because 1 ≤ 6.7897 < 10 and 67,897 = 6.7897 × 10 4 the number is now written in scientific notation, and its order of magnitude is 4. next

15. We could also use exponential notation to express 67,897 in such forms as… Note © 2007 Herbert I. Gross next 67.897 × 10 3 678.97 × 10 2 0. 67897 × 10 5 678,970 × 10 -1

16. However, none of the above forms, while they correctly name the number 67,987, are in scientific notation because in each case the number that modifies the power of 10 is either greater than 10 or less than 1. Note © 2007 Herbert I. Gross next 67.897 × 10 3 678.97 × 10 2 0. 67897 × 10 5 678,970 × 10 -1

17. Express the number 0.00389 in scientific notation. Practice Problem #2 © 2007 Herbert I. Gross Answer: 3.89 × 10 -3 Solution: To be in scientific notation, the number that modifies the power of 10 must be at least 1 but less than 10. The only place we can move the decimal point to achieve this in the given number is between the 3 and the 8. next

18. Practice Problem #2 © 2007 Herbert I. Gross Solution: However, to get from 0.00389 to 3.89 we have to move the decimal point 3 places to the right (that is we have to multiply by 10 3 ). Hence in order not to change the value of the number, we would then have to move the decimal point back 3 places to the left (that is, we would have to multiply by 10 -3 ). next

19. Practice Problem #2 © 2007 Herbert I. Gross Solution: With this in mind we rewrite 0.00389 in the following way… next 0.00389 × 10 3 × 10 -3 ( ) next (0.0 0 3 8 9 × 10 3 ) ×10 -3... next And this tells us that the order of magnitude of 0.00389 is negative 3. More generally, if the order of magnitude is negative, the number is less than 1 but still positive.

20. © 2007 Herbert I. Gross Scientific notation is helpful when we deal with the topic known as significant figures. The following anecdote serves as a non- threatening way to introduce the topic. next A curator at a museum claimed that a particular artifact was approximately 1,000,013 years old. When questioned about how the age could be estimated that closely, he replied, "It isn't all that complicated. When I came to work here I was told that it was approximately 1 million years old; and I've been working here for 13 years!”

21. © 2007 Herbert I. Gross next This anecdote serves as a segue for introducing significant figures. More specifically, the chances are that the 1,000,000 years old estimate may have been measured to the nearest hundred or thousand years, thus making the 13 years insignificant. To relate this to the real world let's look at a statement such as “This piece of string is exactly 6 inches long”. In the ideal world (or in a mathematics textbook) it makes sense to say that a piece of string is exactly 6 inches long. However, in the real world we can never be sure.

22. © 2007 Herbert I. Gross next And even if we decided to declare that the beginning of the 6 inch marking on a ruler was “exactly 6 inches”, we could then use a magnifying glass or a microscope to enlarge the 6 inch marking; and we'd find the same problem occurring. Namely, is it the beginning of the beginning of the 6 inch marking, the middle of the beginning of the 6 inch marking, etc., etc., etc., until eventually we couldn't measure any closer.

23. © 2007 Herbert I. Gross next So in the world of science and engineering we recognize that we can never get an exact measurement. We do, however, have instruments that can measure accurately to a large number of decimal places. In this context, such numbers as 6.0, 6.00, 6.000 (which in pure mathematics are just different ways to express the number 6) mean different things to those who are making real world measurements.

24. © 2007 Herbert I. Gross next Thus, for example, when a technician says that a piece of string is 6.0 inches, it means that measured to the nearest tenth of an inch, the measurement rounds off to 6 inches. In other words, if we let L denote the exact number of inches, in the real world L = 6.0 means that 5.95 < L < 6.05. In this case, either we say that the length is accurate to two significant figures (the 2 figures being the 6 and 0 in 6.0), or that the length is measured accurately to the nearest tenth of an inch.

25. © 2007 Herbert I. Gross next In a similar way if L = 6.00, it means that rounded off to the nearest hundredth of an inch L = 6. That is, 5.995 < L < 6.005. In this case we say that the length is accurate to three significant figures (i.e., 6.00)

26. © 2007 Herbert I. Gross Using significant figures an engineer writes that the dimensions of a rectangle are 6.0 meters by 3.0 meters. What is the least area the rectangle can have? Practice Problem #3a next Answer: 17.5525 meters 2

27. © 2007 Herbert I. Gross Using significant figures an engineer writes that the dimensions of a rectangle are 6.0 meters by 3.0 meters. What is the greatest area the rectangle can have? Practice Problem #3b next Answer: 18.4525 meters 2

28. Practice Problem #3a & b © 2007 Herbert I. Gross Solution: 6.0 meters means that the measurement is accurate to the nearest tenth of a meter. To round off to the nearest tenth we have to look as far as the hundredths place. That is, we may look at 5.9, 6.0 and 6.1 as being 590 centimeters, 600 centimeters and 610 centimeters respectively. Since 595 is midway between 590 and 600, and since 605 is midway between 600 and 610, we see that if the length, L = 6.0 meters (m)… next

29. Practice Problem #3a & b © 2007 Herbert I. Gross Solution: Then… 5.95 m < L < 6.05 m In a similar way if W denotes the width of the rectangle, then… 2.95 m < W < 3.05 m next Hence… 5.95 m < L < 6.05 m × 2.95 m < ×W < × 3.05 m 17.5525 m 2 18.4525 m 2 next L × W< <

30. If the length was exactly 6 meters and if the width was exactly 3 meters, then the area would be exactly 18 square meters. Note © 2007 Herbert I. Gross next However, because of limitations in our ability to measure exactly, in this particular problem the exact area can be any number between 17.5525 square meters and 18.4525 square meters. next

31. What our calculation shows is that in this problem measured to the nearest square meter the area of the rectangle is 18 square meters; but we can't say much beyond this! Note © 2007 Herbert I. Gross next

32. © 2007 Herbert I. Gross next With the above as background, let's relate our earlier anecdote to the statement "The sun is 93,000,000 miles from Earth”. When we talk about the sun being 93 million miles from earth, we don't mean exactly 93 million miles. In fact, we might even be off by 10,000 miles or 100,000 miles. To indicate how accurately our measurement is we use scientific notation to tell us the number of significant figures we have.

33. © 2007 Herbert I. Gross next Using scientific notation, we would ordinarily write 93 million miles as 9.3 × 10 7 miles. From the purely mathematical point of view, there is no difference whether we write 9.3 × 10 7 miles, 9.30 × 10 7 miles or 9.300 × 10 7 miles. However, the number of digits we use to modify the power of 10 tells us the number of significant figures. For example, 9.30 × 10 7 would mean that only the first three digits of 93,000,000 are significant.

34. © 2007 Herbert I. Gross next That is, if the distance is represented as 9.30 × 10 7, then our measurement is accurate only up to 93,0 00,000. In other words, the distance is between 929.5 hundred thousand and 930.5 hundred thousand miles. Hence, if the sun is 9.30 × 10 7 miles from earth, it is at least 92,950,00 miles but not greater than 93,050,00 from earth.

35. © 2007 Herbert I. Gross next For the sake of simplifying computation in the “pre-calculator” era, it was important to use significant figures. However, the concept of significant figures often caused confusion because different people used different rules. With the abundance and availability of calculators, it is now more convenient and less confusing to use such notations as 6 ± 0.05 to indicate that a measurement has to be between 5.95 and 6.05 than to write 6.0 Note

36. © 2007 Herbert I. Gross next That is, in the pre-calculator era it was cumbersome to compute such products as 5.99963 × 2.98742, but with a calculator this isn’t a problem. We can obtain the exact product rather quickly and then round off to the desired degree of accuracy. Note next

37. © 2007 Herbert I. Gross next Note next For example, suppose we know that a number L is 6 ± 0.001 and another number W is 3 ± 0.002, and we want to compute the product L × W. This means that… 5.999 < L < 6.001 2.998 < W < 3.002××× 17.985002 18.015002 L × W << next

38. © 2007 Herbert I. Gross next From the inequalities we see that the exact value of L × W is greater than 17.985002 m 2 but less than 18.015002 m 2. Note Since rounded off to the nearest tenth both 17.985002 and 18.015002 equal 18.0. We can say with certainty that measured to the nearest tenth L × W is exactly 18. However, this is about as certain as we can get. 17.985002 < L × W < 18.015002 next

39. © 2007 Herbert I. Gross next Geometric Note next From a geometric point of view, we can visualize L × W as the area of a rectangle whose length is L and whose width is W. Therefore, if the length of the rectangle is 6±.001 meters, and the width is 3±.002 meters, the rectangle fits inside the rectangle whose length is 6.001m and whose width is 3.002m.

40. © 2007 Herbert I. Gross next In terms of a picture… R denotes the rectangle whose dimensions are 6.001 meters by 3.002 meters. 3.002m R 6.001m Geometric Note A R = Area of R = 6.001m × 3.002 m next

41. © 2007 Herbert I. Gross next S denotes the rectangle whose dimensions are 5.999 meters by 2.998 meters. 2.998m 5.999m Geometric Note A s = Area of S = 5.999m × 2.998 m S next

42. © 2007 Herbert I. Gross next Q denotes the rectangle whose dimensions are L (in meters) by W (in meters). W L Geometric Note A Q = Area of Q = L meters × W meters where 5.998 < L < 6.001 and 3.002 < W < 2.998 Q next

43. Then R, Q, and S can be “nested” as shown below… Using the fact that if one region is contained within another region, the area of the contained region is less than the area of the containing region. R S Q

44. We know that A S < A Q < A R. That is… 5.999m × 2.998m < L × W < 6.001m × 3.002m R S Q 17.985002m 2 < L × W < 18.015002m 2

45. © 2007 Herbert I. Gross next In summary… 5.999 m < L < 6.001 m 2.998 m < W < 3.002 m××× 17.985002 m 2 18.015002 m 2 L × W<< 6 ±.001 m 3 ±.002 m L W Area next

46. © 2007 Herbert I. Gross next Note As the accuracy of our measurements increases, so also does the accuracy of the final product. next For example, suppose the length is 6 ± 0.0000001 m and the width is 3 ± 0.0000001 m, then the area is greater than 5.9999999 m × 2.9999999 m = 17.99999910000001 m 2 but less than 6.0000001 m × 3.0000001 m = 18.00000090000001 m 2 which tells us to the nearest hundred thousandths of a square meter the exact area is 18 square meters.

47. © 2007 Herbert I. Gross next Final Note In many text books the value for π (pi) is often given as either 22 / 7 ( 3 1 / 7 or 3.142857…) or more simply as 3.14. next However, π is equal to 3.14159… What is true, is that measured to the nearest hundredth π equals 3.14 (as well as 22 / 7 ). There are times when the value of π is given to be 3.1416, and in this case the estimate is accurate to the nearest ten thousandth.