3.2 Significance and Uncertain Numbers 3.2.1 The Art of Estimating In physical sciences and in other applications, it is necessary to do calculations with uncertain data, such as measurements and statistics obtained from physical observations. The results of these calculations will carry uncertainty. A full treat- ment requires tools such as statistics and calculus, but we can give a few basic guidelines for making these estimates. Some variations on our treatment are men- tioned in the footnotes and in the exercises. Contents 3.2 Significance and Uncertain Numbers 1 3.2.1 The Art of Estimating . . . . . . . . . . . . . . . . . . . . . . . . 1 3.2.2 Uncertainty in Applications . . . . . . . . . . . . . . . . . . . . . 2 3.2.3 Reporting Uncertain Numbers . . . . . . . . . . . . . . . . . . . 4 3.2.3.1 Intervals of Uncertainty . . . . . . . . . . . . . . . . . . 4 3.2.3.2 Significant Digits. . . . . . . . . . . . . . . . . . . . . . 5 3.2.3.3 Significant Digits and Mathematical Operations . . . . . . 8 3.2.3.4 Significance and Rounding . . . . . . . . . . . . . . . . . 12 3.2.4 Approximate Equality . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.7 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . 19 3.2.8 Appendix: Precision and Accuracy . . . . . . . . . . . . . . . . . 20 1 3.2.2 Uncertainty in Applications In applications of mathematics to real-world problems, small errors are often in- troduced by the act of measurement. Consider the following problem: (Room Area Problem - First Attempt) The floor of a rectangular room is 5.261 meters long and 3.741 meters wide. Determine its area. If the room were perfectly rectangular and the measurements of the length and width were exact, then we could say that the area of the floor is just the length times the width: (5.261 m)(3.741 m) = 19.681401 m2. Units of Measure: The symbol “m2” stands for the square meter, the standard unit of area in the Syst`eme Internationale (SI, commonly known as the metric system). Since a meter is about 3.28 feet, a square meter is about 10.8 square feet. But the measurements probably have errors. Some of these may be purely physi- cal, for example, rulers and tape measures expand slightly in higher temperatures. Some may involve problems with the way the the measuring instrument is used, such as allowing too much slack in a tape measure – or stretching the tape too much. Other errors arise because of the precision of the measuring instrument. For example, my metric tape measure has marks 1 millimeter apart, so it cannot easily be used for measurements much more precise than that. And of course the room is probably not a perfect rectangle. We might need to have a good estimate of the area of our room in order to apply a polyurethane finish to the floor. If the polyurethane is expensive, overesti- mating the room size could be costly. On the other hand, if we underestimate the room size, we will not have enough finishing fluid – we might incur extra costs if the job needs to be redone or if contractors need to be called back for another day’s work. So about how reliable is our estimate of 19.681401 square meters for the area of our floor? Do all those decimal places really provide useful information? (Room Area Problem - Refinement) After taking many independent measurements of the room, I find the length is between 5.15 m and 5.38 m, and the width is between 3.61 m and 3.82 m. What are the smallest and largest possible areas for the room? 2 Using the measurements provided: Amin = Minimum Area = (5.15 m)(3.61 m) = 18.5915 m2 Amax = Maximum Area = (5.38 m)(3.82 m) = 20.5516 m2 It seems silly to provide an answer like 19.681401 square meters when the correct value could be anywhere from roughly 18.6 to 20.6 square meters. Don’t give six extra decimal places when answer is only reliable to the nearest square meter. In addition, the measurements of the length and the width in the refinement of the problem suggest that our measurments are still too precise. How can we approach the problem to reduce the workload and come up with an answer which is as precise as the data allows, but no more? The given measurements for length and width suggest that these measurements are only reliable to about one tenth of a meter. If the width could be anywhere between 5.152 and 5.377 meters, then it is misleading to say that the width is 5.261 meters. It would be better to say that the width is about 5.25 plus or minus about 0.1 meters. The dimensions and the area of our rectangle all seem to be reliable to only about two digits. These considerations give us a new statement of our room area problem and an outline of its solution: (Room Area Problem) The floor of a rectangular room is approximately 5.3 meters long by 3.7 meters wide. Find a good estimate for its area. Calculations: (5.3 m)(3.7 m) = 19.61 m2 = about 20 m2. Answer: 20 square meters . (The underscored zero will be explained in the next section.) There are two technical questions that need to be answered: • How do we decide to round from 19.61 square meters to 20 square meters instead of something else? For example, why not round to 19.6 square meters? • How do we interpret the answer of 20 square meters? Fortunately both questions can be answered with very little additional calculation. We will refer back to this problem several times in the next section. 3 3.2.3 Reporting Uncertain Numbers 3.2.3.1 Intervals of Uncertainty When reporting data, it is important to know how precise the data are. In our room area problem, we were originally given that the length of the room was 5.261 meters. After making additional measurements, we found that the length of the floor was between 5.152 meters and 5.377 meters. Perhaps the room wasn’t quite rectangular, or perhaps the floorboard made it hard to get precise measurements – it would appear our measurements are only precise to about a tenth of a meter. The interval (5.152 m, 5.377 m) could be taken as a rough indication of how precise the room length measurements are. An interval of this sort is called an interval of uncertainty. For example, in measuring the depth of the deep end of a swimming pool, we might find that measurements fall between 12.4 and 12.7 feet. Here the interval of uncertainty is (12.4 ft, 12.7 ft). We can write down an uncertainty interval in another way using the “±” sym- bol. Instead of saying that the measured depth is between 12.4 and 12.7 feet, we might say that the measured depth at the deep end is 12.55±0.15 feet. Definition: When reporting uncertain data: x = c±r means c−r < x < c+r. One way of putting this into words is to say that measurements x that fall outside the interval (c−r,c+r) are significantly in error. Here the value c is the center of the interval and the value r is the radius of the interval.1 Plus or minus the radius (±r) of the uncertainty interval is called the uncertainty. Differences between values inside the uncertainty interval are treated as insignificant. Problem: The power consumption of a handheld electronic game is between 11.2 Watts and 11.7 Watts. Express this as uncertainty interval in the form P±r. Solution: The uncertainty interval is (11.2 W, 11.7 W). The center of the interval or the average power consumption is: P = 11.2+11.72 = 11.45 W. 1The radius is half of the width or the diameter of the interval. 4 The radius or uncertainty is half the width of the interval: r = 11.7−11.22 = 2.5 W. So power consumption is: P±r = 11.45±2.5 W . (The interval is called an uncertainty interval because, although we are reasonably certain that our data point lies somewhere in the interval, we are not certain where in the interval the data point lies. We usually assume that it is more likely to be near the center of the interval than near the ends.2) 3.2.3.2 Significant Digits. One system that works reasonably well and saves a considerable amount of writ- ing is based on the way we write numbers in the base 10 (or decimal) system. For instance 36,500 has three significant positions – the most significant is the ten thousands position (digit 3) and the least significant position is the hundreds position (digit 5). More generally, if an interval of uncertainty is not given, we adopt the following guidelines: • The nonzero digits of any number are always significant. (But some of the the zeros may be significant.) • The most significant position (MSP) in a nonzero base 10 number is the position with first or leftmost nonzero digit. For example, in the numbers 4860 and 0.005210, the MSP’s are thousands and thousandths, respectively. When we want to indicate the most significant position, we will use an overscore, for example 4860 or 0.005210. • For an inexact number, the least significant position (LSP) is found as follows: • If the number is written with a decimal point, the LSP is the position of the rightmost digit, whether zero or not. 2A different approach to uncertainties in data is the confidence interval in statistics. A con- fidence interval is associated with a stated level of confidence that a given data point lies in the interval. With an interval of uncertainty, we are not given a level of confidence. 5 • If the number is written without a decimal point, the LSP is the posi- tion of the rightmost nonzero digit. In the number 4860, the zero is insignificant, so the LSP is the tens position (the digit 6). For 0.005210, the zero in the last position is significant, so the LSP is the millionths position. When we want to indicate the least significant position, we will use an underscore, for example, 4860 or 0.005210. The numbers 4860 (4860) has three significant digits (the 4, 8 and 6 in the thousands, hundreds and tens positions). The number 0.005210 (0.005210) has four significant digits (from left to right: 5, 2, 1 and 0). The number of significant digits is the the number of positions counting from left to right from the most significant position (overscored) to the least significant position (underscored). Then the inexact number 980 indicates a value between about 970 and 990, that is about 980±10 while the inexact number 976 indicates a value of about 976±1. But what about a value like 900±1? That shouldn’t be written as 900 since that suggests a value ranging between 800 and 1000. Instead we could write 900±1 in scientific notation as 9.00×102. The inexact values 100, 1.0×102, 1.00×102 and 1.000×102 have 1, 2, 3, and 4 significant digits, respectively. All of these are ”about 100”, with varying degrees of uncertainty. (The last of these, 1.000×102, could also be written as 100.0.) We can also use the underscore to change the least significant position. In our room area problem, we wrote our answer as 20 square meters – the underscored zero tells us that the digit 0 is significant – the answer has two sig- nificant digits. In the expression 1.234, the underscore indicates that the 4 in the thousandths position is insignificant. Significant digits for negative numbers are handled just as for positive num- bers. So -4860 amperes represents an electric current between about -4870 and about -4850 amperes. Paradoxically, because the uncertain number 0 has no nonzero digits, it has no significant positions. (We can use an underscore to mark a least sigificant position if we need to do that.) To get an uncertainty interval based on significant digits, we added a value of ±1 in the least significant position. So a measured mass of 2.040 grams indicates a mass of about 2.040±0.001 grams.3 Table 1 illustrates these ideas with a few examples. In the room area problem in the previous section, the dimensions of the room were obtained by measurement and given as 5.3 by 3.7 meters. Since we have no 3Some writers prefer an uncertainty of half this value. A measurement of 2.040 grams would be treated as 2.040±0.0005 grams instead of 2.040±0.001 grams. 6 ine xa ct nu mb er mo st sig nifi can t po sit ion lea st sig nifi can t po sit ion sig nifi can t dig its ap pro xim ate un cer tai nty int erv al of un cer tai nty 1200 103 102 2 ±100 1200±100 1230 103 102 2 ±100 1230±100 1200 103 101 3 ±10 1200±10 1230 103 101 3 ±10 1230±10 1030 103 101 3 ±10 1030±10 1 100 100 1 ±1 1±1 1.0 100 10−1 2 ±0.1 1.0±0.1 1.2340 100 10−4 5 ±0.0001 1.2340±0.0001 0.0340 10−2 10−4 3 ±0.0001 0.0340±0.0001 −0.0340 10−2 10−4 3 ±0.0001 −0.0340±0.0001 Table 1: Significant digits and uncertainty intervals additional information about uncertainty, we apply the guidelines above. Using ± notation, the dimensions are 5.3±0.1 by 3.7±0.1 meters. The floor area of the room (about 20 square meters) was obtained by calcula- tion rather than by direct measurement so we have additional information about the interval of uncertainty. (The additional information comes from the measured dimensions of the room.) Using guidelines for multiplication which follow, the area has two significant digits. That suggests an uncertainty interval of about 19 to 21 square meters, in rough agreement with the interval of 18.6 square meters minimum to 20.6 square meters maximum obtained from the refinement of the original problem. 7 3.2.3.3 Significant Digits and Mathematical Operations When inexact numbers are used in mathematical processes (like addition), we can- not use the same kinds of guidelines for estimating the uncertainty of the results. The following guidelines are commonly used in science courses: 1. When adding or subtracting a small collection of inexact numbers, the least significant position of the result is the leftmost of the least significant po- sitions of the terms. Example: In the inexact sum 12.123459+2.3, the tenths position is sig- nificant, but the hundredths position is not. The result has three significant figures. (It would therefore be more appropriate to write the sum as 14.4 rather than 14.423459.) Cautionary Example: In the inexact difference 1.22−1.2 = 0.02, where we are using the overscore to mark the most significant position of the re- sult. Since the most significant position is to the right of the least significant position, the number of significant digits is undefined. The uncertainty in- terval contains both positive and negative real numbers.4 2. When multiplying or dividing a small collection of inexact numbers, the number of significant digits of the result is the smallest of the numbers of significant digits of the factors and divisors. Example: The inexact product 2.117×4.1 has two significant digits. Us- ing a calculator, the product is 8.6797. But since the result has just two significant digits, it would be more appropriate to write the result as 8.7 (or perhaps 8.68). Caution: Don’t divide by an inexact number whose interval of uncertainty includes zero. We have a few final words on the room area problem: • In the room area problem, the dimensions of the floor were about 5.3 by 3.7 meters. Both length and width have 2 significant positions. The number of significant positions in the product (19.61 square meters) is the minimum of 2 (the number of significant digits in the length) and 2 (the number of 4An uncertain number of this sort presents serious problems when used in products and quo- tients. Since the number could be either positive or negative, the sign of the product or quotient is unknown. Since zero is in the interval of uncertainty, division by the number may be undefined. 8 significant digits) in the width. So the area has two significant positions. It can be rounded to 20 square meters with an uncertainty of ±1 square meter. (But since we may be using this value in another calculation, i.e. the amount of floor finish, we might prefer to keep some extra precision and use the uncertainty interval 19.6±1 square meters, that is between 18.6 and 20.6 square meters.) • We can also use the data to estimate the uncertainty interval for the area of the floor with a little more work. Since the length is between 5.2 and 5.4 meters and the width is between 3.6 and 3.8 meters: Amin =(5.2 m)(3.6 m)= 18.72 m2, and Amax =(5.4 m)(3.8 m)= 20.52 m2. The area is between 18.7 and 20.5 square meters. This “squares well” with our estimated area of between 19 and 21 meters, and even better with our alternate estimate of between 18.6 and 20.6 meters. Now let us consider a similar problem with a few complications:. . . a b h Figure 1: Gotham City Community Theatre stage Problem: The stage of the Gotham City Community Theatre has a trapezoidal floor (see Figure 1) with dimensions a = 18.2 meters, b = 15.1 meters and h = 14.3 meters. The floor needs to be refin- ished with a special finish (Formula X) for an upcoming production. If one can of finish will cover c = 0.43 square meters of floor, then how much finish should be ordered? (Do the calculations in two ways: (a) estimate the number of cans needed by using significant digits and (b) find the maximum number of cans of varnish that might be needed.) 9 Solution: The area of the floor is A =h(a+b)/2. Let x be the number of cans needed. Then x cans will cover A = cx square meters of floor. Then x = A/c or: x = h(a+b)2c (Method a) The sum a+b has three significant digits. The number 2 is exact, so the number of significant digits in x is the minimum of the number of significant digits in h, in a+b and in c. Since h and a+b both have three significant digits and c has two significant digits, the result has two significant digits. x = 14.3 m(18.2 m+15.1 m)2(0.43 m2/can) = 553.7 = about 550 cans. Since the uncertainty interval seems to be about 550±10 cans, to be safe, we should order 560 cans. (From the calculations, we expect to use about 554 cans, so we have an expected waste of six cans.5) Algebra Fact: To increase the size of a fraction AB, we in- crease the size of the numerator A and decrease the size of the denominator B. To find the maximum value of AB , we find AmaxBmin . Caution: You have to do some tweaking if some of the num- bers are negative. (Method b) To obtain a better estimate of the maximum value, we need to use the maximum values of the lengths in the numerator and the minimum value of the coverage rate in the denominator. Since the number of cans must be an integer, fractional results should be rounded upward: xmax = 14.4 m(18.3 m+15.2 m)2(0.42 m2/can) = 574.3 = about 575 cans. We would be less likely to run out of varnish with a larger order. However, since we only expect to use 554 cans, we expect to waste twenty-one cans. There is a trade-off here. Since the estimate in part (a) is lower than the estimated maximum in part (b), there is some risk of running out of varnish. On the other 5If we feel uneasy about rounding down, we might use an uncertainty interval of 554±10 cans, making our order 564 cans with an expected waste of 10 cans. 10 hand, there is additional cost for ordering too much varnish. Evaluating this sort of trade-off is outside of the scope of this course. • Notation: We often use a squiggly equals sign to express the idea “equals about”. So instead of writing “x = about 575 cans”, we can save space by writing “x ≈ 575 cans”. Powers of numbers are generally treated as repeated multiplications, but care must be taken. Large exponents, both positive and negative, do cause dramatic loss of significance: even very small errors in the value of the base can cause a loss of significance.6 Example: Suppose that y = an where a = 1.05±0.01. We will look at errors when n = 2 and when n = 100. (a) Suppose that n is exactly 2. Using our guidelines for uncertainty, 1.052 is about 1.10. What happens at the endpoints of the interval of uncertainty for a? ymin = 1.042 = 1.0816, ymax = 1.062 = 1.1236. Both values are slightly outside the estimated interval of uncertainty, with a small error in the least significant position. The uncertainty guidelines give a reasonable estimate of the error. (b) Suppose that n is exactly 100. Since 100 is a large power, the guidelines don’t apply. Rounded to three significant positions, 1.05100 is 132. What happens at the endpoints? ymin = 1.04100 ≈ 50.5, ymax = 1.06100 ≈ 339. Saying that the answer is between 50.5 and 339 is not being very informative. Both values are well outside the interval of uncertainty, with disagreement even in the most significant position. Even with more precision, the situation does not improve much. For example, for 1.04±0.005, ymin = 1.045100 ≈ 81.6, ymax = 1.055100 ≈ 211. You can experiment with 1.04±0.001 by comparing 1.049100 and 1.051100 on your calculator. 6Better estimates of uncertainty for powers can be obtained using differentials, normally cov- ered in a calculus course. 11 3.2.3.4 Significance and Rounding When carrying out calculations, it is best to use all available precision. In short, don’t round intermediate results. However, when reporting results to others, it is important to convey the uncertainty in the calculated data. This can be done by rounding or by giving an uncertainty interval. Sometimes the result of a calculation can serve as both a final result and an intermediate result. In this case, the unrounded value should be used in later calculations, but the rounded value or the uncertainty interval should be reported. When we report an inexact number as a final result, we normally round to the least significant digit. For example: 2.117×4.1 = 8.6797±0.1 ≈ 8.7. In applications such as engineering, handbooks are often used to to supply stan- dard reference values for various quantities. Since these values can be used in many calculations, it is often desirable to retain additional digits, for example 8.68±0.10 or 8.68. As we saw in the room area problem, an extra digit of preci- sion can provide a better center for the uncertainty interval. 12 3.2.4 Approximate Equality The symbol≈ is often used to provide an estimate or an approximation of an exact value. An exact number is approximately equal to (≈) an inexact number if it lies in the uncertainty interval for the inexact number. For example: pi ≈ 3.14, pi ≈ 3.14159, √2 ≈ 1.414, √2 ≈ 1.415. To see that √2 ≈ 1.415, we verify that: 1.414 < √2 < 1.416. Two inexact numbers are approximately equal if their uncertainty intervals overlap. For example: 3.14 ≈ 3.14159, 1.414±0.002≈ 1.416, 1.414 ≈ 1.415. On the other hand: pi negationslash≈ 3.17 and 1.414 negationslash≈ 1.416 Caution Unlike equality, approximate equality is not transitive. For example: 3.14158 ≈ 3.14159 ≈ 3.14160. But: 3.14158 negationslash≈ 3.14160. 13 3.2.5 Summary Guidelines for Locating Significant Digits. In the absence of other information about an inexact decimal number: • The nonzero digits are significant. • The most significant position is the location of the leftmost nonzero digit. • If the number has a decimal point, the least significant position is the loca- tion of the rightmost digit. • If the number does not have a decimal point, the least significant position is the location of the rightmost nonzero digit. Guidelines for Inexact Arithmetic. • When adding or subtracting a small collection of inexact numbers, the least significant position of the result is leftmost least significant position of the terms being added or subtracted. • When multiplying or dividing a small collection of inexact numbers, the number of significant digits in the result is smallest of the numbers of sig- nificant digits of the factors and divisors. Division is illegal if an inexact divisor contains zero in its uncertainty interval. • Two inexact numbers are approximately equal if their intervals of uncer- tainty overlap. When two inexact numbers are approximately equal, their difference is “insignificant” (close to zero). 14 3.2.6 Exercises 1. Express the given uncertainty intervals using ± notation. (a) (2.1,2.4), (b) 1.781, (c) between 2 and 3 teaspoons, (d) about 1.4 inches. 2. Express the given uncertainty intervals using ± notation. (a) (3.31,3.42), (b) 3.1416, (c) between 15.1 and 15.2 inches of rain, (d) about 2.8 square feet. 3. Express the following uncertainty intervals using interval notation. (a) 3.14±0.005, (b) 3.14, (c) 1.5 feet, give or take an inch. 4. Express the following uncertainty intervals using interval notation. (a) 1.421±0.003, (b) 2.718, (c) 1 quart, give or take a cup. In problems 5-6, use the given measurements to determine the area A of a trapezoid with height h and bases a and b. (A = 12h(a + b). In this formula, the constant 12 is treated as an exact number.) Round your answer to the least significant position. 5. (a) h = 1.2 in. a = 1.01 in. b = 2.001 in. (b) h = 1.20 in. a = 1.01 in. b = 2.001 in. (c) h = 1.200 in. a = 1.01 in. b = 2.001 in. 6. (a) h = 1.234 cm. a = 1.221 cm. b = 0.123 cm. (b) h = 1.23 cm. a = 1.221 cm. b = 0.123 cm. (c) h = 1.234 cm. a = 1.22 cm. b = 0.123 cm. 7. The work W done by a constant force F acting over a fixed displacement d is given by W = Fd.7 (a) F = 2.1 lb, d = 39.7 ft. Find the work W done by the force, rounded to the least significant digit. (b) W = 16 J, d = 18.1 m. Find the force F, rounded to the least significant digit. (c) d = 39.7 ft. If the work W needs to have three significant positions, then how many significant positions do we need in our measurement of the 7In English gravitational units, displacements are given in feet [ft], forces in pounds [lb], and work in foot-pounds [ft·lb]. In SI units, displacements are given in meters [m], forces in newtons [N], and work in joules [J]. 15 force F? (d) d = 120 m. If the work W needs to have three significant positions, then how many significant positions do we need in our measurement of the force F? 8. A farmer wishes to paint a 53.1 foot tall silo with a diameter of 37 feet. Ac- cording to the manufacturer, one gallon of paint will cover about 230 square feet of surface. Since the paint is on sale, the farmer wants purchase all the paint now, but he doesn’t want to buy too much paint. (a) How much paint does the farmer expect to use? (Round your answer to the least significant digit.) (b) Use only your answer to part (a) to get a rough estimate of the maximum amount of paint that he may need. (c) Use the original data to estimate the maximum amount of paint that he may need. (d) Briefly answer the following questions: When would the answer to (b) be a better answer to the farmer’s problem? When would (c) be more ap- propriate? 9. For an electrical circuit with two parallel resistors, the total current I in amperes [A] is given by: I =V parenleftbigg 1 R1 + 1 R2 parenrightbigg where V is the potential difference in volts [V] and R1 and R2 are the resis- tances in ohms [Ω]. (a) Find I and Imax if V = 12 V, R1 = 15 Ω, and R2 = 11 Ω. (b) Find I if V = 120 V, R1 = 1000 Ω, and R2 = 11 Ω. (c) Find I if V = 120 V, R1 = 1600 Ω, and R2 = 11 Ω. Is the answer signif- icantly different from your answer in part (b)? Why or why not? 10. A certain rectangle is about w = 15 meters by ℓ = 20 meters. Its area A can be measured with 5 significant digits of precision. (a) How many significant digits are needed in measurements of its width in order to calculate its length with three significant digits of precision? (b) If we calculate length using ℓ = A/w, what is the maximum number of significant digits of precision? 16 For some purposes, it may be better to use relative or percentage uncertain- ties. In the specification for a family of resistors, the manufacturer may state that the resistances have a percentage uncertainty of about 5%. According to the specification, a 40-ohm [Ω] resistor has an uncertainty of: (40 Ω)×0.05 = 2 Ω The uncertainty interval is (38 Ω, 42 Ω) or 40±2 Ω. A weight of 22 pounds (22±1 lb) has a percentage uncertainty of: 1 lb 22 lb = 0.04545 ≈ 0.05 = 5% Problems 11-12 deal with percentage uncertainties. 11. (a) A room has a length of 5 meters, a width of 4 meters and a height of 3 meters, all measured with a percentage uncertainty of 1%. The maximum volume is: Vmax =(5.05)(4.04)(3.03)= 61.81806 m3. Determine the minimum volume Vmin. Based on the values for Vmin and Vmax, which of the following is the best (closest) estimate for the percentage uncertainty for the volume? (i) 0.1% (ii) 1% (iii) 10% (iv) 100% (b) A room has length 5 meters, width 4 meters and height 3 meters. The percentage error in the length and width is 1%, but room height is harder to measure and has a percentage error of 10%. Find values for Vmin and Vmax. Which of the following is now the best estimate for the percentage uncertainty for the volume? (i) 0.1% (ii) 1% (iii) 10% (iv) 100% (c) Using a laser device, we find that a room has a length of 5 meters with a percentage error of 0.1%. Because of the layout of the room, we have to use more conventional means to get a width of 4 meters and a height of 3 meters with percentage errors of 1% and 10%, respectively. Find values for Vmin and Vmax. Which of the following is now the best estimate for the percentage uncertainty for the volume? (i) 0.1% (ii) 1% (iii) 10% (iv) 100% Comment: When multiplying or dividing a small collection of inexact numbers, we can estimate the percentage uncertainty by taking the largest percentage uncertainty. 17 12. (a) The inexact numbers 10.0, 3.14 and 9.00 all have three significant digits. What are their percentage uncertainties? (b) Suppose instead that the measurements 10.0, 3.14 and 9.00 all have a percentage uncertainty of 1%. What are their uncertainty intervals? (c) Using the uncertainty rules in the text, we obtain the estimate: pi2 ≈(3.14)2 ≈ 9.8596±0.01 ≈ 9.86. Uncertainty = 0.01. When doing multiplication or division using relative uncertainties, the rel- ative uncertainty of the result is approximately the larger of the relative uncertainties of the inputs. What is the uncertainty (to one significant digit) if we use this new rule? 13. Ballpark Estimates: Without using a calculator, estimate the following quantities. Round things off to obtain an uncertainty of about one significant digit. (Using scientific notation will make the estimates easier.) (a) A heart rate of about 60 to 100 beats per minute (BPM) is considered normal for most adults. At a rate of 70 BPM, how many times will your heart beat in 10 years? (b) Rip Van Winkle slept for twenty years. In deep sleep, the heart rate drops to about 40 BPM. How times did Rip’s heart beat during his twenty-year nap? (c) The second-nearest star, Proxima Centauri, is about 4.3 light-years from the earth. Light travels at about 190,000 miles per second. About how many miles is it from Earth to Proxima Centauri? (d) A car is travelling 60 miles per hour. What is is speed in feet per second? A foot is 0.3048 meters. What is the speed of the car in meters per second? (e) A fathom is 6 feet. A fortnight is 14 days. If one snail’s pace is one fathom per fortnight, then about how many miles per hour are there in one snail’s pace? About how many snail’s paces are there in one mile per hour? 18 3.2.7 Solutions to Selected Exercises 1. (a) 2.25±0.15, (b) 1.781±0.001, (c) 2.5±0.5 tsp, (d) 1.4±0.1 in. 3. (a) (3.135,3.145), (b) (3.13,3.15), (c) (17 in,19 in). 5. (a) 1.8 in2, (b) 1.81 in2, (c) 1.81 in2. 7. (a) W = 83 ft·lb. (b) F = 0.88 N. (c) three. (d) No solution. Since thre are only two significant positions in the displacement, the calculated force can have at most two significant positions. 9. (a) 1.9 A. (b) 11 A. (c) 11 A. No, the difference in the two answers is insignif- icant because the reciprocals of the resistances R1 are both much less than 0.01. 12. (a) For 10.0: 1%. For 3.14: About 0.3%. (b) For 10: 10.0±0.1. For 3.14: 3.14±0.03. (c) The percentage uncertainty is about 0.318%. The uncertainty is about 0.00318·pi2 ≈ 0.03. 19 3.2.8 Appendix: Precision and Accuracy In informal speech, we often use the terms “accuracy” and “precision” inter- changeably. In technical usage, their meanings are quite different. The following example illustrates the difference. Example: In a recent dart tournament, four players (Al, Bea, Chaz and Don) managed to throw their darts as follows. (They were trying to hit the centers of their targets.) Al Bea Chaz Don precise precise imprecise imprecise accurate inaccurate accurate inaccurate Al’s darts were close to center (accurate) and close to each other (precise). Bea’s darts were also close to each other, but off-center, so Bea was precise but inaccurate. Chaz’s darts were distributed around the center of the target, but not very close to each other. Chaz’s throws were thus accurate but imprecise. Don’s throws were neither distributed around the center nor close to each other, so Don’s throws were both inaccurate and imprecise. More generally, when we say a collection of estimates is accurate, we mean that the average of the collection is close to a true value. But in practice, true values are often unknown. When we say a collection of estimates is precise, we mean that they don’t deviate much from each other or they tend to stay close to their average value. It is not necessary to know a true value in order to talk about precision. In practical situations, we often do not know the true values of the quantities that we are measuring or calculating. In these situations, we usually speak of precision rather than accuracy. 20 Section32.dvi