Plan (Note: Tro covers this material in Section 1.7)

Plan (Note: Tro covers this material in Section 1.7) • Different Kinds of Quantities in Science • Concept of Uncertainty (and “significant figures”) in Measured Quantities • How to properly • write (report) a measured quantity and • interpret a quantity reported by someone else • Counting “Significant Figures” in Reported Qtys • Precision vs. Accuracy • Uncertainty in Calculated Quantities • 2 “Rules” to Apply to Estimate the Uncertainty

Much of the “What Happens” Realm involves quantitative values (quantities) • Important to recognize that there are differences in the fundamental nature of different physical quantities • Not all physical quantities are alike!

I. Three Kinds of Quantities Measured (Raw Data) Exact (not derived from measurement) Calculated (“Worked up” Data)

Measured Quantities Always Have Some Degree of Uncertainty—Exact ones don’t. • Measuring involves • looking at a scale (which involves estimation), or • having some instrument respond to some physical quantity (and electronics have limits, too! Note that even digital outputs are always rounded to some “humanly” programmed decimal place]!) • Exact Quantities have NO uncertainty; 2 types: • “Defined” quantities (1 in = 2.54 cm [exact]) • “Counted” quantities (5 people [no “partials”!]) • Calculated quantities (discussed later)

II. Concept of Uncertainty in Measured Quantities • Try it for yourself!

What is the length of the bar? • How many digits are you certain of? 0 1 m

What is the length of the bar? • How about now? 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m

What is the length of the bar? • And now? 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m

Conclusions and Conventions(including definition of “Significant Figures”) • The level of (un)certainty in a measured quantity depends on the measuring device! • When measuring properly, one writes digits only until the first uncertain (estimated) one. • Any digit after that is “totally” uncertain and thus not meaningful • All of the certain digits plus the first uncertain digit are called “significant figures”(meaningful digits) • Generally speaking, the more digits in a measured quantity, the smaller is the uncertainty, and the greater is the precision.

Example: If a measurement is reported properly as 0.786 m then: • The uncertain (but still significant) digit is the 6 • The value of the quantity appears to be between 0.785 m and 0.787 m • It could be written as 0.786  0.001 m • The magnitude of the uncertainty is approximately  0.001 m • The uncertainty is said to be "in the thousandths place" (because the uncertain digit is in the thousandths place) • There are three significant figures—two certain ones (the 7 and 8) and one uncertain one (the 6)

Example: Consider 124.04 s (Missy Franklin’s WR in the 200-m backstroke in London) • The uncertain (but significant) digit is the ___ • The magnitude of the uncertainty is approximately ______ • The uncertainty is said to be "in the ____________ place" • There are ____ significant figures 4  0.01 s hundredths 5

Adding zeros to indicate precision • What is the proper way to report the lengths of these twobars?  Hint:The uncertainty of any measurement with this ruler should be  _____m because of the markings are every 0.01 m (and you can estimate one more digit) 0.001 (Ans: red bar, 0.500 m; green, 0.610 m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m

Example 1.4 in Tro: Proper reading is…. Ans: 4.56 mL, 4.57 mL, or 4.58 mL And proper reading below is…. Ans: 103.3 F or 103.4 F (Try PS1a, Q3 now!)

Scientific Notation Interlude • “Anatomy of a Base 10 Number”: 504.23 5 hundreds 100 = 102 0 tens 10 = 101 4 ones 1= 100 2 tenths 0.1= 10-1 3 hundredths 0.01 = 10-2 What’s the first digit in this quantity?

Scientific Notation (cont.) • 504.23 in standard scientific notation is: 5.0423 x 102 ___________________ 8 • What is the first digit in: 0.008202 ? • The 8 is in which place? 10? ? = -3 • 0.008202 in standard scientific notation is: 8.202 x 10-3 ___________________ For “drilling” practice with this skill, see: http://academic.umf.maine.edu/magri/PUBLIC.acd/tools/DSconversion.html (Try PS1a, Q4 now!)

III. Counting Significant Figures in Quantities From Somebody Else(i.e., not measured or calculated by you) • In MOST cases: Start with the first (i.e., leftmost) nonzero digit, and count every digit after that until you get to the end of the quantity! • Examples: • 35002 mihas ___ sig figs • 3.44008inhas __ sig figs • 0.004021 ghas __ sig figs 5 (3, 5, 0, 0, 2) 6 (3, 4, 4, 0, 0, 8) 4 (4, 0, 2, 1)

The one exception.... • Trailing zeros when there is no decimal point are not considered significant. • Examples: • 34000 ft has 2 sig figs • 34000.0 ft has 6 sig figs • 1.0540 g has ___ sig figs • 10540 g has ___ sig figs 5 4

Why? (Read on your own, as needed) • 34000 ft has 2 sig figs • the 4 is uncertain; the three zeros are just there to indicate that the 4 is in the thousands place (“placeholder” zeros). • If you didn’t write the zeros, you’d have 34 which is not the same value numerically as 34000. • In scientific notation, you would not “need” the zeros and the 4 would be the last digit “showing”: 3.4 x 104 ft • 34000.0 ft has 6 sig figs • the person writing the value would not have put that last zero there unless s/he was doing so to indicate that the uncertainty was in the tenths place (because the ruler was very precise)

Examples 4 • 3.450 m has ____ SF (sig. figs.) • 93000000 mi has ____ SF • 9.30 x 107 mi has ____ SF • 0.00007045 s has ____ SF • 0.00202020 g has ____ SF • 1500 mL has ____ SF • 1500. mL has ____ SF • 1.50 x 103 has ____ SF 2 3 4 6 2 4 3

For online practice, see: • http://web.mst.edu/~gbert/Aj2.HTML?JAVA/sig1.HTM This professor takes the same approach as I do—I wish more textbook authors would adopt it! (Try PS1a, Q’s 5-7 now!)

IV. Precision is “reproducibility” Accuracy is “correctness” • A measurement (or set of measurements) is precise if a) multiple trials yield values that are close to one another and/OR b) the value of the uncertainty is small. • More digits (sig figs) typically means “more precise” • A measurement is accurate if it is close to the “actual value” of the quantity. • In order to make a judgment about accuracy, you must know (or assume) some “actual” value. • Unless you have a calibrated device, you cannot “know” that your measurements (even very precise ones!) are accurate!

Measured temperature of boiling waterby students (one data point per group) Which is more accurate, the Hg thermometer or the Alcohol thermometer? Which is more precise?

Example on pp 25-26 in Tro Given: Actual value for mass of the measured block is 10.00 g (continued )

(But not that inaccurate, actually. Definitely less precise than B or C)

What is the length of the bar? What if I said it was 0.558 m? Look closely…at left An inaccurate but precise tool is one way to end up with poor accuracy but good precision!  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 m (Try PS1a, Q8 now!)

V. Calculated Quantities • Quantities derived from a calculation will have uncertainty if any of the quantities involved in the calculation have uncertainty • You can’t get “certainty” from “uncertainty”! • The two “significant figure rules” in your text are simply a quick way to estimate which digit in the result of a calculation is the 1st uncertain one(i.e., the last significant one)

Application to Real Life? “Cone” shows the uncertainty involved with path prediction calculations. The initial uncertainty propagates--bigger uncertainly the farther out you predict.1 “in dealing with such complex systems, small variations at one point can cause big changes down the line.”2 1http://matei.org/ithink/2011/08/29/was-hurricane-irene-over-hyped-how-accurate-was-the-noaa-national-hurricane-center-forecast/ 2http://spectrum.ieee.org/tech-talk/computing/software/predicting-hurricane-sandy

Example 1 (multiplication) • What is the area of a rectangle whose sides are 1.9 in and 0.701 in long? • 1.9 in x 0.701 in = 1.3319 in2(on calculator) • Which digit in the calculated result is the first uncertain one? I.e., Which of the following correctly represents the area? • 1.3319  0.0001 in2 • 1.332  0.001 in2 • 1.33  0.01 in2 • 1.3  0.1 in2

“Long Way” (Vary uncertain digit by 1 and see the effect it has!) • 2.0(instead of 1.9) x 0.701 = 1.402 • 1.9 x 0.701 = 1.3319 (original) • 1.8(instead of 1.9) x 0.701 = 1.2618 • Result is somewhere between 1.2 and 1.4 in2 (approximately) **Not exact!** • Value of uncertainty is not ± 0.0001 in2! (as “answer” from calculator would imply if copied!) •  It’s about ±0.1 in2, which is in the second digit  The 2nddigit here is the (first) uncertain one. “Answer” is no more precise than: 1.3 in2(± 0.1in2)

“Long Way”—Part 2 (Does the 0.701 matter?) • 1.9 x 0.701 = 1.3319 (original) • 1.9 x 0.702 = 1.3338 • 1.9 x 0.700 = 1.33 • Results are much closer to one another. • Changing 0.701 has less impact than the change in 1.9 did. • The uncertainty in 1.9 dominates in this calculation and the uncertain digit is the 2nd one. • **Why? Because 1.9, with fewer SFs, is less precise. It is always the least precise quantity that determines the precision in the result. See: http://academic.umf.maine.edu/magri/PUBLIC.acd/tools/SigFigsAndRounding.html#WhyRound

Isn’t there a shorter way?!?(that’s what you were thinking, right?) • Yes, that’s what that SF “rule” in your textbook is used for! (next slide) • But it’s helpful to know that you can always check your answer (from the rules) by doing the “long way”(which isn’t really so long when you do it yourself on a calculator)

The “Rule” for X or (shorter way, gives estimate) • → Rule: The result of a X or  operation will have the same number of significant digits as the original value having thefewest. • 1.9 in x 0.701 in = 1.3319 in2 (on calculator) Q: Which digit in the calculated result is the first uncertain one? Answer: The 2nd(i.e., the “3”), b/c 2 SFs in 1.9 “limits” precision

Procedural Rule in this Class!* • UNDERLINE the first uncertain digit in any result—intermediate or final—which you determine by using the appropriate sig fig rule)* • Then round off (only) the final result so that the proper number of digits (sig figs) is shown • i.e., Round to the decimal place that is underlined: = 1.3 in2 • (1.9 in)(0.701 in) = 1.3319 in2 *Tro states this as Rule 4 on p. 24, and shows its application in the example to the right [see 0.479], but unfortunately does not do this before rounding final answers in his examples. E.g. When he writes 19.1707 = 19 [again, p. 24], I wish he’d underlined the 9 in the 19.1707 to show the application of the X/ rule.

Example 2 (X or ) • What is the density of a sample having a volume of 34.5 mL and a mass of 83.4330 g? 6 S.F. 3  Only ___ S.F. in final result 3 S.F.  Underline the 3rd digit, , and round to that place. = 2.42 g/mL

For online practice: • http://web.mst.edu/~gbert/Aj2.HTML?JAVA/sig2.HTM • NOTE: This web “exercise” is misnamed in my opinion on the web page. A more accurate title would be something like “Applying the mulitplication/division rule, rounding, and writing in scientific notation” (Try PS1a, Q9 now!)

Rule (and example) for Addition and Subtraction • Identify the decimal place of the uncertain digit in each quantity—then determine which of these is farthest to the left(i.e., greatest uncertainty). (Note: you do NOT COUNT THE NUMBER OF SIG FIGS AT ALL IN THESE CASES). • The uncertain digit in the result is in the same decimal placeas the one identified above (farthest left): 1.05atm  0.0896 atm • = 0.96 atm = 0.9604 Uncertainty in the hundredths place Uncertainty in the ten-thousandths place Uncertainty in the hundredths place  NOTE: Each initial qty has 3 SF; result has only 2 SF. +/- rule is not about SFs!

Practice(Underline the last sig. fig. and round) 9.02 g + 3.1 g = __________________ 88.80 cm + 7.391 cm = ______________ 8.08 m x 5.320 m =_________________ 11.1 mi x 24 mi = __________________ *Take values out of scientific notation before adding or subtracting! See: http://faculty.ccbcmd.edu/~cyau/WhatHappenSigFigDuringCalc.htm For “drilling” practice with both rules (one step only), see: http://academic.umf.maine.edu/magri/PUBLIC.acd/tools/RoundComputation.html

Answers to Practice(Underline the last sig. fig. and round) 9.02 g + 3.1 g = 12.12 g = 12.1 g (tenths place) _ • 88.80 cm + 7.391 cm = 96.191 cm (hundredths place) _ = 96.19 cm (3 S.F.) • 8.08 m x 5.320 m = 42.9856 m2 = 43.0 m2 _ • 11.1 mi x 24 mi = 266.4 mi2 = 270 mi2 (2 S.F.) _ (or better yet, 2.7 x 102 mi2) (2 S.F.) (Try PS1a, Q10 now!)

Multi-step Calculations • If ONLY multiplying and/or dividing, you can do whole calculation in one step (on a calculator) and apply the rule “all at once” Example: 2 significant figures is the smallest number of sig figs in all the initial quantities… …Only 2 digits are significant in the result. 

Multi-step Calculations (cont.) • If ONLY adding or subtracting, you can also do the calculation in one step and apply the rule “all at once”: _ → 41.456 g + 0.3 g - 24 g = 17.756 = 18 g • → 41.456 g + 0.3 g - 24 g = 17.756 Uncertainty is in the units (“ones”) place because of the third quantity in the calculation. • Ones place is further “to the left” than the tenths or thousandths place • The uncertainty is greatest there (1 > 0.1 > 0.001)

Multi-step Calculations (cont.) If a calculation involves both arithmetic types, you must do each step separately! • Apply the appropriate rule after each step. • Underline the first uncertain digit after each step.* • Round only at the very END (not after each step!)* *Again, Tro states this as Rule 4 on p. 24, and shows its application in the example to the right on that page [see 0.479], but unfortunately does not do this before rounding final answers in his examples. E.g. When he writes 19.1707 = 19 [again, p. 24], I wish he’d underlined the 9 in the 19.1707 to show the application of the X/ rule. See 1.33b, next slide →

Example • 1.33b (variant) from Zumdahl (old text): NOTE: Assume all qtys are measured quantities with uncertainty in their last digit! I wish they had included units here!!

Example (cont.) • Apply Multiplication Rule: 3 sig figs because of 2.91 18.635 • Underline the 6 1.60 • Apply Subtraction Rule: tenths place (because of 18.7) • Underline the 6 3 • Now look at the fraction: the numerator has ____ sig figs and the denominator has ____ sig figs. (Hint: use your underlines!!) 2 • Thus, before you even punch in the numbers, you should know that your result should have ____ sig figs. 2 • Now do the final division, underline the 2nd digit, and round. 11.647… 12

This takes practice! • You must • Know both rules well • Apply each rule properly after each step • Don’t do whole calc. in calculator in one step! • Use underlines to “keep track” of uncertain digit each step of the way • Keep at least a digit or two after your underline (no need to write down ALL digits!) • Round to your “underline” place • Round only once at the end (Try PS1a, Q11 now! END OF PS1b MATERIAL)

Plan (Note: Tro covers this material in Section 1.7)