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Init fall 2009 by Daniel R. Barnes WARNING: This presentation includes images and other content taken from the world wide web without permission of the owners of that content. Do not copy or distribute this presentation. Its very existence may be illegal.

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What’s the difference between these two pictures? They’re both pictures of the same kitten, but the picture on the left has a much higher resolution than the “lo-res” picture on the right. Perhaps the picture on the left was taken with a much more fancy, expensive camera than what was used to take the picture on the right.

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Just for fun, squint your eyes really tight and look at both of the pictures. Stare with your eyes squinted really tight for about twenty seconds. Now, open your eyes wide again. Looks different, doesn’t it?

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... explain why reporting numbers to the correct number of significant digits is good practice.

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The practice of reporting the correct number of significant digits is an exercise in humility. When you report your numerical data using only the correct number of significant digits, you are admitting that your measuring devices are not perfect. Maybe I can illustrate this idea with an imaginary example... When scientists report numerical data from an experiment, they have to report the data using the correct number of digits so that other scientists know just how exact their measurements and calculations are.

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Imagine that you are driving from Los Angeles to San Francisco and you want to calculate your average speed when you’re done with the trip.

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Your odometer reads “73,294.8” when you start your trip in Los Angeles.

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Your digital wristwatch reads “10:58:50 PM” when you leave.

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At the end of the trip, when you reach San Francisco, your odometer reads “73,676.4”, and your digital wristwatch says “5:16:52 AM”. (You drove all night in the dark and arrived just in time to see the sun rise over the Golden Gate Bridge.)

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Let’s do the math...

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odo i = 73,294.8 milesodo f = 73,676.4 miles t i = 10:58:50 PMt f = 5:16:52 AM the next morning distance = (73,676.4 – 73,294.8) miles distance = miles t = 6 h, 18 min, 2 seconds = hours speed = distance / time= mi / hours speed = mi / hours speed = mi/h That’s a very fancy-looking answer, but here’s the problem...

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distance = milestime = hours speed = mi/h Do we really have any business reporting our speed by giving a number that has ten digits in it? Our odometer only measures down to a tenth of a mile, so our distance calculation has only four digits in it. A chain is only as strong as its weakest link, so our answer really only deserves to have four digits in it, too. xxxxxx “60.56 mi/h” ! 7

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distance = milestime = hours speed = mi/h Do we really have any business reporting our speed by giving a number that has ten digits in it? Our odometer only measures down to a tenth of a mile. A chain is only as strong as its weakest link, so our answer really only deserves to have four digits in it, too. xxxxxx “60.57 mi/h” so our distance calculation has only four digits in it.

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Let’s look at another imaginary example.

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Let’s say your sister is on a diet and she’s just lost some weight, so she calls you on the phone from across town and says she weighs lbs.

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Curious to see how you compare, you weigh yourself on your very different bathroom scale and turn out to be 114 lbs.

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Your sister mentions how nice it is you’ve both lost so much weight, but claims that if you both climbed onto your daddy’s shoulders at the same time, your combined weight would still be too much for him to carry. You pull your calculator out of your purse and punch in the numbers. 114 lbs lbs = lbs... right?

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Once again, there’s a bit of a problem. Your sister’s scale measures down to a tenth of a pound. However, your more old-fashioned scale isn’t that exact. No matter how much you squint at that needle, you don’t feel confident estimating your weight to the closest tenth of a pound. Heck, when you’re standing on the scale, your eyes are so far above that dial that you’re not even sure if you’re closer to 114 lbs or 115 lbs!

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Considering all this, when you report the combined weight of your sister and you, do you really have any business claiming that you know your combined mass down to a tenth of a pound? 114 lbs lbs = lbs, but you better just report the total as 249 lbs.

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Enough intro. Let’s learn some skills.

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... determine how many significant digits there are in any number put before them.

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Q: How many significant figures are there in the measurement y A: 4 significant figures

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Q: How many significant figures are there in the measurement ft A: 3 significant figures

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Q: How many significant figures are there in the measurement ,000, K A: 11 significant figures

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Q: How many significant figures are there in the measurement amu A: 4 significant figures

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Q: How many significant figures are there in the measurement g A: 2 significant figures xx x x

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Q: How many significant figures are there in the measurement days A: 4 significant figures

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Q: How many significant figures are there in the measurement ms A: 4 significant figures xx x

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Q: How many significant figures are there in the measurement y A: 4 significant figures

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Q: How many significant figures are there in the measurement nm A: 5 significant figures xx x x

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Q: How many significant figures are there in the measurement mm A: 7 significant figures xx x x x x

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Q: How many significant figures are there in the measurement amu A: 4 significant figures

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Q: How many significant figures are there in the measurement x molecules/mole A: 4 significant figures Only the coefficient counts, Not the power of ten.

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Q: How many significant figures are there in the measurement inches/foot A: infinite significant figures HUH? Well, because this isn’t a measured number, but is a declared number, it is exact and 100% certain. If you were to write “12” with an inifinite number of significant digits, what would it look like?

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Q: How many significant figures are there in the measurement × grams A: 9 significant figures Only the coefficient counts, Not the power of ten.

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Q: How many significant figures are there in the measurement lb A: 1 significant figure

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Q: How many significant figures are there in the measurement... 75,106,200 mol A: 6 significant figures Well, it could be up to 8, but not likely.

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Q: How many significant figures are there in the measurement ,000 mi/s A: 3 significant figures Well, it could be up to 6, but not likely. xx x

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Q: How many significant figures are there in the measurement x 10 5 mi/s A: 3 significant figures With no ambiguity : )

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Q: How many significant figures are there in the measurement... 93,000,000 miles A: 2 significant figures Well, it could be up to 8, but not likely. xxxxx x

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Q: How many significant figures are there in the measurement x 10 4 mi/s A: 4 significant figures With no ambiguity : )

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Q: How many significant figures are there in the measurement x 10 6 mi/s A: 2 significant figures With no ambiguity : )

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Q: How many significant figures are there in the measurement m/km A: infinite significant figures Remember, it is 100% certain with perfect exactitude that there are EXACTLY one thousand meters in a kilometer. If you were to write “1000” with an inifinite number of significant digits, what would it look like? This is not a measured number. It is a declared number. The guys who invented the metric system decided that a kilometer would be exactly 1000 meters.

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... round the answers of calculations to the correct number of significant digits.

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23.43 g g = ? g Okay. That’s true, but how should we write the answer so that it has the correct number of significant digits? For addition and subtraction, it helps to arrange the numbers vertically. DON’T ADD ANY PLACEHOLDER ZEROS! I like to put an “x” in any position where a number doesn’t have a digit, but other numbers do have a digit. x x Any column that has even one “x” in it is insignifcant. x x Two is less than five, so the one digit that remains, the “2” in front, is not rounded up. Also, any column is insignificant if it has even one insignificant zero in it. x x 200 g X X

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23.43 g g = ? g Yeah. I know that looks weird, but that’s the kind of results you get sometimes when you follow the rules x x x x x x 200 g In real life, this kind of situation might arise where you make two different mass measurements with two different scales. One of the scales, maybe, measures down to the centigram, but the other scale is so coarse in its level of detail that it only measures to the closest 100 g. (0.1 kg) The “low-res” scale is probably a much larger scale used for measuring much heavier objects (like people), whereas the scale that measured the g object is a much smaller, more sensistive scale, used for measuring small piles of dust. Or something.

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34, – 40 = ? Okay. That’s true, but how should we write the answer so that it has the correct number of significant digits? = 34, Just like with addition, it helps to arrange numbers vertically, in columns, when subtracting x As with addition, I like to put an “x” in any position where a digit is missing. xx This answer’s last three digits, are, therefore, insignificant. x x x However, the zero in 40 is also insignificant, so its whole column is insignificant as well. x Because the insignificant “6” in the answer is five or greater, it rounds the previous digit up as its last, dying act. = 34,800

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78.1 x 32,510,000 = ? = 2,539,031,000 The multiplication may be correct, but we’re going to need to trim off some insignificant digits. Luckily, with multiplication and division, you don’t have to re-write the numbers with their digits all lined up in the correct columns. With multiplication and division, all you have to do is count the number of significant digits in each number. The answer gets to have as many significant digits as the ingredient number with the smallest number of significant digits. As with addition and subtraction, the chain is, once again, only as strong as its weakest link. This time, however, it’s easier to deal with. = 2,540,000,000

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78.1 x 32,510,000 = ? = 2,539,031,000 How many significant digits are in the answer? = 2,540,000,000 Why are there only three significant digits in the answer? Notice that although “78.1” goes all the way to the tenths digit, this doesn’t mean squat in multiplication or division. In multiplication and division, it doesn’t matter what column the digits are in, it just matters how many digits there are. Notice, also, that the leftmost insignificant digit in “2,539,031,000” is a “9”. Remember that if the leftmost insignificant digit in the answer is five or greater, it causes the rightmost significant digit to round up

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32,000 / = ? = Of course, as always, we need to trim some digits off of this answer. How do we handle significant figures during division? Our most primitive ingredient number, “32,000”, has only two significant digits, so it’s the weakest link. Therefore, our answer only gets to keep two significant digits. Will there be any rounding as we trim off insignificant digits? Eight is greater than or equal to five, so, yes, we do. = 66 We don’t even have a decimal point anymore. That’s how low-res this answer has become.

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Q: The calculator says that 37.2 x 58, = 2,169, How should this number be reported? A: There are three sig figs in the first #, and seven sig figs in the 2 nd #, so the answer only deserves to have three. As the nine disappears, it rounds the “6” up to a “7”, yielding... “2,170,000” as the properly-reported answer.

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Q: The calculator says that = How should this answer be reported? A: The first number has no hundredths or tenths digit, so those two columns are insignificant. However, all three numbers have a ones digit, so the ones column and every column to the left of it (tens, hundreds) are significant. As the “.37” disappears, no rounding occurs because three is less than 5. The properly-reported answer, therefore, is... “428”.

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Q: The calculator says that / = How should this number be reported? A: There are four sig figs in both the divisor and the dividend, so the quotient gets to have four also. As the leftmost insignificant digit, “2”, disappears, it causes no rounding because it’s less than 5, yielding... “ ” as the properly-reported answer.

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Q: The calculator says that 26, – 1,200 = 25, How should this answer be reported? A: The first number has a digit in every position all the way to the hundredths, but the second number only goes to the hundreds. It’s got a tens digit and a ones digit, but they’re both just placeholder zeros whose only purpose is to let us know where the decimal point goes. Therefore, the hundreds column is the farthest-right significant column. The tens column in the answer has a “9”, so as it disappears, it rounds the “1” up to a “2”, yielding: “25,200” as the properly-reported answer.

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The next practice problem is kind of tricky, so make sure you’re good at all the easy problems first before trying it.

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Consider the following molar mass calculation for stearic acid: C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol Get together with your seating group and discuss how this calculation should be done if the rules for significant figures are properly taken into account. Be ready to present your group’s conclusions in two minutes.

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol The numbers of atoms of each element per molecule are known with 100% certainty, so these numbers do not affect the number of significant digits in the answer. Every stearic acid moleulce has exactly 18 carbon atoms, so the number of carbon atoms is: The number of significant digits in a certain number is infinity.

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol “18” may have an infinite number of significant digits, but “12.01” has only four, so it’s the weak link in the chain. Since this is multiplication, the answer only gets to have four digits, also. HOWEVER, since “216.18” is what we call an “intermediate result” (a result of a calculation, but not our final answer), we’re not going to do any rounding right now. We will, however, mark the fifth digit as insignificant... for later...

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol “36” may have an infinite number of significant digits, but “1.01” has only three, so it’s the weak link in the chain. Since this is multiplication, the answer only gets to have three digits, also. HOWEVER, since “36.36” is an “intermediate result”, we’re not going to do any rounding right now. We will, however, mark the fourth digit as insignificant... for later...

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol “2” has an inifinite number of significant digits, and “16.00” has four. Therefore, the answer, according to the calculator, “32”, really deserves to have two more digits. Usually, rounding to the correct number of significant digits is a matter of getting rid of digits, not adding, them, but in this case, we do add a couple of zeros..00

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol When you’re adding or subtracting, it’s always best to line up your columns. This problem, as it appears, has bad column alignment, so let’s slide the intermediate results and the final result around until they all line up right....00

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol There. That’s better..00 Now, all the tenths digits are in the same column, all the hundredths digits are in the same column, et cetera. Now that all the columns are lined up, we can start disqualifying columns, starting from the right and moving left. Is the hundredths column significant? Nope. The top two numbers don’t have a significant digit in this column. That disqualifies the whole column.

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol.00 Is the tenths column any good? All three numbers have a significant digit in this column, so the tenths digit is significant. We get to keep it. Since we’ve gotten to the first column that isn’t disqualified, our job disqualifying columns is done. We know now that the only digit in the final result that we have to get rid of is the last one.

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TIME’S UP! Your instructor will now raise the projector screen and choose a random person to come to the whiteboard and show the rest of the class what his/her group decided. C 18 H 36 O 2 : C: 18 x = H: 36 x 1.01 = O: 2 x = g/mol.00 As the “4” disappears, does it round up the “5” next to it? Nope. If it were a “5” or higher, we’d round up, but it isn’t, so we don’t. Our final answer, expressed to the correct number of significant digits, is g/mol

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Read pp 66 – 71, including practice problems 1, 2, 3, 4, 5, 6, 7, and 8. Answers are on page R Section Assessment, pg 72: questions 11, 12, 14, and 15. Answers not in book! Click the red button to go to Barnes’ ch 3 section assessment answers power point. Chapter 3 Assessment, pg 96: questions 58, 59, 60, and 61. Answers to odd #’d questions are on page R84. For all answers, click the yellow button to get a helpful pdf. Push me for answers

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