Init fall 2009 by Daniel R. Barnes

Significant Figures 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.

a.k.a. Significant Digits

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.

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?

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

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. When you report your numerical data using only the correct number of significant digits, you are admitting that your measuring devices are not perfect. The practice of reporting the correct number of significant digits is an exercise in humility. Maybe I can illustrate this idea with an imaginary example . . .

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.

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

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

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.)

Let’s do the math . . .

odoi = 73,294.8 miles odof = 73,676.4 miles distance = (73,676.4 – 73,294.8) miles distance = 381.6 miles ti = 10:58:50 PM tf = 5:16:52 AM the next morning Dt = 6 h, 18 min, 2 seconds = 6.300555556 hours speed = distance / time = 381.6 mi / 6.300555556 hours speed = 381.6 mi / 6.300555556 hours speed = 60.56608765 mi/h That’s a very fancy-looking answer, but here’s the problem . . .

distance = 381.6 miles time = 6.300555556 hours ! x x x x x x D speed = 60.56608765 mi/h “60.56 mi/h” 7 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.

distance = 381.6 miles time = 6.300555556 hours x x x x x x speed = 60.56608765 mi/h “60.57 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.

Let’s look at another imaginary example.

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 135.2 lbs.

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

+ = ? 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 + 135.2 lbs = 249.2 lbs . . . right?

+ = ? 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!

+ = ? 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 + 135.2 lbs = 249.2 lbs, but you better just report the total as 249 lbs.

Enough intro. Let’s learn some skills.

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

Q: How many significant figures are there in the measurement. . . 2009 y A: 4 significant figures

Q: How many significant figures are there in the measurement. . . 3.14 ft A: 3 significant figures

Q: How many significant figures are there in the measurement. . . 386,000,000.10 K A: 11 significant figures

Q: How many significant figures are there in the measurement. . . 55.85 amu A: 4 significant figures

Q: How many significant figures are there in the measurement. . . x x x x 0.00076 g A: 2 significant figures

Q: How many significant figures are there in the measurement. . . 29.25 days A: 4 significant figures

Q: How many significant figures are there in the measurement. . . x x x 0.003009 ms A: 4 significant figures

Q: How many significant figures are there in the measurement. . . 1776 y A: 4 significant figures

Q: How many significant figures are there in the measurement. . . x x x x 0.00071120 nm A: 5 significant figures

Q: How many significant figures are there in the measurement. . . x x x x x x 0.000002870500 mm A: 7 significant figures

Q: How many significant figures are there in the measurement. . . 14.01 amu A: 4 significant figures

Q: How many significant figures are there in the measurement. . . 6.022 x 1023 molecules/mole A: 4 significant figures Only the coefficient counts, Not the power of ten.

Q: How many significant figures are there in the measurement. . . 12 inches/foot 12.0000000000000000000000000000000000000000000000 . . . 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?

Q: How many significant figures are there in the measurement. . . 9.10938188 × 10-28 grams A: 9 significant figures Only the coefficient counts, Not the power of ten.

Q: How many significant figures are there in the measurement. . . 0.000000000000002 lb A: 1 significant figure

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.

Q: How many significant figures are there in the measurement. . . x x x 186,000 mi/s A: 3 significant figures Well, it could be up to 6, but not likely.

Q: How many significant figures are there in the measurement. . . 1.86 x 105 mi/s A: 3 significant figures With no ambiguity : )

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

Q: How many significant figures are there in the measurement. . . 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. 1000 m/km 1000.00000000000000000000000000000000000000000000 . . . 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?

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

23.43 g + 200 g = ? 223.43 g 200 g addition 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! 23.43 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 + 200 X X 223.43 x x x x Any column that has even one “x” in it is insignifcant. Also, any column is insignificant if it has even one insignificant zero in it. Two is less than five, so the one digit that remains, the “2” in front, is not rounded up.

23.43 g + 200 g = ? 223.43 g 200 g addition Yeah. I know that looks weird, but that’s the kind of results you get sometimes when you follow the rules. In real life, this kind of situation might arise where you make two different mass measurements with two different scales. 23.43 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) x x + 200 223.43 x x x x The “low-res” scale is probably a much larger scale used for measuring much heavier objects (like people), whereas the scale that measured the 23.43 g object is a much smaller, more sensistive scale, used for measuring small piles of dust. Or something.

subtraction 34,836.982 – 40 = ? = 34,800 = 34,796.982 Okay. That’s true, but how should we write the answer so that it has the correct number of significant digits? Just like with addition, it helps to arrange numbers vertically, in columns, when subtracting. 34836.982 As with addition, I like to put an “x” in any position where a digit is missing. x x x - 40 34796.982 x x x x This answer’s last three digits, are, therefore, insignificant. However, the zero in 40 is also insignificant, so its whole column is insignificant as well. Because the insignificant “6” in the answer is five or greater, it rounds the previous digit up as its last, dying act.

78.1 x 32,510,000 = ? Multiplication = 2,539,031,000 = 2,540,000,000 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.

1 2 3 4 1 2 3 78.1 x 32,510,000 = ? Multiplication 2 3 1 = 2,539,031,000 = 2,540,000,000 How many significant digits are in the answer? 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.

Init fall 2009 by Daniel R. Barnes