Significant Figures in Calculations Chapter 1 Measurements Significant Figures in Calculations

To learn how uncertainty in a measurement arises Objectives To learn how uncertainty in a measurement arises To learn to indicate a measurement’s uncertainty by using significant figures To learn to determine the number of significant figures in a calculated result To learn the difference between accuracy and precision of a measurement

Significant Figures When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. There are 2 different types of numbers Exact Measured Exact numbers are infinitely important Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR. When you use your calculator your answer can only be as accurate as your worst measurement… 

Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

Measured Numbers Do you see why Measured Numbers have error…you have to make that Guess! All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

Measurement and Significant Figures Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place.

? 8.00 cm or 3 (2.2/8)

A. Uncertainty in Measurement A measurement always has some degree of uncertainty. How long is this nail?

A. Uncertainty in Measurement Different people estimate differently. Record all certain numbers and one estimated number. Whenever we measure in chemistry we always record one extra decimal place (estimated number) beyond the markings on the measurement device.

Significant Digits Significant digits only apply to measurements; therefore there must be a number and a unit. Without a unit there are no “significant digits” Significant digits include all measured values and one estimated value The last number is considered to be estimated

Significant Figures in Measurement The numbers reported in a measurement are limited by the measuring tool Significant figures in a measurement include the known digits plus one estimated digit

Learning Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by

Solution 2. counting 3. definition B. Measured numbers are obtained by A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool

Learning Check Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C.

Solution Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required.

Significant Figures 100 m __ sig figs 410100 L __ sig figs Rules for Counting Significant Figures 100 m __ sig figs 410100 L __ sig figs 100. mm __ sig figs 100.8 s __ sig figs 100.80 μm __ sig figs 0.00287 kg __ sig figs 0.002870 cg __ sig figs 1.0805 x 104 mm __ sig figs 1.0805 x 10-12 km __ sig figs 1.205670 x 10-6 mg __ sig figs

Significant Figures Rules for Counting Significant Figures Exact numbers - unlimited significant figures Exact counts have unlimited (infinite) sig figs; they are obtained by simple counting (no uncertainty): 3 apples, 12 eggs, 1 mole Conversion Factors have unlimited sig figs also: 1 in. = 2.54 cm 1 mole = 6.022 x 1023 atoms

Significant Figures Rules for Counting Significant Figures Nonzero integers always count as significant figures. 1457 mm 4 significant figures

Significant Figures Rules for Counting Significant Figures Zeros Leading zeros - never count 0.0025 m 2 significant figures Captive zero (zeros between 2 non-zero numbers) - always count 1.008 cm 4 significant figures Trailing zeros - count only if the number is written with a decimal point 100 L 1 significant figure Zeros between a non-zero and a decimal point 100. kg 3 significant figures Zeros after a non-zero and a decimal point 120.0 mg 4 significant figures

Rounding Off 5 or more to the right of the number being rounded Raise the score 4.678 cm rounded to the nearest hundredth 4.68 cm 4 or less to the right of the number being rounded Let it rest!!! 3.421 cm rounded to the nearest hundredth 3.42 cm

Rounding Off Numbers Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. How do you decide how many digits to keep? Simple rules exist to tell you how.

Note the 4 rules When reading a measured value, all nonzero digits (1, 2, 3, 4, 5, 6, 7, 8, and 9) should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.

RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.

Significant Figures Rules for Addition and Subtraction The number of decimal places in the result is the same as in the measurement with the smallest number of decimal places (not sig figs!). m m m m m m m m m

Significant Figures Rules for Multiplication and Division The number of significant figures in the answer is the same as in the measurement with the smallest number of significant figures.

Below are two measurements of the mass of the same object Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

Practice Rule #1 Zeros 6 45.8736 cm 3 .000239 L 5 .00023900 kg All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal 0’s between digits count as well as trailing in decimal form 45.8736 cm .000239 L .00023900 kg 48000. mm 48000 m 3.982106 mm 1.00040 cg

Counting Significant Figures Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___ Complete: All non-zero digits in a measured number are (significant or not significant).

Leading Zeros 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ Complete: Leading zeros in decimal numbers are (significant or not significant)

Sandwiched Zeros Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ Complete: Zeros between nonzero numbers are (significant or not significant).

Trailing Zeros Number of Significant Figures 25,000 in. 2 200 yr 1 48,600 gal 3 25,005,000 g ____ Complete: Trailing zeros in numbers without decimals are (significant or not significant) if they are serving as place holders.

Significant Figures VITALLY IMPORTANT: Rules for Counting Significant Figures VITALLY IMPORTANT: For the rest of the year, all calculations must include the correct number of significant figures in order to be fully correct! on all homework, labs, quizzes, and tests, etc. even if the directions don’t specifically tell you so

Significant Figures Numbers recorded in a measurement. All the certain numbers plus the first estimated number Accuracy of a measurement – how close your number comes to the actual value similar to hitting the bull's-eye on a dart board Precision of a measurement – how close your repeated measurements come to each other (not necessarily the actual value) how closely grouped are your 3 darts on the board (even if they’re not close to the bull's-eye) It is possible for measurements to be precise but not accurate, just as it is possible to be accurate but not precise

Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 g becomes 2.4 gwhen rounded off to two significant figures because the first dropped digit (a 2) is 4 or less. RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 m is 4.6 m when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. If a calculation has several steps, it is best to round off at the end.

Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig 4965.03 kg   780,582 mm 1999.5 L 4965 kg 780,600 mm 2000. L

Practice Rule #2 Rounding Make the following into a 3 Sig Fig number 1.5587 m 0.0037421 m 1367 m 128,522 m 1.6683 106 m Your Final number must be of the same value as the number you started with, 129,000 m and not 129 m 1.56 m 0.00374 m 1370 m 129,000 m 1.67 106 m

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.

Significant Numbers in Calculations A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

Multiplication and division 46.4 m/s 0.05985 m2 1.586 107 m2 1.000 m2 32.27 m  1.54 m = 49.6958 m2 3.68 m  .07925 s = 46.4353312 m/s 1.750 m  .0342000 m = 0.05985 m2 3.2650106 m  4.858 m= 1.586137  107 m2 6.0221023 m  1.66110-24 m = 1.000000 m2

Addition/Subtraction 25.5 L 32.72 cm 320 m +34.270 L ‑ 0.0049 cm + 12.5 m 59.770 L 32.7151 cm 332.5 m 59.8 L 32.72 cm 330 m

Addition and Subtraction 0.71 g 82000 mL 0.1 g 0 g Look for the last important digit 0.56 g + 0.153 g = 0.713 g 82000 mL + 5.32 mL = 82005.32 mL 10.0 m - 9.8742 m = 0.12580 m 10 g – 9.8742 g = 0.12580 g __ ___ __

Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 m 2) 0.00476 m 3) 4760 m B. All the zeros are significant in 1) 0.00307 m 2) 25.300 m 3) 2.050 x 103 m C. 534,675 rounded to 3 significant figures is 1) 535 m 2) 535,000 m 3) 5.35 x 105 m

Solution A. Which answers contain 3 significant figures? 2) 0.00476 m 3) 4760 m B. All the zeros are significant in 2) 25.300 m 3) 2.050 x 103 m C. 534,675 rounded to 3 significant figures is 2) 535,000 m 3) 5.35 x 105 m

Learning Check 1) 22.0 m and 22.00 m 2) 400.0 m and 40 m In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 m and 22.00 m 2) 400.0 m and 40 m 3) 0.000015 m and 150,000 m

Solution In which set(s) do both numbers contain the same number of significant figures? 3) 0.000015 m and 150,000 m

Learning Check SF3 A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

Solution A. 0.030 m 2 B. 4.050 L 4 C. 0.00008 g 1 D. 3.00 m 3 E. 2,080,000 bees 3

Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 mm one decimal place + 1.34 mm two decimal places 26.54 mm answer 26.5 mm one decimal place

Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 m + 19.6 m + 2.1 m = 1) 256.75 m 2) 256.8 m 3) 257 m B. 58.925 m - 18.2 m = 1) 40.725 m 2) 40.73 m 3) 40.7 m

Solution A. 235.05 m + 19.6 m + 2.1 m = 2) 256.8 m B. 58.925 m - 18.2 m = 3) 40.7 m

Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

Learning Check A. 2.19 m X 4.2 m = 1) 9 m2 2) 9.2 m2 3) 9.198 m2 B. 4.311m2 ÷ 0.07 m = 1) 61.58 m 2) 62 m 3) 60 m C. 2.54 m X 0.0028 m2 = 0.0105 m X 0.060 m 1) 11.3 m 2) 11 m 3) 0.041 m

Solution A. 2.19 m X 4.2 m = 2) 9.2 m2 4.311 m2 ÷ 0.07 m = 3) 60 m C. 2.54 m X 0.0028 m2 = 2) 11 m 0.0105 m X 0.060 m Continuous calculator operation = 2.54 m x 0.0028 m3  0.0105 m  0.060 m

Scientific Notation

Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point. The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures. Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4. Scientific notation can make doing arithmetic easier.

Scientific Notation Scientific notation is a convenient way to write a very small or a very large number. Numbers are written as a product of a number between 1 and 9, times the number 10 raised to power. 215 is written in scientific notation as: 215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102

Scientific Notation When your exponent is negative: Going from scientific notation to standard form you will move the decimal to the left Going from standard form to scientific notation you will move your decimal to the right When the exponent is positive: Going from scientific notation to standard form you will move your decimal to the right Going from standard form to scientific notation you will move your decimal to the left

Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8

An easy way to remember this is: If an exponent is positive, the number gets larger, so move the decimal to the right. If an exponent is negative, the number gets smaller, so move the decimal to the left.

The exponent also tells how many spaces to move the decimal: In this problem, the exponent is +3, so the decimal moves 3 spaces to the right.

The exponent also tells how many spaces to move the decimal: In this problem, the exponent is -3, so the decimal moves 3 spaces to the left.

When changing from Standard Notation to Scientific Notation: 1) First, move the decimal after the first whole number: 3 2 5 8 2) Second, add your multiplication sign and your base (10). 3 . 2 5 8 x 10 3) Count how many spaces the decimal moved and this is the exponent. 3 . 2 5 8 x 10 3 3 2 1

Two examples of converting standard notation to scientific notation are shown below.

Two examples of converting scientific notation back to standard notation are shown below.

Two examples of converting scientific notation back to standard notation are shown below.

Positive Exponents 101 = 10 102 = 10X10= 100 103 = 10X10X10 = 1000

Negative Exponents 10-1 = 1/10 = 0.1 10-2 = 1/100 = 0.01 10-3 = 1/1000 = 0.001 10-4 = 1/10000 = 0.0001

Scientific Notation We use the idea of exponents to make it easier to work with large and small numbers. 10,000 = 1 X 104 250,000 = 2.5 X 105 Count places to the left until there is one number to the left of the decimal point. 230,000 = ? 35,000 = ?

Scientific Notation Continued 0.00006 = 6 X 10-5 0.00045 = 4.5 X 10-4 Count places to the right until there is one number to the left of the decimal point 0.003 = ? 0.0000025 = ?

Adding or Subtracting with Scientific Notation The exponents are like denominators When adding or subtracting fractions, the denominators must be the same When adding or subtracting in scientific notation the exponents must be the same 4.53 x 105 cm 0.453 x 106 cm 453000 cm + 2.2 x 106 cm + 2.2 x 106 cm + 2200000 cm 2.653 x 106 cm 2653000 cm Because the least certainty lies in the tenths place you must round to the tenth place for significant digits Answer= 2.7 x 106 cm

Adding and Subtracting Significant Figures 1913.0 cm 1.9130 x 103 cm 1913.0 cm - 4.6 x 103 cm - 4.6 x 103 cm - 4600 cm - 2.687cm x 103 cm - 2687 cm Because the least certainty lies in the tenths place you must round to the tenth place for significant digits Answer= -2.7 x 103 cm

Multiplying with Scientific Notation Add the Exponents 102 X 103 = 105 100 X 1000 = 100,000

Multiplying with Scientific Notation (2.3 X 102)(3.3 X 103) 230 X 3300 Multiply the Coefficients 2.3 X 3.3 = 7.59 Add the Exponents 102 X 103 = 105 7.59 X 105 759,000

Multiplying with Scientific Notation (4.6 X 104) X (5.5 X 103) = ? (3.1 X 103) X (4.2 X 105) = ?

Try this one (6 x 102)(3 x 107) 6 x 3 = 18 102 x 107 = 109 18 x 109 Now convert to Scientific Notation 1.8 x 1010

Try this one 2.5 x 107 (5 x 103)2 = (5 x 103) (5 x 103) 5 x 5 = 25

Dividing with Scientific Notation Subtract the Exponents 104/103 = 101 10000/1000 = 10

Dividing with Scientific Notation (3.3 X 104)/ (2.3 X 102) 33000 / 230 = 143.4783 Divide the Coefficients 3.3/ 2.3 = 1.434783 Subtract the Exponents 104 / 102 = 102 1.4347823 X 102 143.4783

Dividing with Scientific Notation (4.6 X 104 cm2) / (5.5 X 103cm) = 0.83 x 101cm = 8.3 x 100 cm (3.1 X 103 cm2) / (4.2 X 105 cm) =0.74 x 10-2 cm = 7.4 x 10-3 cm

Using a calculator Enter 9.1 in your calculator. Use a calculator to perform the indicated operation. Write your result in correct scientific notation. Enter 9.1 in your calculator. Press the key marked EXP or EE on your calculator. If this is written above another key, then you will have to press SHIFT or 2nd before pressing the EXP or EE key. Enter the value of the exponent. Press the times key. Enter 4.2 Repeat steps 2 and 3. Press Enter or =. You should get 3.822x10-1

Use a calculator to evaluate:. 7. 2 x 10-9. 1 Use a calculator to evaluate: 7.2 x 10-9 1.2 x 102 On the calculator, the answer is: 6.E -11 The answer in scientific notation is 6 x 10 -11 The answer in decimal notation is 0.00000000006