Significant Figures In Measurements

Significant Figures At the conclusion of our time together, you should be able to: Explain what significant figures are in a measurement Determine the number of significant figures in any measurement

Significant Figures The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated. The numbers reported in a measurement are limited by the measuring tool.

How many sig figs are there in a given measurement?

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.

To indicate the precision of a measurement, the value recorded should use all the digits known with certainty, plus one additional estimated digit that usually is considered uncertain by plus or minus 1. No further insignificant digits should be recorded. The total number of digits used to express such a measurement is called the number of significant figures. All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.

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.

Does 1 = 2??? assume a = b multiply by b ab = b2 subtract a2 ab – a2 = b2 – a2 factor each a(b – a) = (b + a)(b – a) divide by (b – a) a = 2a divide by a therefore: 1 = 2

An Easier Way to have 1 = 2

Reading a Meterstick . l2. . . . I . . . . I3 . . . .I . . . . I4. . cm First digit (known) = 2 2.?? cm Second digit (known) = 0.7 2.7? cm Third digit (estimated) between 0.03- 0.05 Length reported = 2.74 cm or 2.73 cm or 2.75 cm

Known + Estimated Digits In 2.74 cm… Known digits 2 and 7 are 100% certain The third digit 4 is estimated (uncertain) In the reported length, all three digits (2.74 cm) are significant including the estimated one

. l8. . . . I . . . . I9. . . . I . . . . I10. . cm Learning Check What is the length of the line? 1) 9.3 cm 2) 9.32 cm 3) 9.33 cm How does your answer compare with your neighbor’s answer? Why or why not?

Zero as a Measured Number . l3. . . . I . . . . I4 . . . . I . . . . I5. . cm What is the length of the line? First digit 5.?? cm Second digit 5.0? cm Last (estimated) digit is 5.00 cm

Always estimate ONE place past the smallest mark! 11.5 mL

So how many sig figs are there in a given measurement? 52.8 mL

Beware of Identity Theft!!

How to Determine Significant Figures in a Problem Use the following rules:

Rule #1 Every nonzero digit is significant Examples: 24 = 2 3.56 = 3 24 = 2 3.56 = 3 7 = 1

Rule #2 – Sandwiched 0’s Zeros between non-zeros are significant Examples: 7003 = 4 40.9 = 3

Rule #3 – Leading 0’s Zeros appearing in front of non-zero digits are not significant Act as placeholders Can’t be dropped, show magnitude Examples: 0.00024 = 2 0.453 = 3

Rule #4 – Trailing 0’s with DP Zeros at the end of a number and to the right of a decimal point are significant. Examples: 43.00 = 4 1.010 = 4 1.50 = 3

Rule #5 – Trailing 0’s without DP Zeros at the end of a number and to the left of a decimal point aren’t significant Examples: 300 = 1 27,300 = 3

Interesting Answers to Catholic Elementary School Bible Questions: Many religions teach that you are to have only one wife This is called Monotony.

Easier Way to do Sig Figs!! Pacific/Atlantic P A If a decimal point is present, start on the Pacific (P) side and draw an arrow through the number until you hit a non-zero digit. Count all numbers without an arrow through them. If a decimal is absent, start on the Atlantic (A) side and draw an arrow through the number until you hit a non-zero digit.

Examples: 123.003 grams decimal present, start on “P” side, draw arrow, count digits without an arrow through it. Answer = 6 10,100 centimeters Decimal absent, start on “A” side, draw an arrow, count digits without an arrow through it. Answer = 3

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

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

Significant Figures and Numbers Some numbers are exact: There are 60 seconds in 1 minute 25 cents in 1 quarter 12 eggs in one dozen There is no uncertainty in any of these numbers. In other words there are 12.0000000000000000000000000000000000 eggs in 1 dozen (add as many zeros as you like)

Counting Numbers Counting numbers have infinite sig figs. Ex: 3 apples

Significant Figures Lets’ see if you can: Explain what significant figures are in a measurement Determine the number of significant figures in any measurement

Learning Check 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

And Mathematical Calculations Significant Figures And Mathematical Calculations

Significant Figures At the conclusion of our time together, you should be able to: Determine the number of significant figures needed for an answer involving calculations. Round math problems properly

Rules for Rounding Off If the digit to be removed is less than 5, the preceding digit stays the same is equal to or greater than 5, the preceding digit is increased by 1 In a series of calculations, carry the extra digits to the final result and then round off Don’t forget to add place-holding zeros if necessary to keep value the same!! 13 13

Significant Figure Math Rules Addition / Subtraction Problem: Penny Example = 0.019 m using meter stick 0.0192 m using ruler 0.0191 m using calipers 0.019046 m using micrometer To find the total = 0.076346 m But most of my measurements have fewer decimal places than my best tool!!!

Significant Figure Math Rules Addition / Subtraction: Answers can’t have more numbers to the right of the decimal point than the number in the problem with the least amount of numbers to the right of the decimal point. Example = 24.1 m + 3.35 m + 2.23 m Calculator says: 29.68 m (wrong) Answer: 29.7 m

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

Significant Figure Math Rules Let’s Try a Multiplication / Division Problem: Find the volume? 0.041m high 0.091 m wide 0.034 m deep 0.0001269 m3 What should my answer be??

Significant Figure Math Rules Multiplication / Division Problem: 14.1 cm 3.3 cm 4.23 cm2 42.3 cm2 46.53 cm2 What should my answer be??

Significant Figure Math Rules Multiplication / Division: Your answer can’t have more sig figs than the number in the problem with the least amount of sig figs Example = 60.56227892 cm x 35.25 cm Calculator says: 2134.890832 cm2 (wrong) Answer: 2135 cm2

Chemical Compound Quiz Which is denser: ice or water? Water Why?? Ice Expands 1/7th!!!

Significant Figures Lets’ see if you can: Determine the number of significant figures needed for an answer involving calculations. Round math problems properly

Significant Figure Math Rules Remember this Problem: Penny Example = 0.019 m using meter stick 0.0192 m using ruler 0.0191 m using calipers 0.019046 m using micrometer To find the total = 0.076346 m 0.076 m

Significant Figure Math Rules Remember This One: 14.1 cm 3.3 cm 4.23 cm2 42.3 cm2 46.53 cm2 What should my answer be?? 47 cm2

Significant Figure Math Rules How About This One: Find the volume? 0.041m high 0.091 m wide 0.034 m deep 0.0001269 m3 What should my answer be?? 0.00013 m3

Learning Check 1. 2.19 m X 4.2 m = A) 9 m2 B) 9.2 m2 C) 9.198 m2 2. 4.311 m ÷ 0.07 m = A) 61.58 B) 62 C) 60 3. 2.54 m X 0.0028 m = 0.0105 m X 0.060 m A) 11.3 B) 11 C) 10

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

Euphemisms in Science We all know that some politicians and government spokesmen use certain euphemistic phrases to give an aura of respectability to descriptions of events or actions which would be offensive when expressed in plain English. The following is a list of Euphemisms in Science and their translations into plain English. “Handled with extreme care during the experiments...” Not dropped on the floor!