AIM: Significant FiguresAIM: Significant Figures

►What are significant figures?

►On a blank sheet of paperOn a blank sheet of paperChapter Two 1

AIM: Significant FiguresAIM: Significant Figures

►The number of significant figures is the The number of significant figures is the number of digits believed to be correct number of digits believed to be correct by the person doing the measuring. It by the person doing the measuring. It includes one includes one estimatedestimated digit digit

Chapter Two 2

Lab ReportLab ReportTitle page Title page is present

and contains the student's name, class, date, teacher's name and title of the lab.Members of your group and boxes tested

Title page is present but is missing one or more of the following components: student's name, class, date, teacher's name, members of your group and title of the lab.

No title page provided

Problem Inquiry problem is clearly stated.

Inquiry problem is not clearly stated.

No inquiry problem provided.

Experimental Hypothesis

Hypothesis for each box provides an educated guess that is clear and reasonable based on what has been studied.

Hypothesis provides an educated guess but is not logical.

No hypothesis has been stated.

Materials All materials and setup used in the experiment are clearly and accurately described.

Almost all materials and the setup used in the experiment are clearly and accurately described.

Many materials are described inaccurately OR are not described at all.

3

Lab ReportLab ReportProcedures Procedures are

listed in clear steps. Each step is numbered and is a complete sentence.

Procedures are listed in a logical order, but steps are not numbered and/or are not in complete sentences.

Procedures do not accurately list the steps of the experiment.

Results Results give a detailed description of what happened when you performed your tests for each box. You may include tables that summarize your results for each box tested

Results give some description of what happened when you performed your tests.

No results provided.

Conclusion Conclusion includes what was predicted to be inside the boxes and how whether this prediction was correct or not.

Conclusion includes what was predicted to be inside the boxes but does not indicate if this prediction was correct or not.

No conclusion was included in the report OR shows little effort and reflection.

4

Significant FiguresSignificant 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 so these numbers have ERROR.► When you use your calculator your answer can only be as

accurate as your worst measurement…

Chapter Two 5

6

Exact NumbersExact Numbers

An exact number is obtained when you count objects or use a defined relationship.

Counting objects are always exact2 soccer balls4 pizzas

Exact relationships, predefined values, not measured1 foot = 12 inches1 meter = 100 cm

For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

7

Learning Check 1 Learning Check 1

A. Exact numbers are obtained by 1. using a measuring tool

2. counting3. definition

B. Measured numbers are obtained by 1. using a measuring tool

2. counting3. definition

8

SolutionSolution

A. Exact numbers are obtained by 2. counting

3. definition

B. Measured numbers are obtained by 1. using a measuring tool

9

Learning Check 2Learning Check 2

Classify each of the following as an exact or ameasured number.

1 yard = 3 feet

The diameter of a world cup soccer ball is 6 x 106

cm.

There are 6 hats on the shelf.

Gold melts at 1064°C.

10

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

SolutionSolution

2.4 Measurement and Significant 2.4 Measurement and Significant FiguresFigures

► 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.

Chapter Two 11

12

What is the Length?

►We can see the markings between 1.6-1.7cm►We can’t see the markings between the .6-.7►We must guess between .6 & .7►We record 1.67 cm as our measurement►The last digit an 7 was our guess...stop there

Learning Check 3Learning Check 3

What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

14

8.00 cm or 3 (2.2/8)?

Measured NumbersMeasured 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.

15

Chapter Two 16

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.

Note the 4 rulesNote the 4 rules

When reading a measured value, all nonzero digits 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 0.02907 mL has four.

17Chapter Two

► 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.

Chapter Two 18

Practice Rule #1 Zeros

45.8736

.000239

.00023900

48000.

48000

3.982106

1.00040

6

3

5

5

2

4

6

•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

•All digits count

•0’s between digits count as well as trailing in decimal form

2.5 Scientific Notation2.5 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 10, 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

Chapter Two 20

Chapter Two 21

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

Chapter Two 22

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

► 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. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A.

Chapter Two 23

2.6 Rounding Off Numbers2.6 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.

Chapter Two 24

► 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 becomes 2.4 when 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 is 4.6 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.

Chapter Two 25

Practice Rule #2 RoundingPractice Rule #2 Rounding

Make the following into a 3 Sig Fig numberMake the following into a 3 Sig Fig number

1.5587

.0037421

1367

128,522

1.6683 106

1.56

.00374

1370

129,000

1.67 106

Your Final number must be of the same value as the number you started with,129,000 and not 129

Examples of RoundingExamples of RoundingFor example you want a 4 Sig Fig number

4965.03

780,582

1999.5

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

780,600

2000.

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

Chapter Two 28

►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.

Chapter Two 29

Multiplication and divisionMultiplication and division

32.27 1.54 = 49.6958

3.68 .07925 = 46.4353312

1.750 .0342000 = 0.05985

3.2650106 4.858 = 1.586137 107

6.0221023 1.66110-24 = 1.000000

49.7

46.4

.05985

1.586 107

1.000

Addition/SubtractionAddition/Subtraction

25.5 32.72 320 +34.270 0.0049‑ + 12.5 59.770 32.7151 332.5

59.8 32.72 330

__ ___ __

Addition and SubtractionAddition and Subtraction

.56 + .153 = .713

82000 + 5.32 = 82005.32

10.0 - 9.8742 = .12580

10 – 9.8742 = .12580

.71

82005

.1

0

Look for the last important digit

Mixed Order of OperationMixed Order of Operation

8.52 + 4.1586 18.73 + 153.2 =

(8.52 + 4.1586) (18.73 + 153.2) =

239.6

2180.

= 8.52 + 77.89 + 153.2 = 239.61 =

= 12.68 171.9 = 2179.692 =

Homework Due 10/09/2013Homework Due 10/09/2013

►Use your calculator to do problems 16-19 on the Use your calculator to do problems 16-19 on the practice problems worksheetpractice problems worksheet

34Chapter Two

Practice problemsPractice problems

35