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INTRODUCTION

IV. Significant Figures

A. Purpose of Sig Figs

Units of Measurement:•Measurements indicate the magnitude of something•Must include:

– A number– A unit

A. Purpose of Significant Figures

• There are 2 different types of numbers in our world– Exact numbers (counting numbers)– Measured numbers

• Measured numbers are measured with a tools; these numbers have ERRORS and are not exact.

• Errors (called the degree of uncertainty) are a result of:– Human error reading the instrument

(not intentional error)– Limitation in precision of instruments

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1. Exact NumbersAn exact number is obtained when you count

objects or use a defined relationship.

Counting objects are always exact

2 soccer balls4 pizzasExact 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.

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Learning Check

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

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Solution

A. Exact numbers are obtained by 2. counting3. definition

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

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Learning Check

Classify each of the following as an exact or a measured number.

1. 1 yard = 3 feet

2. The diameter of a red blood cell is 7.8

um.

3. There are 6 hats on the shelf.

3. Gold melts at 1064°C.

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Classify each of the following as an exact (1) or a measured(2) number.

1. This is a defined relationship.2. A measuring tool is used to determine

length.3. The number of hats is obtained by

counting.4. A measuring tool is required.

Solution

2. Measurement and Significant Figures

• Every measurement has a degree of uncertainty.

• The volume, V, at right is certain in the 1’s place: 17mL<V<18mL

• A best guess is needed for the tenths place.

• Two people may estimate to the tenth differently

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• We can see the measurement is between 1 and 2 cm

• We can’t see the markings between • We must guess between 1 and 2• I estimate the measurement to be 1.6 cm• The last digit in a measurement is an

estimate• You may estimate differently

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What is the Length?

• We can see the measurement is between 1.6-1.7cm• We can’t see the markings between 1.6-1.7• We must guess between .6 & .7• I estimate the measurement to be 1.67 cm• The last digit in a measurement is an estimate• You cannot make a more precise estimate than to

the hundreth using this ruler

Learning Check

What is the length of the wooden stick?

1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

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?

Measured Numbers• Measurements have a degree of uncertainty

that results from– _________________– _________________

• Significant figures are the numbers represented in a measurement. “Sig Figs” include:– All of the numbers recorded are known with

certainty – One estimated digit.

• To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

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The mass of an object is measured using two different scales. The same quantity is being described at two different levels of precision or certainty.

B. Using Significant Figures

Timberlake lecture plus 16

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1. Counting Significant Figures

Non Zero Numbers Number of Significant

Figures

38.15 cm 45.6 ft 265.6 lb ___122.55 m ___

All non-zero digits in a measured number are significant

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Leading Zeros

Number of Significant Figures

0.008 mm 1

0.0156 oz 3

0.0042 lb ____

0.000262 mL ____

Leading zeros in decimal numbers

(less than 1) are not significant.

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Sandwiched Zeros

Number of Significant Figures

50.8 mm 3

2001 min 4

0.702 lb ____

0.00405 m ____

Zeros between nonzero numbers are significant

Zeros between a non zero and followed by a decimal are significant

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Trailing Zeros

Number of Significant Figures 25,000 in.

2

200 yr 1

48,600 gal 3

25,005,000 g ____ Trailing zeros in numbers without decimals

are not significantThey are significant if followed by or preceded by a decimal

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Learning Check

A. Which answers 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

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Solution

A. Which answers contain 3 significant figures?

2) 0.00476 3) 4760

B. All the zeros are significant in

2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

2) 535,000 3) 5.35 x 105

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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

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Solution

In which set(s) do both numbers contain the same number of significant figures?

3) 0.000015 and 150,000

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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

Learning Check

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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

Solution

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2. 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

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a. Adding and Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places. The uncertain digit must reflect precision of instruments.

25.2 one decimal place

+ 1.34 two decimal places

26.54answer 26.5 one decimal place

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Learning Check

In each calculation, round the answer to the correct number of significant figures.A. 235.05 + 19.6 + 2.1 =

1) 256.75 2) 256.8 3) 257

B. 58.925 - 18.2 =1) 40.725 2) 40.73 3) 40.7

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Solution

A. 235.05 + 19.6 + 2.1 = 2) 256.8

B. 58.925 - 18.2 =3) 40.7

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b. Multiplying and Dividing

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

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Learning Check

A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60

C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

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Solution

A. 2.19 X 4.2 = 2) 9.2

B. 4.311 ÷ 0.07 = 3) 60

C. 2.54 X 0.0028 = 2) 11 0.0105 X 0.060

Continuous calculator operation = 2.54 x 0.0028 0.0105

0.060

ReviewWhen 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 0.029 07 mL has four.

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• 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.

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Practice Rule #1 Zeros

45.8736

0.000239

0.00023900

48000.

48000

3.982106

1.00040

6

3

5

5

2

4

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•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

Practice Rounding

Make the following into a 3 Sig Fig number

1.5587

0.0037421

1367

128,522

1.6683 106

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

Practice Rounding

Make the following into a 3 Sig Fig number

1.5587

0.0037421

1367

128,522

1.6683 106

1.56

0.00374

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 RoundingFor example you want a 4 Sig Fig number

4965.03

 

780,582

 

1999.5

Examples 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.

Multiplication and division

32.27 1.54 =

3.68 .07925 =

1.750 .0342000 =

3.2650106 4.858 =

6.0221023 1.66110-24 =

Multiplication 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

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46.4

.05985

1.586 107

1.000

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Addition and Subtraction

.56 + .153 =

82000 + 5.32 =

10.0 - 9.8742 =

10 – 9.8742 =

Look for the last important digit

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Addition and Subtraction

.56 + .153 = .713

82000 + 5.32 = 82005.32

10.0 - 9.8742 = .12580

10 – 9.8742 = .12580

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82000

.1

0

Look for the last important digit