Significant Figures and Scientific Notation

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

► 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 2

3

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.

4

Learning CheckLearning Check

Classify each of the following as an exact or ameasured 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.

5

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 length.The number of hats is obtained by counting.A measuring tool is required.

SolutionSolution

Measurement and Significant FiguresMeasurement and Significant Figures

► Every experimental measurement has a degree of uncertainty.

► The volume 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 6

7

What is the Length?

1 2 3 4 cm

►We can see the markings between 1.6-1.7cm►We must guess between .6 & .7►We record 1.67 cm as our measurement

Learning CheckLearning Check

What is the length of the wooden stick?A. 4.5 cm B. 4.54 cm C. 4.547 cm

Chapter Two 9

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 0.029 07 mL has four.

10Chapter 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 11

Practice

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

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.

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

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 15

Chapter Two

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

►RULE 2. In carrying out an addition or subtraction, the answer cannot have more significant digits BEFORE or AFTER the DECIMAL point than either of the original numbers.

Chapter Two 17

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

82000

.1

0

Look for the last important digit

Chapter Two

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 =

How wide is our universe?210,000,000,000,000,000,000,000 miles

(22 zeros)

This number is written in decimal notation. When numbers get this large,

it is easier to write them in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is in the form

a x 10n

where a is between 1 and 10

and n is an integer

Write the width of the universe in scientific notation.

210,000,000,000,000,000,000,000 miles

Where is the decimal point now?

After the last zero.

Where would you put the decimal to make this number be between 1 and 10?

Between the 2 and the 1

2.10,000,000,000,000,000,000,000.

How many decimal places did you move the decimal?

23When the original number is more than 1,

the exponent is positive.The answer in scientific notation is

2.1 x 1023

Express 0.0000000902 in scientific notation.

Where would the decimal go to make the number be between 1 and 10?

9.02The decimal was moved how many places?

8When the original number is less than 1, the

exponent is negative.9.02 x 10-8

Write 28750.9 in scientific notation.

A. 2.87509 x 10-5

B. 2.87509 x 10-4

C. 2.87509 x 104

D. 2.87509 x 105

Express 1.8 x 10-4 in decimal notation.

0.00018

Express 4.58 x 106 in decimal notation.

4,580,000

On the calculator, scientific notation is done with the button.

4.58 x 106 is typed 4.58 6

Use a calculator to evaluate: 4.5 x 10-5

1.6 x 10-2

Type 4.5 -5 1.6 -2

You must include parentheses if you don’t use those buttons!!

(4.5 x 10 -5) (1.6 x 10 -2)

0.0028125Write in scientific notation.

2.8 x 10-3

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 6.0 x 10 -11

The answer in decimal notation is 0.000000000060

Write (2.8 x 103)(5.1 x 10-7) in scientific notation.

A. 14.28 x 10-4

B. 1.4 x 10-3

C. 14.28 x 1010

D. 1.428 x 10-3

Write in PROPER scientific notation.(Notice the number is not between 1 and 10)

234.6 x 109

2.346 x 1011

0.0642 x 104

6.42 x 10 2

Write 531.42 x 105 in scientific notation.

A. .53142 x 102

B. 5.3142 x 103

C. 53.142 x 104

D. 531.42 x 105

E. 53.142 x 106

F. 5.3142 x 107

G. .53142 x 108