Chapter 2

Measurements

Measured Numbers and Significant Figures

Measured Numbers

When you use a measuring tool is used to determine a quantity such as your height or weight, the numbers you obtain are called measured numbers.

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 0.05- 0.07

Length reported = 2.75 cm

or 2.76 cm

or 2.77 cm

Known + Estimated Digits

Known digits 2 and 7 are 100% certain

The third digit 6 is estimated (uncertain)

In the reported length, all three digits

(2.76 cm) are significant including the

estimated one

Learning Check

. l8. . . . I . . . . I9. . . .I . . . . I10. . cm

What is the length of the line?

1) 9.2 cm

2) 9.22 cm

3) 9.23 cm

How your answer compare with your neighbor’s

answer? Why or why not?

Solution

. l8. . . . I . . . . I9. . . .I . . . . I10. . cmEstimate to the hundreth’s place (0.01 cm)

2) 9.22 cm

3) 9.23 cm

The estimated digit may be slightly different.

Both readings are acceptable.

Learning Check

l5. . . . I . . . . I6. . . .I . . . . I7. . cm

What is the length of the line?

1) 6.0 cm

2) 6.06 cm

3) 6.60 cm

Solution

l5. . . . I . . . . I6. . . .I . . . . I7. . cm

Estimate to the hundreth’s place (0.01 cm). The reading is between 6.0 and 6.1. You may have estimated 6.07 cm as well.

2) 6.06 cm

Zero as a Measured Number

. l3. . . . I . . . . I4 . . . . I . . . . I5. . cm

What is the length of the line?

First digit 4.?? cm

Second digit 4.5? cm

Last (estimated) digit is 4.50 cm

(not to the left or right of .5)

Exact Numbers

Obtained when you count objects2 soccer balls

1 watch

4 pizzasObtained from a defined relationship

1 foot = 12 inches

1 meters = 100 cmNot obtained with measuring tools

Learning Check

A. Exact numbers are obtained by

1. measuring

2. counting3. definition

B. Measured numbers are obtained by

1. measuring

2. counting

3. definition

Solution

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 (1)

or a measured (2) number.

A.___Gold melts at 1064°C

B.___1 yard = 3 feet

C.___A red blood cell with diameter 6 x 10-4 cm

D.___There were 6 hats on the shelf

E.___A can of soda contains 355 mL of soda

Solution

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

A. 2 Requires a thermometer(measuring tool)

B. 1 From a definition in U.S. system

C. 2 Need measuring tool to determine

D. 1 Counted the hats

E. 2 Measured

Significant Figures

• Significant Figures – The digits that indicate the precision with which a measurement is made; all digits of a measured quantity are significant, including the last digit, which is uncertain.

Sig Fig Inquiry

• Determine the rules for finding the number of significant digits.

Rules1. All non-zero numbers are significant.

2. Zeros at the end without a decimal point are not significant.

3. Zeros at the end with a decimal point are significant.

4. Zeros in the middle of other numbers are significant.

5. Zeros at the beginning are NOT significant

Timberlake lecture plus 18

Adding and Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2 one decimal place

+ 1.34 two decimal places

26.54

answer 26.5 one decimal place

Addition/Subtraction with Sig Figs

• Your answer must have the same number of decimal places as the value with the fewest decimal places28.0 + 23.538 + 25.68 = 77.218

Round so that the answer is 77.2

Timberlake lecture plus 20

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

Timberlake lecture plus 21

Solution

A. 235.05 + 19.6 + 2.1 =

2) 256.8

B. 58.925 - 18.2 =

3) 40.7

Timberlake lecture plus 22

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.

Multiply/Divide with Sig Figs

• Your answer must have the same number of sig figs as the measurement with the fewest number of sig figs.Calculate the volume if L = 3.65 cm, W = 3.20

cm, and H = 2.05 cm

V = 23.944 cm3, after rounding to the correct number of sig figs, V=23.9 cm3

Timberlake lecture plus 24

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

Timberlake lecture plus 25

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