2.4 Significant Figures in 2.4 Significant Figures in MeasurementMeasurement

The The significant figuressignificant figures in a in a measurement include measurement include all the digits all the digits that are known precisely plus one that are known precisely plus one last digit that is estimatedlast digit that is estimated.. Example: With a thermometer that has 1° Example: With a thermometer that has 1°

intervals, you may determine that the intervals, you may determine that the temperature is between 24°C and 25°C and temperature is between 24°C and 25°C and estimate it to be 24.3°C.estimate it to be 24.3°C. You know the first two digits (2 and 4) with You know the first two digits (2 and 4) with

certainty, and the third digit (3) is a “best guest”certainty, and the third digit (3) is a “best guest” By estimating the last digit, you get By estimating the last digit, you get

additional information additional information

RulesRules

1. Every nonzero digit in a recorded 1. Every nonzero digit in a recorded measurement measurement is significantis significant..- Example: 24.7 m, 0.743 m, and 714 m all - Example: 24.7 m, 0.743 m, and 714 m all have three sig. figs.have three sig. figs.

2. Zeros appearing between nonzero digits 2. Zeros appearing between nonzero digits are are significantsignificant..

- Example: 7003 m, 40.79 m, and 1.503 m all - Example: 7003 m, 40.79 m, and 1.503 m all have 4 sig. figs.have 4 sig. figs.

RulesRules

3. Zeros appearing in front of all nonzero digits 3. Zeros appearing in front of all nonzero digits are not significantare not significant; they act as placeholders ; they act as placeholders and cannot arbitrarily be dropped (you can get and cannot arbitrarily be dropped (you can get rid of them by writing the number in scientific rid of them by writing the number in scientific notation).notation).- Example: 0.0071 m has two sig. figs. And can - Example: 0.0071 m has two sig. figs. And can be written as 7.1 x 10be written as 7.1 x 10-3-3

4. Zeros at the end of the number and to 4. Zeros at the end of the number and to the right of a decimal point the right of a decimal point are always are always significantsignificant..- Example: 43.00 m, 1.010 m, and 9.000 - Example: 43.00 m, 1.010 m, and 9.000 all have 4 sig. figs.all have 4 sig. figs.

RulesRules

5. Zeros at the end of a measurement and 5. Zeros at the end of a measurement and to the left of the decimal point to the left of the decimal point are not are not significant unless they are significant unless they are measured valuesmeasured values (then they are (then they are significant). Numbers can be written in significant). Numbers can be written in scientific notation to remove ambiguity.scientific notation to remove ambiguity.

- Example: 7000 m has 1 sig. fig.; if - Example: 7000 m has 1 sig. fig.; if those zeros were measured it could be those zeros were measured it could be written as 7.000 x 10written as 7.000 x 1033

RulesRules

6. Measurements 6. Measurements have an unlimited have an unlimited number of significant figuresnumber of significant figures when when they they are counted or if they are are counted or if they are exactly defined quantitiesexactly defined quantities..

- Example: 23 people or 60 minutes = - Example: 23 people or 60 minutes = 1 hour1 hour

* You must recognize exact values to * You must recognize exact values to round of answers correctly in round of answers correctly in calculations involving measurements.calculations involving measurements.

Significant Figures – Significant Figures – Example 1Example 1

How many significant figures are in How many significant figures are in each of the following measurements?each of the following measurements?a. 123 ma. 123 m

b. 0.123 cmb. 0.123 cm

c. 40506 mmc. 40506 mm

d. 9.8000 x 10d. 9.8000 x 1044 m m

e. 4.5600 me. 4.5600 m

f. 22 meter sticksf. 22 meter sticks

g. 0.07080 mg. 0.07080 m

h. 98000 mh. 98000 m

3 (rule 1)3 (rule 1)

3 (rule 3)3 (rule 3)

5 (rule 2)5 (rule 2)5 (rules 4 and 5)5 (rules 4 and 5)

5 (rule 4)5 (rule 4)

Unlimited (rule 6)Unlimited (rule 6)4 (rules 2, 3, and 4)4 (rules 2, 3, and 4)

2 (rule 5)2 (rule 5)

2.4 Concept Practice2.4 Concept Practice

7. Write each measurement in scientific 7. Write each measurement in scientific notation and determine the number of notation and determine the number of significant figures in each.significant figures in each.

a. 0.05730 ma. 0.05730 m

b. 8765 dmb. 8765 dm

c. 0.00073 mmc. 0.00073 mm

d. 12 basketball playersd. 12 basketball players

e. 0.010 kme. 0.010 km

f. 507 thumbtacksf. 507 thumbtacks

5.730 x 105.730 x 10-2-2 m, 4 m, 4

8.765 x 108.765 x 103 3 dm, 4dm, 4

7.3 x 107.3 x 10-4-4 mm, 2 mm, 2

1.2 x 101.2 x 101 1 BB players, unlimitedBB players, unlimited

1.0 x 101.0 x 10-2-2 km, 2 km, 2

5.07 x 105.07 x 1022 thumbtacks, unlimited thumbtacks, unlimited

Significant Figures in Significant Figures in CalculationsCalculations

The number of significant figures in a The number of significant figures in a measurement refers to the precision of measurement refers to the precision of a measurement; a measurement; an answer cannot be an answer cannot be more precise than the least precise more precise than the least precise measurement from which it was measurement from which it was calculated.calculated. Example: The area of a room that measures Example: The area of a room that measures

7.7 m (2 sig. figs.) by 5.4 m (2 sig. figs.) is 7.7 m (2 sig. figs.) by 5.4 m (2 sig. figs.) is calculated to be 41.58 mcalculated to be 41.58 m2 2 (4 sig. figs.) – you (4 sig. figs.) – you must round the answer to 42 mmust round the answer to 42 m22

Rounding – The Rule of 5Rounding – The Rule of 5

If the digit to the right of the last sig. fig If the digit to the right of the last sig. fig is less than 5, all the digits after the last is less than 5, all the digits after the last sig. fig. are dropped.sig. fig. are dropped. Example: 56.212 m rounds to 56.21 m (for 4 Example: 56.212 m rounds to 56.21 m (for 4

sig. figs.)sig. figs.)

If the digit to the right is 5 or greater, the If the digit to the right is 5 or greater, the value of the last sig. fig. is increased by 1.value of the last sig. fig. is increased by 1. Example: 56.216 m rounds to 56.22 m (for 4 Example: 56.216 m rounds to 56.22 m (for 4

sig. figs.)sig. figs.)

Rounding – Example 2Rounding – Example 2

Addition and SubtractionAddition and Subtraction

The answer to an addition or The answer to an addition or subtraction problem subtraction problem should be should be rounded to have the same number rounded to have the same number of decimal places as the of decimal places as the measurement with the least number measurement with the least number of decimal places.of decimal places.

Multiplication and Multiplication and DivisionDivision

In calculations involving multiplication and In calculations involving multiplication and division, the answer is rounded off to the division, the answer is rounded off to the number of significant figures in the least number of significant figures in the least precise term (least number of sig. figs.) in precise term (least number of sig. figs.) in the calculationsthe calculations