Chapter 2Significant Digits

Taking Measurements

•All measurements involve one estimation.

•If the measuring device is digital it will take the estimation for you.

Electronic Measuring Devices

• Digital readout measuring diameter of 1.0420 inches.

• The last zero is the estimated digit.

Taking Measurements

•All measurements involve one estimation.

•If the measuring device is scaled you must take the estimation yourself.

Scaled Measuring Devices

Scaled Measuring Devices

•Bottom ruler gives a measurement of 8.? cm.

Scaled Measuring Devices

•Measure the length of the metal using the top ruler.

How to read a meniscus.

Read the Volume in mL

Significant Digits

Rules for Determining the Number of Significant Digits in a Measurement

1. Digits other than zero are always significant.

96g 2 significant digits 61.42g 4 significant digits 0.538g 3 significant digits

2. One or more trailing zeros after the decimal point are always significant. 4.00000s 6 significant digits 0.05600s 4 significant digits 0.002 1 significant digit

3. Zeros between two other significant digits are always significant. 5.029km 4 significant digits 30600km 3 significant digits 0.050060km 5 significant digits

4. A zero which is not significant by any of the above rules can be made significant by placing a bar over it. However we usually write numbers such as these in scientific notation to show the number of significant figures.

1

0 ml 2 significant digits 1.0 x 102 ml

100

0 ml 4 significant digits 1.000 x 103 ml

10

0 000 ml 3 significant digits 1.00 x 105 ml When determining the number of significant digits in a measurement it is often helpful to use the following strategy.

1. Find the leftmost significant digit. It will always be your leftmost nonzero digit.

2. Find your rightmost significant digit. It will be the rightmost nonzero digit unless you have a zero which is further to the right and is (a) after the decimal point and at the end of the number or (b) has a bar over it.

3. All digits in between the leftmost significant digit and the rightmost significant digit are significant.

Examples:

70.12 L –270.8 °C 0.000800 mg 1000 mL 82.003 µm 42,729.00 cm 27.0 km 225 beans 50 people 99.294 dm 1.002cm 0.06900 m

2

0 0 kg 3,200,

0 00 µL

Rules for Significant Digits

• All non-zero digits are significant.

• “Trailing” zeros after the decimal point ARE significant.

• Zeros between significant digits are significant.

• All other zeros are NOT significant unless indicated to be so by having a bar placed over them.

How to Determine Significant Digits

•Underline the leftmost nonzero digit.

•Use the rules for significant digits to determine the rightmost significant digit.

•Every digit in between the leftmost and rightmost significant digits are significant as well.

Counting or Exact Numbers

Counting numbers: If there are 10 people in a room there are not 9.5 or 10.76 people in the room. Counting numbers are exact.

Ones in Conversion Factors: 1 kilometer = 1000 meters.  Exactly 1 km is equal to exactly 1000m. The 1 is considered to be an exact number and so is the 1000.

• Since Counting numbers and metric conversions are exact they have an infinite number of significant digits.

Determine the Significant Digits(Examples in Notebook)

• 70.12 L • 0.000800 mg • 82.003 µm • 27.0 km • 50 people • 1.002 cm • • 200 kg

• -270.8 ºC • 1000 mL • 42,729.00 cm • 225 beans • 99.294 dm • 0.06900 m • • 3,200,000 µL

Determine the Significant Digits (Examples in Notebook)

• 70.12 L 4• 0.000800 mg 3• 82.003 µm 5• 27.0 km 3• 50 people infinite• 1.002 cm 4• • 200 kg 2

• -270.8 ºC 4• 1000 mL 1• 42,729.00 cm 7• 225 beans infinite• 99.294 dm 5• 0.06900 m 4• • 3,200,000 µL 5

Calculating in Chemistry Class

Multiplying and Dividing

Rule: When multiplying and dividing your answer should always contain the number of significant digits that are in the original number with the least number of significant digits.

Example: When multiplying 22.37 cm x 3.10 cm x 85.75 cm = 5946.50525 cm3

We look to the original problem and check the number of significant digits in each of the original measurements:

22.37 shows 4 significant digits.

3.10 shows 3 significant digits.

85.75 shows 4 significant digits.

Our answer can only show 3 significant digits because that is the least number of significant digits in the original problem. Our answer is 5946.50525 which has 9 significant digits, we must round this to show only three significant digits. To do this start at the leftmost significant digit and count 3 places to the right. This would place us at the “4” which occupies the tens place. We must therefore round our answer to the tens place in order to show only 3 significant digits. Our final answer becomes 5950 cm3.

Examples:

a) 24 mm x 31.8 mm =

b) 8.40 g ÷ 4.2 ml =

c) 200 dm x 3.58 dm =

d) 5500 km ÷ 55.0 s =

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.

• (17.3 cm)(28 cm) = 484.4 cm2

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.

• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.

• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2

• 708g ÷ 4.700ml = 150.63829 g/ml =

Math Operations with Significant Digits

• When multiplying and/or dividing your answer must reflect the smallest number of significant digits.

• (17.3 cm)(28 cm) = 484.4 cm2 = 480 cm2

• 708g ÷ 4.700ml = 150.63829 g/ml = 151g/ml

Adding and Subtracting

RULE: When adding or subtracting your answer should always contain the number of decimal places that are in the original number with the least number of decimal places.

Example: When we add 3.76 g + 14.83 g + 2.1 g = 20.69 g

We look to the original problem to see the number of decimal places shown in each of the original measurements. 2.1 shows the least number of decimal places. We must round our answer, 20.69, to one decimal place (the tenth place). Our final answer is 20.7 g

Examples:

a) 49.1 g + 8.001 g =

b) 81.350 m − 7.35 m =

c) 4.60 s + 3 s =

d) 67.5 cm − 0.009 cm =

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g =

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g = 10g

Addition and/or Subtraction reflects the fewest decimal places

decimal places.

• 24.6 cm − 17.01 cm = 7.59 cm = 7.6 cm

• 8.5g + 1.32g + 0.18g = 10g = 10.0g

Homework

Worksheet – Significant Digits & Rounding.