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Daily question

What is the difference between accuracy and precision?

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Chapter 3

Scientific Measurement

3.1 Using and Expressing

Measurements

Significant Figures

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Learning targets

I can state the number of significant figures in a number and why they are significant.
I can calculate problems and have the correct number of significant figures in the answer.

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Significant Figures

You can easily read the temperature on this thermometer to the nearest degree.

You can also estimate the temperature to about the nearest tenth of a degree by noting the closeness of the liquid inside to the calibrations.

Suppose you estimate that the temperature lies between 22°C and 23°C, at 22.9°C.

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Significant Figures

This estimated number, 22.9°C, has three digits.

The first two digits (20 and 2) are known with certainty, while the rightmost digit (9) has been estimated and involves some uncertainty.

These reported digits all convey useful information and are called significant figures.

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Significant Figures

The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

Measurements must always be reported to the correct number of significant figures because calculated answers cannot be more precise than measured data.

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Significant Figures

The width of the door can be expressed as:

0.8 m for meter stick “a” because 0 is known and .8 is estimated
0.77 m for meter stick “b” because 0.7 is known and 0.07 is estimated
0.772 m for meter stick “c” because 0.77 is known and .2 is estimated

a.

b.

c.

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Significant Figures

Every nonzero digit is significant.

Each of these measurements has three significant figures:

24.7 meters

0.743 meter

714 meters

Determining Significant Figures in Measurements

To determine whether a digit in a measured value is significant, you need to apply the following rules.

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Significant Figures

2. Zeros appearing between nonzero digits are significant. These are called “sandwich zeros”.

Each of these measurements has four significant figures: Why?

7003 meters

40.79 meters

1.503 meters

Determining Significant Figures in Measurements

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Significant Figures

Rule 3: Trailing zeros follow a non zero digit and are significant only if there is a decimal point.

examples of this rule with the zeros this rule affects in boldface:

0.00500

0.03040

2.30 x 10^-5

4.500 x 10¹²

100.000

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Significant Figures

Each of these measurements has four significant figures: WHY? (find the captured & trailing zeros)

43.00 meters

1.010 meters

9.000 meters

Determining Significant Figures in Measurements

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Which digits are significant figures?

All non zero digits (1 to 9)
sandwich zeros – which are between non-zero digits
Example 1,001

3. Trailing zeros – with a decimal point

Example 1,001.000

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Which ZERO digits are NOT significant figures?

Leading zeros before real numbers
Example 0.000231
Trailing zeros after real numbers if no decimal is written- Example 1,000,000

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Significant Figures

Each of these measurements has only two significant figures:

0.0071 meter = 7.1 x 10^-3 meter

0.42 meter = 4.2 x 10^-1 meter

0.000 099 meter = 9.9 x 10^-5meter

Determining Significant Figures in Measurements

These are leading zeros

All digits are significant using scientific notation

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Significant Figures

The zeros in these measurements are not significant: WHY?

300 meters (one significant figure)

7000 meters (one significant figure)

27,210 meters (four significant figures)

No captured zeros nor trailing zeros, no decimal points!

Determining Significant Figures in Measurements

?

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Significant Figures

Unlimited number of significant figures

There are two situations in which numbers have an unlimited number of significant figures. Counting and Equivalents

The first involves counting.

23 people in your classroom

This measurement is a counted value, so it has an unlimited number of significant figures.

The second situation involves equivalents like those found within a system of measurement.

Each of these numbers has an unlimited number of significant figures.

60 min = 1 hr

100 cm = 1 m

Counting and equivalents do not limit the significant figures in your calculations!

Determining Significant Figures in Measurements

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Significant figures

Rules for significant figures

Rule 1: Non-zero digits are always significant.

Rule 2: “sandwich zeros” – any zeros between two significant digits are significant.

Rule 3: Trailing zeros are significant if there is a decimal point

Rule 4 – Unlimited significant figures

Counted values
equivalents

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Suppose that the winner of a 100-meter dash finishes the race in 9.98 seconds. The runner in second place has a time of 10.05 seconds. How many significant figures are in each measurement? Give the rules 1-4.

CHEMISTRY & YOU

There are three significant figures in 9.98

Rule 1- every nonzero digit is significant

There are four significant figures in 10.05

Rule 1- every nonzero digit is significant

Rule 2 – sandwich zeros between nonzero digits are significant

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Sample Problem 3.3

Counting Significant Figures in Measurements

How many significant figures are in each measurement? Give the rule(s) for counting significant figures

123 m

40,506 mm

9.8000 x 10⁴ m

3 sig-figs -– all nonzero numbers

5 sig-figs -– all nonzero numbers

– sandwich zeros

5 sig-figs -– trailing zeros with a decimal are significant

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Counting Significant Figures in Measurements

How many significant figures are in each measurement? Give the rule(s) for counting significant figures

22 metersticks
0.070 80 m

98,000 m

Unlimited- counting

4 sig-figs - all nonzero numbers are significant

-sandwich zeros are significant

– trailing zeros with a decimal are significant

2 sigfigs -– all nonzero numbers

No decimal so zeros are placeholders

No captured & no trailing zeros

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Significant Figures

Rounding

To round a number, first decide how many significant figures the answer should have.

Then round to that many digits, counting from the left.

“4 & below, let it go”
“5 & above, give it a shove”

Significant Figures in Calculations

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Page 69 # 6 Rounding Measurements

Round off each measurement to the number of significant figures shown in parentheses.

a. 314.721 meters (four)

b. 0.001 775 meter (two)

c. 8792 meters (two)

Sample Problem 3.4

314.7

0.001 8

8800

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Sample Problem 3.4

Page 69 # 6 Rounding Measurements

The arrow points to the digit immediately following the last significant digit. Write your answer using scientific notation.

a. 314.721 meters – round to 4 significant figures

↑

2 is less than 5, so you do not round up.

314.7 meters = 3.147 x 10² meters

2

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Sample Problem 3.4

The arrow points to the digit immediately following the second significant digit. Write your answer using scientific notation.

b. 0.001 775 meters - round to 2 significant figures

↑

7 is greater than 5, so round up.

0.0018 meter = 1.8 x 10^-3 meter

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Sample Problem 3.4

Solve Apply the concepts to this problem.

The arrow points to the digit immediately following the second significant digit. Write your answer using scientific notation.

c. 8792 meters - round to 2 significant figures

↑

9 is greater than 5, so round up.

8800 meters = 8.8 x 10³ meters

2

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Significant Figures

A calculated answer cannot be more precise than the least precise measurement from which it was calculated.

The calculated value must be rounded to make it consistent with the measurements from which it was calculated based on the number of significant figures.

A die measures 1.8 cm on each side.
Its volume is 1.8 cm x 1.8 cm x 1.8 cm = 5.832cm³.
This is incorrect because the measurements have tenths and the answer is expressed using thousandths.

Significant Figures in Calculations

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Significant Figures

Addition and Subtraction

The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

Significant Figures in Calculations

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Page 70

Sample Problem 3.5

Significant Figures in Addition and Subtraction

Give the answer to the correct number of significant figures.

12.52 meters + 349.0 meters + 8.24 meters

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Sample Problem 3.5

Solve Apply the concepts to this problem.

12.52 meters

349.0 meters

+ 8.24 meters

369.76 meters

2

349.0 meters has the fewest decimal places, just one. The answer must be rounded to one decimal place

369.8 meters

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Significant Figures

Multiplication and Division

In calculations involving multiplication and division, round the answer to the same number of significant figures as the measurement with the least number of significant figures.

Significant Figures in Calculations

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Sample Problem 3.6

Significant Figures in Multiplication and Division

Give the answers to the correct number of significant figures.

a. 7.55 meters x 0.34 meter

b. 2.10 meters x 0.70 meter

c. 2.4526 meters² ÷ 8.4 meters

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Sample Problem 3.6

Solve Apply the concepts to this problem.

a. 7.55 meters x 0.34 meter

The second measurement (0.34 meter) has the least number of significant figures (two). So, the answer must be rounded to two significant figures.

a. 7.55 meters x 0.34 meter = 2.567 meters²

= 2.6 meters²

2

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The second measurement (0.70 meter) has the least number of significant figures (two). So, the answer must be rounded to two significant figures.

b. 2.10 meters x 0.70 meter = 1.47 meters²

= 1.5 meters²

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Sample Problem 3.6

Solve Apply the concepts to this problem.

c. 2.4526 meters² ÷ 8.4 meters

The second measurement (8.4 meters²) has the least number of significant figures (two). So, the answer must be rounded to two significant figures.

c. 2.4526 meters² ÷ 8.4 meters = 0.291 076 meter

= 0.29 meter

2

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In what case are zeros significant in a measured value?

trailing zeros with a decimal point
Sandwich zero’s – between real numbers

Sig Fig Rule when + or - numbers?

Round to fewest decimal places

Sig Fig Rule when x or ÷ numbers?

Round to fewest significant figures

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End of Significant Figures!
Time for practice!
Practice here
http://science.widener.edu/svb/tutorial/sigfigurescsn7.html

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Glossary Terms

measurement: a quantitative description that includes both a number and a unit

scientific notation: an expression of numbers in the form m x 10ⁿ, where m is equal to or greater than 1 and less than 10, and n is an integer

accuracy: the closeness of a measurement to the true value of what is being measured

precision: describes the closeness, or reproducibility, of a set of measurements taken under the same conditions

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Glossary Terms

accepted value: a quantity used by general agreement of the scientific community

experimental value: a quantitative value measured during an experiment

error: the difference between the accepted value and the experimental value

percent error: the percent that a measured value differs from the accepted value

significant figure: all the digits that can be known precisely in a measurement, plus a last estimated digit

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