BELL RINGER – Complete on a sheet of paper and TURN IN before working on notes!

A student needed to calibrate a graduated cylinder [a device to measure liquids]. She collected the following data: Trail #1 99.98 mL Trial #2 100.02 mL Trial #3 99.99 mL The accepted value of the cylinder ’s volume is 100.00 mL. What is the PERCENT ERROR of her measurements?

• • Average = 99.98 + 100.02 + 99.99 = 99.99

3 • Error = 99.99 – 100.00 = -.01

• • Percent Error = 0.1 x 100 = 0.01 % error 100.00

Significant Figures

Dealing with uncertainty in measurements.

What values are shown below?

• Why is it difficult to be certain about some of the measurements you make?

– All measurements have SOME DEGREE OF UNCERTAINTY due to limits associated with the measuring device.

– Generally, uncertainty begins with the LAST DIGIT of the measurement.

• In a measurement, ALL THE DIGITS KNOWN FOR CERTAIN plus the first ESTIMATED DIGIT are known as the SIGNIFICANT FIGURES of the measurement.

• It is generally accepted that when a measurement is given, ALL NON-ZERO DIGITS are considered SIGNIFICANT . For example 175.4 grams Digits known for certain.

First estimated digit.

The Problem with Zero

• While all NON-ZERO DIGITS are considered significant, ZEROS present a particular problem.

– Zeros can be measurements – Zeros can be place holders • How do you decide whether or not a zero is significant?

•

Rules for Significant Figures

• 1. ALL NON-ZERO digits are considered significant. Examples 125.45

5648 1.1211

• • 2. Zeros IN THE MIDDLE OF NUMBERS are SIGNIFICANT parts of a measurement.

Examples 5 00 5 12 0 3 0 1

• 3. Zeros AT THE BEGINNING OF A NUMBER are not significant.

Examples 0.

00000 3432 0.

00 21111 • 4. Zeros AT THE END OF A NUMBER are only significant IF THE FOLLOW A DECIMAL or a BAR is placed over a zero… when this occurs, ALL digits up to and including the zero with the bar are significant.

Example 45.23

000 1.

000 505.32

000 _ 475 00 00

•

NOTE

– If the number is in SCIENTIFIC NOTATION only consider the COEFFICIENT when determining Significant Figures.

• Example 4.965

x 10 16

Practice Problems

• Determine how many figures are significant in each of these measurements: • 1. 375 • 3. -0.00032

2. 89.000

4. 4300 • 5. 12.0900

6. 0.00003200

• 7. 900001 8. 2.34 x 10 4 _ • 9. -0.000212000 10. 4002000

Mathematical Operations with Significant Figures

• When completing math calculation, the final answer must be reported rounded to the appropriate number of significant figures.

• The answer is rounded according to the

LAST

mathematical operation completed.

Rules

• 1. Complete calculations following the order of operations.

• 2. If the FINAL step is MULTIPLICATION or DIVISION: – A. Look at each value given in the problem and find the one with the LEAST number of significant figures.

– B. Round the FINAL ANSWER to the same number of significant figures.

– DO NOT ROUND UNTIL THE FINAL STEP!

Mult/Div Examples

• 4.59 X 1.22 = 5.5998 = 5.59

98 = 5.60

• 3 sf 3sf 3sf 3sf • • • 3 sf • 4 sf 45.6 0.002454

= 18581.90709

= 185 87.90709 3sf = 18600 3sf

ADD/SUBTRACT

• Complete calculations following order of operations.

• If the FINAL step is addition or subtraction: – A. Only consider digits to the

RIGHT

decimal.

of the – B. Determine the fewest SF to the right of the decimal.

– C. Round final answer to this number of SF.

ADD/SUBTRACT EXAMPLES

25.4 (1 sf) 63.66 (2 sf) + 102.44

(2 sf) 191.5

0 = 191.5

15.000 – 2.3791 = 12.620

9 (3 sf) (4 sf) = 12.621