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 5005 120301

• 3. Zeros AT THE BEGINNING OF A

NUMBER are not significant.

Examples 0.000003432 0.0021111

• 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 45.23000 1.000 505.32000 4750000

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

• Example 4.965 x 1016

Practice Problems

• Determine how many figures are significant in each of these measurements:

• 1. 375 2. 89.000

• 3. -0.00032 4. 4300

• 5. 12.0900 6. 0.00003200

• 7. 900001 8. 2.34 x 104

• 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 = 5.5998 = 5.60

• 3 sf 3sf 3sf 3sf

• 3 sf 45.6 = 18581.90709

• 4 sf 0.002454

• = 18587.90709 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 of the

decimal.– 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) 15.000 – 2.3791 = 12.6209

63.66 (2 sf) (3 sf) (4 sf) = 12.621 + 102.44 (2 sf)

191.50

= 191.5