BELL RINGER Complete on a sheet of paper

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 m. L Trial #2 100. 02 m. L Trial #3 99. 99 m. L The accepted value of the cylinder’s volume is 100. 00 m. L. 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. 60 • 3 sf 3 sf 3 sf • 3 sf • 4 sf • • 45. 6 0. 002454 = 18581. 90709 = 18587. 90709 3 sf = 18600 3 sf

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) 63. 66 (2 sf) + 102. 44 (2 sf) 191. 50 = 191. 5 15. 000 – 2. 3791 = 12. 6209 (3 sf) (4 sf) = 12. 621