Measurement and calculating using Significant figures and Sci notation

Chapter 2 Part 1: Measuring and Calculating Using Significant Figures: 

Chapter 2 Part 1: Measuring and Calculating Using Significant Figures 1 SIGNIFICANT FIGURES... Y U HAUNT ME???

Significant Figures: 

Significant Figures When using our calculators we must determine the correct answer. There are 2 different types of numbers Exact Measured Exact numbers are infinitely important Measured number = they are measured with a measuring device so these numbers have ERROR. When you use your calculator your answer can only be as accurate as your worst measurement. 2

Exact Numbers: 

Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

Practice: 

4 Practice 1. Exact numbers are obtained by A. using a measuring tool B. counting C. definition 2. Measured numbers are obtained by A. using a measuring tool B. counting C. definition

PowerPoint Presentation: 

5 Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10 -4 cm. There are 6 hats on the shelf. Gold melts at 1064°C. exact measured measured exact

Measurement and Significant Figures: 

Measurement and Significant Figures Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place. 6

What is the Length?: 

What is the Length? We can see the markings between 1.6-1.7cm We can’t see the markings between the .6-.7 We must guess between .6 & .7 We record 1.67 cm as our measurement The last digit an 7 was our guess...stop there

Practice: 

Practice What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

PowerPoint Presentation: 

9 8.00 cm ?

Measured Numbers: 

Measured Numbers Do you see why Measured Numbers have error…you have to make that Guess ! All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. To indicate the precision of a measurement, the value recorded should use all the digits known with certainty. 10

PowerPoint Presentation: 

11 Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

Note the 4 rules: 

Note the 4 rules When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.

PowerPoint Presentation: 

RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. 13

: 

Practice 45.8736 .000239 .00023900 48000. 48000 3.982  10 6 1.00040 6 3 5 5 2 4 6 All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal All digits count 0’s between digits count as well as trailing in decimal form

Scientific Notation: 

Scientific Notation Scientific notation is a convenient way to write a very small or a very large number. Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power. 215 is written in scientific notation as: 215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 10 2

PowerPoint Presentation: 

16 Two examples of converting scientific notation back to standard notation are shown below.

PowerPoint Presentation: 

Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point. The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures. Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 10 8 indicates 2 and writing it as 1.500 x 10 8 indicates 4. Scientific notation can make doing arithmetic easier. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A . 17

Rounding Off Numbers: 

Rounding Off Numbers Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. How do you decide how many digits to keep? Simple rules exist to tell you how. 18

PowerPoint Presentation: 

Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less. RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. If a calculation has several steps, it is best to round off at the end.

Practice: 

Practice Make the following into a 3 Sig Fig number 1.5587 .0037421 1367 128,522 1.6683  10 6 1.56 .00374 1370 129,000 1.67  10 6

Examples of Rounding: 

Examples of Rounding For example you want a 4 Sig Fig number 4965.03 780,582 1999.5 0 is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig 4965 780,600 2000.

PowerPoint Presentation: 

RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers. 22 Significant Figures and Calculations

PowerPoint Presentation: 

RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers. This will reflect the reliability of the least precise operation. 23

Practice: 

Practice 32.27  1.54 = 49.6958 3.68  .07925 = 46.4353312 1.750  .0342000 = 0.05985 3.2650  10 6  4.858 = 1.586137  10 7 6.022  10 23  1 .661  10 -24 = 1.000000 49.7 46.4 .05985 1.586  10 7 1.000

Addition/Subtraction: 

Addition/Subtraction 25.5 32.72 321 +34.270 ‑ 0.0049 + 12.5 59.770 32.7151 333.5 59.8 32.72 334

Addition and Subtraction: 

Addition and Subtraction .56 + .153 = .713 82000 + 5.32 = 82005.32 10.0 - 9.8742 = .12580 10 – 9.8742 = .12580 .71 82005 .1 0 Look for the last important digit

Mixed Order of Operation: 

Mixed Order of Operation 8.52 + 4 .1586  18.73 + 153.2 = (8.52 + 4 .1586)  (18.73 + 153.2) = 239.6 2180. = 8.52 + 77.89 + 153.2 = 239.61 = = 12.68  171.9 = 2179.692 =

Accuracy and Precision: 

Accuracy and Precision 28 Accuracy refers to the degree to which a measurement is true or correct. A measurement can be precise without being accurate. This can occur, for example, when a measuring instrument needs adjustment, so that the measurements obtained, no matter how precisely measured, are inaccurate.

Accuracy and Precision: 

29 Precision refers to the degree to which a measurement is repeatable and reliable; that is, consistently getting the same data each time the measurement is taken. The precision of a measurement depends on the magnitude of the smallest measuring unit used to obtain the measurement (for example, to the nearest meter, to the nearest centimeter, to the nearest millimeter, and so on). In theory, the smaller the measurement unit used, the more precise the measurement. Accuracy and Precision

PowerPoint Presentation: 

30 The amount of error involved in a physical measurement is the approximate error of the measurement. The maximum possible error of a measurement is half the magnitude of the smallest measurement unit used to obtain the measurement. Accuracy and Precision

Examples: 

Examples 31 Suppose a lab refrigerator holds a constant temperature of 38.0 F. A temperature sensor is tested 10 times in the refrigerator. The temperatures from the test yield the temperatures of: 39.4, 38.1, 39.3, 37.5, 38.3, 39.1, 37.1, 37.8, 38.8, 39.0. Neither accurate nor precise Suppose a lab refrigerator holds a constant temperature of 38.0 F. A temperature sensor is tested 10 times in the refrigerator. The temperatures from the test yield the temperatures of: 37.8, 38.3, 38.1, 38.0, 37.6, 38.2, 38.0, 38.0, 37.4, 38.3 accurate but not precise

Examples continued: 

Examples continued 32 Suppose a lab refrigerator holds a constant temperature of 38.0 F. A temperature sensor is tested 10 times in the refrigerator. The temperatures from the test yield the temperatures of : 39.2, 39.3, 39.1, 39.0, 39.1, 39.3, 39.2, 39.1, 39.2, 39.2 Precise but not accurate Suppose a lab refrigerator holds a constant temperature of 38.0 F. A temperature sensor is tested 10 times in the refrigerator. The temperatures from the test yield the temperatures of: 38.0, 38.0, 37.8, 38.1, 38.0, 37.9, 38.0, 38.2, 38.0, 37.9. Accurate and precise!!

Percent Error and Percent Yield: 

Percent Error and Percent Yield 33 This formula is used to calculate how much product you are able to produce in a lab. We will use this formula in the future so be familiar with the calculation. You will sometimes see the numerator in this formula switched. Both calculations are valid. **negative values with this calculation means that your experimental value was less than the theoretical value.