Numbers and Uncertainty

Transcription

1 Significant Figures

2 Numbers and Uncertainty Numbers express uncertainty. Exact numbers contain no uncertainty. They are obtained by counting objects (integers) or are defined, as in some conversion factors (ex min hours). Inexact numbers contain uncertainty. They are obtained from measurements. The un-certainty in a measurement is shown by the number of digits recorded.

3 Measurements and Uncertainty Every measurement has UNITS. Every measurement has UNCERTAINTY. Uncertainty refers to the errors in measurement The uncertainty of the measurement is determined by the scale of the measuring device. The smaller the unit you use to measure with, the more precise the measurement is 2 types of errors can lead to uncertainty in measurements

4 2 types of Error Systematic error: come from the measuring instrument There is something wrong with the instrument or its data handling system\ the instrument is wrongly used by the experimenter Examples: worn out instrument. For example, a plastic tape measure becomes slightly stretched over the years, resulting in measurements that are slightly too high. An incorrectly calibrated or tared instrument, like a scale that doesn t read zero when nothing is on it. A person consistently takes an incorrect measurement. For example, they might think the 3/4 mark on a ruler is the 2/3 mark.

5 Random error: caused by unknown and unpredictable changes in the experiment. These changes may occur in the measuring instruments or in the environmental conditions. Example: You measure the mass of a ring three times using the same balance and get slightly different values: g, g, g Minimizing random errors: Take more data.

6 A few rules to observe in the lab: 1) a mechanical instrument (ruler, triple beam balance, graduated cylinder etc) record all the digits that are marked on the instrument s scale and estimate one more (and only one more) digit. "read between the lines" for last digit. That digit is an estimate, and contains uncertainty. Example: 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.

7 2) an electronic instrument record all the digits on the readout. Consider the last digit to be approximate. 3) Round calculated answers only once, at the end of the calculation, so that the number of significant digits reflects the precision of the original measurements.

8 Precision vs. Accuracy What is the difference? Precision- how close individual measurements agree with each other. Accuracy- how close individual measurements agree with the true or accepted value.

9 When making a measurement you must always estimate 1 place past the smallest division. Ruler example 9.5 cm is all you can read off of a cm ruler, you can estimate the 2 nd decimal and say the pencil is 9.50 cm but to say that the pencil was would give the impression that a very precise tool had been used not a ruler.

10 Significant figures Significant Figures are used to indicate the precision of a measured number or to express the precision of a calculation with measured numbers. In any measurement the digit farthest to the right is considered to be estimated

11 Which digits are significant? Rule #1: All non-zero digits are significant 24 has two sig figs, 24.1 has 3 sig figs Rule #2: All zeros bounded (trapped) by non-zero integers are significant 2004 has four sig figs, also has 4 sig figs

12 Rule #3: Zeros placed before other digits (leading zeros) are not significant has 2 sig figs Rule #4: Zeros at the end of a number are significant ONLY if they come after a decimal point 2.40 has three sig figs, 240 only has 2 sig figs Hint: Change the number to scientific notation. It is easier to see Example: 2.4 x 10 2 has 2 sig. figs., while 2.40 x 10 2 has 3 sig. figs.

13 Measurement # of Sig Figs 1) ) 300 3) ) ) )

14 QUESTIONS If Jenn measures a line to be 12.0 cm, what number is she doubtful about and how many sig. figs. are there? If Darren measures a mass to be 1300 g, what number is he doubtful about and how many sig. figs. are there?

15 Significant Figures in Calculations RULE 1. MULTIPLICATION AND DIVISION the answer cannot have more significant figures than either of the original numbers.

16 RULE 2. ADDITION AND SUBTRACTION the answer cannot have more digits after the decimal point than either of the original numbers. The last digit retained is set by the first doubtful digit. Two digits after the decimal 3.18 L L L 3.19 L Five digits after the decimal Two digits after the decimal

17 Rule 3: Rounding If the digit removed is 4 or less, drop it and all following digits Ex becomes 2.4 when rounded to 2 significant figures If the digit removed is 4 or greater, round the digit preceding number up Ex becomes 4.6 when rounded to 2 significant figures

18 Rule 4: Working with exact and inexact numbers: Exact numbers don't have sig figs because they don t introduce uncertainty. Just use sig figs in inexact numbers 0.16 km x 60 min = 9.6 km min hr hr 60 min/hour is an exact conversion

19 Rule 5: 5. If your calculator gives you fewer sig figs than the value should have, add zeroes Example x = = 1.800

20 Rule 6: If your calculator gives you more digits to the left of the decimal than are significant, use scientific notation x 24.8 x 675 = = 1.26 x 10 6

21 Examples of Rounding Round the following to 4 significant figures is dropped, it is <5 780, , 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

22 Practice: Rounding Make the following into a 3 Sig Fig number , ,000 Your Final number must be of the same value as the number you started with, 129,000 and not

23 Addition/Subtraction Practice Add/Subtract the following. Remember to record your answer using significant figures Look for the last important digit

24 Addition and Subtraction Add/Subtract the following. Remember to record your answer using significant figures = = = =

25 Multiplication and division Multiply/divide the following. Remember to record your answer using significant figures = = = = x =