Significant Figures and Measurement Geneva High School
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Transcript of Significant Figures and Measurement Geneva High School
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Significant Figures and Significant Figures and MeasurementMeasurement
Geneva High SchoolGeneva High School
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Exact & Measured Numbers• There are 2 different types of numbers
– Exact– Measured
• Measured number = they are measured with a measuring device so these numbers have ERROR.
• Also, we all read measuring instruments slightly differently.
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How many donuts do you see in this picture?
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Hopefully you counted four donuts.This is an example of an exact number.An exact number is obtained when you count objects or use a defined relationship.
Counted objects are always exact, for example:2 soccer balls4 pizzas
Exact relationships with predefined values are also exact numbers. For example:
1 foot = 12 inches1 meter = 100 cm
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Check Your Learning
A. Exact numbers are obtained by 1. using a measuring tool
2. counting3. definition
B. Measured numbers are obtained by 1. using a measuring tool
2. counting3. definition
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Check Your Learning
A. Exact numbers are obtained by 2. counting
3. definition
B. Measured numbers are obtained by 1. using a measuring tool
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Check Your Learning
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.
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Check Your Learning
Classify each of the following as an exact (1) or a measured(2) number.
This is a defined relationship. A measuring tool is used to determine
length. The number of hats is obtained by
counting. A measuring tool is required.
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Measurements
• All scientific measurements have a number and a unit
• Ex: 11.2 km 65.304 mg
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Accuracy
• Accuracy is the closeness of a measurement to the actual value of what is being measured.
• How close a measured value is to the real value/target.
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Precision
• Precision is a gauge of how exact a measurement is.
• How repeatable is a measurement?
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Accuracy and Precision
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Significant Figures
• Significant digits are all the digits that are known in a measurement, plus the last digit that is estimated.
• The more precise a measurement is, the more significant digits the measurement contains.
• You cannot record a piece of data that is more precise than the measuring instrument allows.
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Tomorrow’s Quiz
• Scientific Notation• Metric Conversions• Identifying the number of significant
figures (yes, you may use your rules sheet I gave you_
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For example…• Suppose you measure the mass of a piece of iron to be 34.73
grams on a digital scale.• You then measure the volume to be 4.42 cubic centimeters.• The calculated density of that iron would be:
Density = 34.73 g = 7.857466g/cm3
4.42 cm3
This calculated answer can’t be recorded like this, because it contains seven significant figures, while the volume of the sample only had three significant figures in the measurement.
You cannot report a calculation or measurement that is more precise than the data used in the calculation.
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2.4 Measurement and 2.4 Measurement and Significant FiguresSignificant 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.
Chapter Two 16
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17
What is the Length?
1 2 3 4 cm
• 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
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Learning CheckLearning Check
What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
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8.00 cm or 3 (2.2/8)?
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Measured NumbersMeasured 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.
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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.
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Significant Figure Rule 1
• All non-zero digits are significant• Zeros between non-zeros are significant
34.591.0910.3001
All have 4 sig figs
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Rule 2
• If number ≥ 1 with a decimal point, then…all digits are significant
7.10010078.00
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Rule 3
• If digit ≥ 1 and no decimal point, then all zeros to right of non-zeros are not significant
32,000890400290
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Note the 4 rulesNote the 4 rulesWhen 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.
25Chapter Two
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• 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.
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Practice Rule #1 Zeros
45.8736.000239 .00023900 48000. 48000 3.982106 1.00040
6355246
•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
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2.5 Scientific Notation2.5 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
102
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Chapter Two 29
Two examples of converting standard notation to scientific notation are shown below.
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Chapter Two 30
Two examples of converting scientific notation back to Two examples of converting scientific notation back to standard notation are shown below. standard notation are shown below.
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• 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 108 indicates 2 and writing it as 1.500 x 108 indicates 4.
• Scientific notation can make doing arithmetic easier. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A.
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2.6 Rounding Off Numbers2.6 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.
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• 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.
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Practice Rule #2 RoundingPractice Rule #2 RoundingMake the following into a 3 Sig Fig numberMake the following into a 3 Sig Fig number
1.5587
.0037421
1367
128,522
1.6683 106
1.56
.00374
1370
129,000
1.67 106
Your Final number must be of the same value as the number you started with,129,000 and not 129
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Examples of RoundingExamples of RoundingFor 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.
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RULE 1. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.
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•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.
Chapter Two 37
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Multiplication and divisionMultiplication and division
32.27 1.54 = 49.6958
3.68 .07925 = 46.4353312
1.750 .0342000 = 0.05985
3.2650106 4.858 = 1.586137 107
6.0221023 1.66110-24 = 1.000000
49.7
46.4
.05985
1.586 107
1.000
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Addition/SubtractionAddition/Subtraction
25.5 32.72 320 +34.270 ‑ 0.0049 + 12.5 59.770 32.7151 332.5
59.8 32.72 330
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__ ___ __
Addition and SubtractionAddition and Subtraction
.56 + .153 = .713
82000 + 5.32 = 82005.32
10.0 - 9.8742 = .12580
10 – 9.8742 = .12580
.71
82000
.1
0
Look for the last important digit
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Mixed Order of OperationMixed 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 =
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Rule 4
• If number < 1, all zeros to left of non-zeros are not significant
0.0980.00030.90020
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Rule 5
• Any number that represents a numerical count or exact definition has an infinite # of sig figs
32 students in the class
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Sig Figs in Addition/Subtraction
• Answer is rounded to same # of decimal places as the measurement with the least # of decimal places.
• Ex: 13.1 + 4.25
63.408 This answer would be 80.758 rounded to 80.8
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Sig Figs in Multiplication/Division
• Round your answer to the measurement with the least # of sig figs.
Ex: 4.3 x 2 = 8.6 = 9
18.75 ÷ 3.5 = 5.357 = 5.4
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Error
Error = experimental value – accepted value
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Percent Error
Percent Error = error / accepted value x 100