Measuring

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Lesson 4 Measuring Significant Digits
Topic 1 Units of Measurement Metric
Prefixes Topic 2 Scientific Notation Topic 3
Accuracy, Precision and Error Topic 4
Significant Digits
Anything in black letters write it in your
notes (knowts)
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Topic 1 Units of Measurement Metric Prefixes
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Measurements without units are useless!
I walked 5 today. The speed of light is
186,000 I weigh 890 20 of water
All measurements need units!
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SI International System of Units
SI Base Units SI Base Units SI Base Units
Quantity SI base unit Symbol
Length meter m
Mass kilogram kg
Temperature kelvin K
Time second s
Number of Things mole mol
Luminous intensity candela cd
Electric current ampere A
We will use all of these in this class
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Commonly Used Metric Prefixes Commonly Used Metric Prefixes Commonly Used Metric Prefixes Commonly Used Metric Prefixes
Prefix Symbol Meaning Factor
mega M 1 million times larger than the base 106
kilo k 1000 times larger than the base 103
BASE Base Unit (meter, second, gram, etc)
deci d 10 times smaller than the base 10-1
centi c 100 times smaller than the base 10-2
milli m 1000 times smaller than the base 10-3
micro µ 1 million times smaller than the base 10-6
nano n 1 billion times smaller than the base 10-9
pico p 1 trillion times smaller than the base 10-12
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Volume -
Amount of space occupied by an object (remember?)
Normal units used for volume Solids m3 or
cm3 Liquids Gases liters (L) or
milliliters (ml)
1 L 1000 mL 1 mL 1 cm3
The volume of a material changes with
temperature, especially for gases.
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Mass -
Measure of inertia (remember?)
Weight -
Force of gravity on a mass measured in pounds
(lbs) or Newtons.
Weight can change with location, mass does not
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Energy
Ability to do work or produce heat.
Normal units used for energy SI joule
(J) non-SI calorie (cal)
1 cal 4.184 J
How many joules are in a kilojoule? How many
calories are in a kilocalorie?
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Temperature
measure of how cold or hot an object is.
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Temperature
measure of the average kinetic energy of
molecules.
Normal units used for temp SI kelvin (K)
non-SI celsius (C) or Fahrenheit (F)
yucky!
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(No Transcript)
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Normal units for density g/cm3, g/mL, g/L
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Densities of Some Common Materials Densities of Some Common Materials Densities of Some Common Materials Densities of Some Common Materials
Solids and Liquids Solids and Liquids Gases Gases
Material Density at 20C (g/cm3) Material Density at 20C (g/L)
Gold 19.3 Chlorine 2.95
Mercury 13.6 Carbon dioxide 1.83
Lead 11.3 Argon 1.66
Aluminum 2.70 Oxygen 1.33
Table sugar 1.59 Air 1.20
Corn syrup 1.351.38 Nitrogen 1.17
Water (4C) 1.000 Neon 0.84
Corn oil 0.922 Ammonia 0.718
Ice (0C) 0.917 Methane 0.665
Ethanol 0.789 Helium 0.166
Gasoline 0.660.69 Hydrogen 0.084
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Topic 2 Scientific Notation
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We will often work with really large or really
small numbers in this class.
Scientific Notation
Standard Notation
872,000,000 grams 0.0000056 moles
8.72 x 108 grams 5.6 x 10-6 moles
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exponent
coefficient
6.02 x 1023
The coefficient must be a single, nonzero digit,
exponent must be an integer.
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Multiplication and Division
To multiply numbers written in scientific
notation, multiply the coefficients and add the
exponents. (3 x 104) x (2 x 102) (3 x 2) x
1042 6 x 106 (2.1 x 103) x (4.0 x 107)
(2.1 x 4.0) x 103(7) 8.4 x 104
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To divide numbers written in scientific notation,
divide the coefficients and subtract the
exponents (top bottom)
Coefficient needs to be between 1 and 10
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Addition and Subtraction
When adding or subtracting in Sci. Not., the
exponents must be the same.
(5.4 x 102) (8.0 x 102)
(5.4 8.0) x 102
13.4 x 102
1.34 x 103
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• Example
• Solve each problem and express the answer in
  scientific notation.
• a. (8.0 x 102) x (7.0 x 105)
• b. (7.1 x 102) (5 x 102)

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a. Multiply the coefficients and add the
exponents.
(8.0 x 102) x (7.0 x 105)
(8.0 x 7.0) x 102 (5)
56 x 107
5.6 x 106
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b. Rewrite one of the numbers so that the
exponents match. Then add the coefficients
(7.1 x 102) (5 x 102)
(7.1 5) x 102
12.1 x 102
1.21 x 101
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Topic 3 Accuracy, Precision and Error
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Accuracy -
closeness of a measurement to the actual or
accepted value.
Precision -
closeness of repeated measurements to each other
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Accuracy and Precision
Darts on a dartboard illustrate the difference
between accuracy and precision.
Poor Accuracy, Poor Precision
Good Accuracy, Good Precision
Poor Accuracy, Good Precision
The closeness of a dart to the bulls-eye
corresponds to the degree of accuracy. The
closeness of several darts to one another
corresponds to the degree of precision.
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Error
Suppose you measured the melting point of a
compound to be 78C Suppose also, that the actual
melting point value (from reference books) is
76C. The error in your measurement would be 2C.
Error is always a positive value
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How far off you are in a measurement doesnt tell
you much. For example, lets say you have
1,000,000 in your checking account. When you
balance your checkbook at the end of the month,
you find that you are off by 175 error
175 Now, lets be more realistic, you have 225
in your checking account and after balancing you
are off by 175!
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In both cases, there is an error of 175. But in
the first, the error is such a small portion of
the total that it doesnt matter as much as the
second. So, instead of error, percent error is
more valuable.
Percent error compares the error to the size of
the measurements.
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Topic 4 Significant Digits
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In any measurement, the last digit is estimated
30.2C
The 2 is estimated (uncertain) by the
experimenter, another person may say 30.1 or 30.3
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9.3 mL
0.72 cm
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Increasing Precision
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The significant figures in a measurement are the
numbers that are part of the measurement. Zeros
that are NOT significant are called placeholders.
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Rules for determining Significant Figures 1.
Every nonzero digit in a reported measurement is
assumed to be significant. 2. Zeros appearing
between nonzero digits are significant. 3.
Leftmost zeros appearing in front of nonzero
digits are not significant. They act as
placeholders. By writing the measurements in
scientific notation, you can eliminate such
placeholding zeros. 4. Zeros at the end of a
number and to the right of a decimal point are
always significant. 5. Zeros at the rightmost end
of a measurement that lie to the left of an
understood decimal point are not significant if
they serve as placeholders to show the magnitude
of the number. 5 (continued). If such zeros were
known measured values, then they would be
significant. Writing the value in scientific
notation makes it clear that these zeros are
significant. 6. There are two situations in which
numbers have an unlimited number of significant
figures. The first involves counting. A number
that is counted is exact. 6 (continued). The
second situation involves exactly defined
quantities such as those found within a system of
measurement.
HOLY SMOKES!!
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Line Through Method for Counting Sig Figs
1. If there is a decimal, start from the left and
draw a line through any zeros, the numbers
remaining are significant.
2. If there is no decimal, start the line from
the right.
A Shorter Method
SOURCE Skylar Morben, 2014 MRHS Graduate
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• How many significant digits?
• 100 1.00 0.23
• 0.0034 1.01 1005.4
• 0.10 100.0 54.0

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How many significant digits are in the following
measurements? a) 150.31 grams b) 10.03 mL c)
0.045 cm d) 4.00 lbs e) 0.01040 m f)
100.10 cm g) 100 grams h) 1.00 x 102 grams
i) 11 cars j) 2 molecules
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An answer cant be more accurate than the
measurements it was calculated from
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Rules for Add/Subtracting Sig Figs The answer to
an /- calculation should be rounded to the same
number of decimal places as the measurement with
the least number of decimal places.
3.2 cm
37.10 g
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Rules for Mult/Division Sig Figs The answer to a
x/ calculation should be rounded to the same
number of sig figs as the measurement with the
least number of sig figs.
3.8 cm2
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Always round your final answer off to the correct
number of significant digits.
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• Draw a box around the significant digits in the
  following measurements.
• 2.2000 b) 0.0350 c) 0.0006
• 0.0089 e) 24,000 f) 4.360 x 104
• 0.0708 h) 1200 i) 0.6070
• k) 21.0400 l) 0.007 m) 5.80 x 10-3

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• Round off each of the following numbers to two
  significant figures.
• a) 86.048 b) 29.974 c) 6.1275
• 0.008230 e) 800.7 f) 0.07864
• g) 0.06995 h) 7.096 i) 8000.10

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• Express each of the following numbers in standard
  scientific notation with the correct number of
  significant digits.
• 0.00000070
• 25.3
• 825,000
• 826.7
• 43,500
• 65.0
• 0.000320
• 0.0432

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• Perform the following arithmetic. Round the
  answers to the proper number of sig. figs. Box
  in your final answer and dont forget units!
• 2.41 cm x 3.2 cm b) 4.025 m x 18.2 m
• c) 81.4 g ? 104.2 cm3

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Perform the following arithmetic. Round the
answers to the proper number of sig. figs. Box
in your final answer and dont forget units! d)
822 mi ? 0.028 hr e) 10.89 g / 1 mL