Download Chapter 2 - Lyndhurst School District

The objectives of this tutorial are:
• —Explain the concept of signficant figures.
• —Define rules for deciding the number of significant figures
in a measured quantity.
• —Explain the concept of an exact number.
• —Define rules for determining the number of significant
figures in a number calculated as a result of a mathematical
operation.
• —Explain rules for rounding numbers.
• —Present guidelines for using a calculator.
• —Provide some exercises to test your skill at significant
figures.
Uncertainty in Measurements
• Measurements always involve a comparison. When
you say that a table is 6 feet long, you're really saying
that the table is six times longer than an object that is
1 foot long. The foot is a unit; you measure the length
of the table by comparing it with an object like a
yardstick or a tape measure that is a known number of
feet long.
• The comparison always involves some
uncertainty. If the tape measure has marks
every foot, and the table falls between the sixth
and seventh marks, you can be certain that the
table is longer than six feet and less than seven
feet. To get a better idea of how long the table
actually is , though, you will have to read
between the scale division marks. This is done
by estimating the measurement to the nearest
one tenth of the space between scale divisions.
A measurement is a quantity that
has both a number and a unit.
2.34 g
36.1 mL
16.5 Years Old
–Measurements are fundamental to the experimental sciences.
For that reason, it is important to be able to MAKE
measurements and to decide whether a measurement is
CORRECT.
–-- Are you “certain” that your measurement is correct? HOW
“certain” are you???
Accuracy
 Accuracy refers to how closely a measurement matches
the true or actual values
 To be accurate only requires the true value (bulls eye) &
one measurement (for the arrow to hit the target)
 Highly accurate data can be costly and difficult to
acquire
Precision
• Precision refers to the reproducibility of the
measurement and exactness of
description in a number.
• To decide on precision, you need several
measurements (notice multiple arrow holes), and you
do not need to know the true value (none of the
values are close to the target but all the holes are
close together.)
Accuracy & Precision
• In order to be accurate and precise, one
must pay close attention to detail to
receive the same results every time as
well as “hit the target”.
Comparing Accuracy & Precision
• Notice the difference in these pictures.
• To win the tournament the archers must hit
the target the most times. The winner must
show accuracy &BAD
precision.
•
•
•
•
1st
BAD
The
archer has _____ accuracy & ____ precision.
BAD
GOOD
nd
The 2 archer has _____
accuracy & GOOD
____ precision.
GOOD
The 3rd archer has _____ accuracy & ____ precision.
GOOD
BAD
th
The 4 archer has _____ accuracy & _____ precision
Example 1
• A sample is known to weigh 3.182 g. Jane
weighed the sample five different times
with the resulting data. Which
measurement was the most accurate?
– 3.200 g
– 3.180 g
– 3.152 g
– 3.189 g
Example 2
Consider the data (in cm) for the length of
an object as measured by three students.
The length is known to be 14.5 cm.
Which student had the most precise
work, and which student had the most
accurate work?
Trial 1
Student 14.8
A
Student 14.7
B
Student 14.4
C
Trial 2
Trial 3
Trial 4
Trial 5
14.7
14.8
14.7
14.8
14.2
14.6
14.6
14.8
14.4
14.5
14.4
14.5
Error = experimental value (EV) - accepted value (AV)
(measured by student) - (correct value)
–Determining Error
– The experimental value (EV) is the value measured in
the lab. (by the student)
– The accepted value (AV) is the correct value based on
reliable references.
– The error is the difference between the experimental
value and the accepted value
What is the error in your measurement
of the age of my cat?
Percent Error
Percent Error = EV - AV x 100%
AV
•The percent error is an absolute value
(there is no positive or negative value.)
What is the percent error in your
measurement of the age of my cat?
Error
VS
Percent Error
What is the weight of my car?
Experimental Value (EV) = 3,585 kg
Actual Value (AV) = 3,580 kg
Error = EV – AV = 3,585 kg – 3,580 kg =
5 kg
Percent Error = EV - AV x 100%
AV
= 3,585 kg – 3,580 kg x 100 % =
3,580 kg
0.14%
Significant Figures
All measurements are approximations—no measuring device can
give perfect measurements without experimental uncertainty. By
convention, a mass measured to 13.2 g is said to have an absolute
uncertainty of plus or minus 0.1 g and is said to have been
measured to the nearest 0.1 g. In other words, we are somewhat
uncertain about that last digit—it could be a "2"; then again, it
could be a "1" or a "3". A mass of 13.20 g indicates an absolute
uncertainty of plus or minus 0.01 g.
What is a "significant figure"?
The number of significant figures in a
result is simply the number of figures
that are known with some degree of
reliability. The number 13.2 is said to
have 3 significant figures. The number
13.20 is said to have 4 significant
figures.
Rules for deciding the number of significant
figures in a measured quantity:
• (1)All nonzero digits are significant:
1.234 g has 4 significant figures,
1.2 g has 2 significant figures.
(2) Zeroes between nonzero digits are
significant:
1002 kg has 4 significant figures,
3.07 mL has 3 significant figures.
(3) Leading zeros to the left of the first
nonzero digits are not significant; such
zeroes merely indicate the position of the
decimal point:
0.001 oC has only 1 significant figure,
0.012 g has 2 significant figures.
(4) Trailing zeroes that are also to the
right of a decimal point in a number are
significant:
0.0230 mL has 3 significant figures,
0.20 g has 2 significant figures.
(5) When a number ends in zeroes that
are not to the right of a decimal point, the
zeroes are not necessarily significant:
190 miles may be 2 or 3 significant figures,
50,600 calories may be 3, 4, or 5 significant
figures.
The potential ambiguity in the last rule
can be avoided by the use of standard
exponential, or "scientific," notation.
For example, depending on whether the
number of significant figures is 3, 4, or 5,
we would write 50,600 calories as:
5.06 × 104 calories (3 significant
figures) 5.060 × 104 calories (4
significant figures), or 5.0600 × 104
calories (5 significant figures).
Rules for mathematical operations
• In carrying out calculations, the general rule is that
the accuracy of a calculated result is limited by the
least accurate measurement involved in the
calculation.
• (1) In addition and subtraction, the result is
rounded off to the last common digit occurring
furthest to the right in all components. Another
way to state this rule is as follows: in addition
and subtraction, the result is rounded off so
that it has the same number of digits as the
measurement having the fewest decimal places
(counting from left to right).
For example:
• 100 (assume 3 significant figures) + 23.643 (5
significant figures) = 123.643,which should be
rounded to 124 (3 significant figures). Note, however,
that it is possible two numbers have no common
digits (significant figures in the same digit column).
• (2) In multiplication and division, the result should be
rounded off so as to have the same number of
significant figures as in the component with the least
number of significant figures. For example,
• 3.0 (2 significant figures ) × 12.60 (4 significant
figures) = 37.8000which should be rounded to 38 (2
significant figures).
What is an "exact number"?
• Some numbers are exact because they are known with
complete certainty.
• Most exact numbers are integers: exactly 12 inches are in a
foot, there might be exactly 23 students in a class. Exact
numbers are often found as conversion factors or as counts of
objects.
• Exact numbers can be considered to have an infinite number of
significant figures. Thus, the number of apparent significant
figures in any exact number can be ignored as a limiting factor
in determining the number of significant figures in the result of
a calculation.
Rules for rounding off numbers
• (1) If the digit to be dropped is greater than 5, the last
retained digit is increased by one. For example,
12.6 is rounded to 13.
• (2) If the digit to be dropped is less than 5, the last
remaining digit is left as it is. For example,
12.4 is rounded to 12.
(3) If the digit to be dropped is 5, and if any digit
following it is not zero, the last remaining digit is
increased by one.
For example: 12.51 is rounded to 13.
(4) If the digit to be dropped is 5 and is followed only by
zeroes, the last remaining digit is increased by one if it is
odd, but left as it is if even. For example,
11.5 is rounded to 12,
12.5 is rounded to 12. This rule
means that if the digit to be dropped is 5 followed only by
zeroes, the result is always rounded to the even digit. The
rationale for this rule is to avoid bias in rounding: half of
the time we round up, half the time we round down.
Example 1:
• How would you record this measurement?
How many significant digits
would be recorded?
How many significant digits
would be recorded?
Meters, Grams and Liters
The Metric System
• The metric system is a measurement
system based on our decimal (base 10)
number system.
• Other countries and all scientists and
engineers use the metric system for
measurement.
DID YOU KNOW?
It’s a metric
world
The United States is the only
western country not
presently using the metric
system as its primary system
of measurement. The only
other countries in the world
not using metric system as
their primary system of
measurement are Yemen,
Brunei, and a few small
islands; see
Fig. 8.15.
DID YOU KNOW?
In 1906, there was a major effort to convert to the metric system
in the United States, but it was opposed by big business and the
attempt failed.
The Trade Act of 1988 and other legislation declare the metric
system the preferred system of weights and measures of the
U.S. trade and commerce, call for the federal government to
adopt metric specifications, and mandate the Commerce
Department to oversee the program. The conversion is currently
under way; however, the metric system has not become the
system of choice for most Americans’ daily use.
DID YOU KNOW
Lost in space
In September 1999, the United
States lost the Mars Climate
Orbiter as it approached Mars.
The loss of the $125 million
spacecraft was due to scientists
confusing English units and
metric units.
Two spacecraft teams, one at
NASA’s Jet Propulsion Lab (JPL)
in Pasadena, CA, and the other
at a Lockheed Martin facility in
Colorado, where the spacecraft
was built, were unknowingly
exchanging some vital
information in different units.
The missing Mars Climate Orbiter
DID YOU KNOW
Lost in space
On Jan. 3, 1999, NASA launched the $165 million Mars
Polar Lander. All radio contact was lost Dec. 3 as the
spacecraft approached the red planet.
A NASA team that investigated the loss of the Mars Polar Lander
concluded a rocket engine shut off prematurely (due to programming
error) during landing, leaving the spacecraft to plummet about 130 feet
to almost certain destruction on the Martian surface.
Metric Prefixes
• Metric Units
• The metric system has prefix modifiers that are multiples of
10.
Prefix Symbol
Factor Number Factor Word
Kilo-
k
1000
Thousand
HectoDecaUnit
DeciCentiMilli-
h
da or dk
m, l, or g
d
c
m
100
10
1
.1
.01
.001
Hundred
Ten
One
Tenth
Hundredth
thousandth
Place Values of Metric Prefixes
dm
dg
dL
cm
cg
cL
Thousandth
m
g
L
Hundredth
Tenth
dkm
dkg
dkL
One
hm
hg
hL
Ten
Hundred
Thousand
km
kg
kL
mm
mg
mL
Meters
• Meters measure
length or distance
• One millimeter is
about the thickness of
a dime.
Meters
• One centimeter is
about the width of a
large paper clip
•or your fingernail.
Meters
• A meter is about the width of a doorway
Meters
• A kilometer is about six city blocks or 10
football fields.
• 1.6 kilometers is about 1 mile
Gram
• Grams are used to
measure mass or the
weight of an object.
Grams
• A milligram weighs
about as much as a
grain of salt.
Grams
• 1 gram weighs about
as much as a small
paper clip.
• 1 kilogram weighs
about as much as 6
apples or 2 pounds.
Liters
• Liters measure liquids
or capacity.
Liter
• 1 milliliter is about the
amount of one drop
Liter
• 1 liter is half of a 2
liter bottle of Coke or
other soda
Liter
• A kiloliter would be
about 500 2-liter
bottles of pop
Which unit would you use to
measure the length of this bicycle?
km
m
cm
mm
Which unit would you use to
measure the mass of a penny?
km
g
cL
mg
Which unit would you use to
measure the water in an aquarium?
L
m
cL
mg
Which unit would you use to
measure the mass of a feather?
L
m
cL
mg
Which unit would you use to
measure the mass of a student
desk?
kg
g
cL
mg
Which unit would you use to
measure the mass of a whole
watermelon?
kg
g
cL
mg
Which unit would you use to
measure the mass of an egg?
kg
g
mm
mg
Which unit would you use to
measure a can of soup?
kL
L
mm
mL
Which unit would you use to
measure this glass of milk?
kL
L
cL
mL
Which unit would you use to
measure the distance across
Kansas?
km
m
cm
mL
Which unit would you use to
measure the height of a tree?
km
m
cm
mm
Which unit would you use to
measure the length of a bracelet?
km
m
cm
mm
Changing Metric Units
• To change from one unit to another in the
metric system you simply multiply or divide
by a power of 10.
Multiples of Basic Unit
Subdivisions of the Basic Unit
Equivalence Statements vs Conversion
Factors
Fractions in which the numerator and denominator are
EQUAL quantities expressed in different units
Example: 1 in. = 2.54 cm (eq. statement)
Factors:
1 in.
2.54 cm
and
2.54 cm
1 in.
Both conversion factors are equal to 1. This is why they can be
multiplied or divide by anything out there because they do not
change the value of that thing.
How many minutes are in 2.5 hours?
Conversion factor
2.5 hr x
60 min
1 hr
= 150 min
cancel
By using dimensional analysis / factor-label method, the UNITS
ensure that you have the conversion right side up, and the UNITS
are calculated as well as the numbers!
Sample Problem
• You have $7.25 in your pocket in
quarters. How many quarters do you
have?
7.25 dollars X
4 quarters = 29 quarters
1 dollar
Learning Check
How many seconds are in 1.4 days?
Unit plan: days
hr
min
1.4 days x 24 hr 60xmin x 60 s
1 hr
1 min
1 day
seconds
= 1.2 x 105 s
Steps to Problem Solving

Read problem
 Identify data
 Make a unit plan from the initial unit to the
desired unit (good practice at beginning, not
necessary as you get comfortable with this)
 Select conversion factors
 Change initial unit to desired unit
 Cancel units and check
 Do math on calculator
 Give an answer using significant figures
Temperature Scales
Fahrenheit
Water boils
Celsius
Kelvin
_____°F
_____°C
______K
Water freezes _____°F
_____°C
______K
70
Units of Temperature between
Boiling and Freezing
Fahrenheit
Water boils
212°F
180°
Water freezes 32°F
Celsius Kelvin
100°C
100°C
0°C
373 K
100K
273 K
Fahrenheit Formula
180°F
100°C
=
Zero point:
9°F
5°C
=
1.8°F
1°C
0°C = 32°F
°F
= 9/5 T°C + 32
°F
= 1.8 T°C + 32
or