Transcript Chapter 2b A Mathematical Toolkit

Chapter 2b A Mathematical
Toolkit
Measurement
Système Internationale
d̀ Unité́ s/Metric System
Accuracy and Precision
Significant Figures
Visualizing Data/Graphing
The Problem
Area of a rectangle = length x width
We measure:
Length = 14.26 cm Width = 11.70 cm
Punch this into a calculator and we find the area as:
14.26 cm x 11.70 cm = 166.842 cm2


But there is a problem here!
This answer makes it seem like our measurements
were more accurate than they really were.

By expressing the answer this way we
imply that we estimated the thousandths
position, when in fact we were less
precise than that!
Significant Figures
Every measurement has some degree of
uncertainty because the last digit is assumed
to be estimated.
Significant figures (“sig figs”): the digits in a
measurement that are reliable (or precise). The
greater the number of sig figs, the more precise
that measurement is.
A more precise instrument will give more sig
figs in its measurements.
Significant Figures
To help keep track of (and communicate
to others) the precision and accuracy of
our measurements, we use Significant
Figures
 These are the digits in any measurement
that are known with certainty plus one
digit that is uncertain (but usually
assumed to be accurate ± 1)

Rules for Significant Figures
1.
2.
3.
4.
Digits from 1-9 are always significant.
Zeros between two other significant digits
are always significant
One or more additional zeros to the right of
both the decimal place and another
significant digit are significant.
Zeros used solely for spacing the decimal
point (placeholders) are not significant.
Counting Significant Figures
RULE 1. All non-zero digits in a measured
number are significant. Only a zero could
indicate that rounding occurred.
Number of Significant Figures
38.15 cm
5.6 ft
65.6 lb
122.55 m
4
2
___
___
Leading Zeros
RULE 2. Leading zeros in decimal numbers
are NOT significant.
Number of Significant Figures
0.008 mm
1
0.0156 oz
3
0.0042 lb
____
0.000262 mL
____
Sandwiched Zeros
RULE 3. Zeros between nonzero numbers are
significant. (They can not be rounded unless
they are on an end of a number.)
Number of Significant Figures
50.8 mm
3
2001 min
4
0.702 lb
____
0.00405 m
____
Trailing Zeros
RULE 4. Trailing zeros in numbers without
decimals are NOT significant. They are only
serving as place holders.
Number of Significant Figures
25,000 in.
2
200. yr
3
48,600 gal
____
Examples
EXAMPLES
# OF SIG. DIG.
COMMENT
453
3
All non-zero digits are always significant.
5057
4.06
4
3
Zeros between two significant digits are
significant.
5.00
106.00
114.050
3
5
6
Additional zeros to the right of decimal and a
significant digit are significant.
0.007
1
Placeholders are not significant
12000
2
Trailing zeros in numbers with no decimal
point are not significant (= placeholder)
Learning Check
A. Which answers contain 3 significant
figures?
1) 0.4760
2) 0.00476
3) 4760
B. All the zeros are significant in
1) 0.00307
2) 25.300
3) 2.050 x 103
C. 534,675 rounded to 3 significant figures is
1) 535
2) 535,000
3) 5.35 x 105
Learning Check
In which set(s) do both numbers
contain the same number of
significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150,000
Learning Check
State the number of significant figures in
each of the following:
A. 0.030 m
1
2
3
B. 4.050 L
2
3
4
C. 0.0008 g
1
2
4
D. 3.00 m
1
2
3
E. 2,080,000 bees
3
5
7
Practice
How many significant digits in the following?
Number
1.4682
# Significant Digits
5
110256.002
0.000000003
114.00000006
110
9
1
11
2
120600
4
When are digits “significant”?
“PACIFIC”
Decimal point
is PRESENT.
Count digits
from left side,
starting with
the first
nonzero digit.
PACIFIC
PACIFIC
The “Atlantic-Pacific” Rule
40603.23 ft2 = 7 sig figs
0.01586 mL = 4 sig figs
When are digits “significant”?
“ATLANTIC”
Decimal point
is ABSENT.
Count digits
from right
side, starting
with the first
nonzero digit.
3 sig figs = 40600 ft2
1 sig fig = 1000 mL
ATLANTIC
ATLANTIC
Examples



0.00932
Decimal point present → “Pacific” → count digits
from left, starting with first nonzero digit = 3 sig
figs
4035
Decimal point absent → “Atlantic” → count digits
from right, starting with first nonzero digit = 4 sig
figs
27510
Decimal point absent → “Atlantic” → count digits
from right, starting with first nonzero digit = 4 sig
figs
Write the following measurements in
scientific notation, then record the
number of sig figs.
3 sig figs
7.89*102 g
1. 789 g
9.6875*104 mL 5 sig figs
2. 96,875 mL
1.33*10-5 J
3 sig figs
3. 0.0000133 J
8.915 atm
4 sig figs
4. 8.915 atm
-1 °C
2 sig figs
9.4*10
5. 0.94°C
The Problem
Area of a rectangle = length x width
We measure:
Length = 14.26 cm Width = 11.70 cm
Punch this into a calculator and we find the area as:
14.26 cm x 11.70 cm = 166.842 cm2




But there is a problem here!
This answer makes it seem like our measurements were more
accurate than they really were.
By expressing the answer this way we imply that we estimated the
thousandths position, when in fact we were less precise than that!
A much better answer would be that the area is 166.84 cm2
because that keeps the same accuracy as our original
measurements.
Significant Numbers in Calculations
A calculated answer cannot be more
precise than the measuring tool.
 A calculated answer must match the
least precise measurement.
 Significant figures are needed for final
answers from
1) adding or subtracting
2) multiplying or dividing

Multiplication and Division with Significant Digits
The rule for multiplying or dividing significant digits
is that the answer must have only as many significant
digits as the original measurement with the least
number of significant digits.
Our measurements, 14.26 and 11.70 each have four significant digits. Our calculator told us
the answer was 166.842, but we need to round it off. Do we round up or do we round it down?
166.842
166.84
If our original measurements had been 14.26 and 11.7, what happens?
166.842
167
How many significant digits would the answer to each of these have?
Problem
114.6 cm x 2.0004 cm
#Sig. Digits in Result?
4
0.0006 cm x 14.63 cm
1
12.901 cm2 / 6.23 cm
3
Multiplying and Dividing
Round (or add zeros) to the
calculated answer until you
have the same number of
significant figures as the
measurement with the fewest
significant figures.
Learning Check
A. 2.19 X 4.2 =
1) 9
2) 9.2
B.
C.
3) 9.198
4.311 ÷ 0.07 =
1) 61.58
2) 62
2.54 X 0.0028
0.0105 X 0.060
1) 11.3
2) 11
3) 60
=
3) 0.041
Addition and Subtraction with Significant Digits
The rule for adding or subtracting with significant
digits is that the answer must have only as many
digits past the decimal point as the measurement
with the least number of digits past the decimal.
How many significant digits would the answer to each
of these have?
Problem
#Digits Past the Decimal?
114.6g + 2.0004g
1
0.0006g + 14.63g
2
12.901g - 6.23g
2
Adding and Subtracting
The answer has the same number
of decimal places as the
measurement with the fewest
decimal places.
25.2
one decimal place
+ 1.34 two decimal places
26.54
answer 26.5 one decimal place
Learning Check
In each calculation, round the answer to
the correct number of significant
figures.
A. 235.05 + 19.6 + 2.1 =
1) 256.75 2) 256.8
3) 257
B.
58.925 - 18.2
=
1) 40.725 2) 40.73
3) 40.7
Rounding
Round to the
nearest tenth
 6.7512
 6.7777
 6.7499
 6.9521
6.8
6.8
6.7
7.0
After you have determined
to what decimal place
(or how many digits)
your reported answer
must be rounded,
Look at digit following
specified rounding
value. If it is 5 or
greater, then round up.
If not, truncate (cut off
the rest of the
numbers).
Rounding Rules



If the first digit to be dropped is less than 5, that digit and all digits
that follow it are simply dropped. Thus, 62.312 rounded off to three
sig. figures becomes 62.3
If the first digit to be dropped is greater than 5 or a 5 followed by
digits other than 0, the excess digits are all dropped and the last
retained digit is increased in value by one unit. Thus, 62.36 rounded
off to three sig. figures becomes 62.4.
If the first digit to be dropped is a 5 not followed by any other digit
or a 5 followed only by zeros, an odd-even rule applies. If the last
retained digit is odd, that digit is increased in value by one unit after
dropping the 5 and any zeros that follow it. If the last retained digit
is even, its value is not changed, and the 5 and any zeros that follow
are simply dropped. Thus 62,150 and 62.450 rounded to 3 sig. figures
become 62.2 (odd rule) and 62.4 (even rule).
Reading Vernier Calipers
Introduction

These are the main features of a typical
vernier caliper:
Small
jaws (for inside
Depth
gauge
Metric
vernierMetric
scale fixed scale
measurements)
Beam
Jaws (for
English
outside
vernier
scalescale
English
fixed
measurements)
applet
Reading a Caliper: metric
 You
only need to make two
readings: one from the fixed
scale and one from the
vernier portion.
Reading a caliper: metric

Start by obtaining a measurement from
the fixed scale...
This is the fixed scale used
for the metric readings.
Reading a caliper: metric

Use the zero line on the vernier to
locate your position on the fixed scale.
Reading a caliper: metric
So based upon the two readings (one from the
fixed scale, and one from the ruler) the length
must be 63 mm + .50 mm = 63.50 mm
63 mm
+ .50 mm
63.50 mm
2.3 Visualizing Data
A Proper Graph
Plotting Line Graphs





Identify Independent and Dependent Variable.
Independent variable gets plotted on x-axis
(time is usually on x-axis)
Determine range of independent variable
Decide whether origin (0,0) is a valid data point
Spread data as much as possible, use a
consistent scale
Number and label x-axis
Plotting Line Graphs




Repeat previous steps for y-axis, except plotting
the dependent variable
Plot all data points on the graph
Draw “Best fit” line or curve. Line does NOT
have to go through each point, but does have to
approximate the “trend” of the data
Give your graph a title, usually an expression of
Independent vs. dependent variables (
Linear Relationships

Whenever data
results in a straightline graph, it is
referred to as a linear
relationship.
Follows general equation
Y = mx + b
Where m = slope
b = y intercept
m = rise/run or Δy/ Δx
Slope m = 1N/1.5 cm
Y=0
Non-Linear Relationships

Quadratic
relationship

Y = ax2 + bx + c
Y varies as a function of
the square of x
Non-Linear Relationships

Inverse
Relationships


Y = a/x
Y varies as a
function of the
inverse of x
Factor-label method of problem
solving
Conversion Factors
Fractions in which the numerator and
denominator are EQUAL quantities
expressed in different units
Example:
1 in. = 2.54 cm
Factors: 1 in.
2.54 cm
and
2.54 cm
1 in.
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
Write conversion factors that relate
each of the following pairs of units:
1. Liters and mL
2. Hours and minutes
3. Meters and kilometers
Learning Check
A rattlesnake is 2.44 m long. How
long is the snake in cm?
a) 2440 cm
b) 244 cm
c) 24.4 cm
Learning Check
How many seconds are in 1.4 days?
Unit plan: days
hr
1.4 days x 24 hr
1 day
min
x
seconds
??
Solution
Unit plan: days
hr
min
seconds
1.4 day x 24 hr x 60 min x 60 sec
1 day
1 hr
1 min
= 1.2 x 105 sec
Wait a minute!
What is wrong with the following
setup?
1.4 day
x 1 day
24 hr
x
60 min
1 hr
x 60 sec
1 min
English and Metric Conversions
If you know ONE conversion for each type
of measurement, you can convert
anything!
 You must memorize and use these
conversions:
 Mass: 454 grams = 1 pound
 Length:
2.54 cm = 1 inch
 Volume:
0.946 L = 1 quart

Learning Check
An adult human has 4.65 L of blood. How
many gallons of blood is that?
Unit plan: L
qt
gallon
Equalities: 1 quart = 0.946 L
1 gallon = 4 quarts
Your Setup: