Chapter 2:
A Mathematical Toolkit
Dr. Zalesinsky
2.2 Measurements with
Uncertainties
Sig Figs
2.1 Scientific
Measurement
SI Units
Conversions
Scientific
Notation
Operations
with Sig Figs
2.3
Visualizing
Data
• Graphs
• Relationships
The Metric System
• Length or distance is
measured in meters
• Mass is measured in
grams
• Volume or capacity is
measured in liters
• Time is measured in
seconds
• Energy is measured in
Joules
• Temperature is
measured in Kelvin (not
degrees)
• Quantity is measured in
moles
The Liter (L)
The meter is
slightly
longer than a
yard
The Meter (m)
1 inch = 2.54
cm (this is the
only English
to metric
conversion
you must
know
Gram (g)
A gram’s
mass is
approximately
the mass of 2
paperclips.
Many
medicines are
massed by
their active
ingredients in
grams or
parts of a
gram.
The really small and really large
SCIENTIFIC NOTATION
Scientific Notation
Scientific
notation is the
idea of writing a
very large or
small number as
a power of 10.
Move the
decimal so that
there’s only ONE
non-zero digit in
front of the
decimal and the
number of places
it has been
moved is the
exponent.
Examples to
follow
Scientific Notation Examples
• Change to Scientific Notation:
985,000,000
0.0000674
9.85 x 108
6.74 x 10-5
Change to Standard Notation:
8.27 x 106
9.565x10-2
8,270,000
.09565
Using Scientific Notation with Calculator
Find the EE or EXP
button on your calculator
(not the 10x nor the SCI
buttons)
TI Graphing Calculator
Examples
• Do this calculation: (don’t type in the “x 10
part—use the EE or EXP button)
Multiply:(8.76 x 10-10 )(7.9 x1011) =
692 or 6.92x102
Divide: (9.43x1043)/(7.33x1023) =
(approximate) 1.3 x1020 be careful to
turn your display into correct scientific notation
1.320 is not the same!
Uncertainty in Measurement
SIGNIFICANT FIGURES
English to Metric
Comparisons
•
1.
2.
3.
4.
5.
6.
7.
Which is larger?
Meter or yard
Mile or Kilometer
Gallon or liter
Pound or gram
Quart or liter
Centimeter or inch
Gram or ounce
•
1.
2.
3.
4.
5.
6.
What is the
abbreviation for
each unit?
Meter = _____
Gram = _____
Liter = ______
Second = ______
Joule = _______
Mole = _______
Metric Abbreviations
•
The BASE units for the metric system are
gram, liter, meter, second, Kelvin, Joule and
mole.
•
There are larger and smaller portions of
each of these
•
Their abbreviations come in front of the
unit’s abbreviation (ex. centimeter = cm)
Larger and Smaller Prefixes
• Larger
• 1 billion = GIGA (G)
• 1 million = MEGA
(M)
• 1,000 = kilo (k)
• 100 = hecto (h)
• 10 = deca (D, dk, or
da)
•
•
•
•
•
•
•
Smaller
1trillion = pico (p)
1 billion = nano (n)
1 million = micro (m)
1,000 = milli (m)
100 = centi (c )
10 = deci (d)
Match the abbreviation with the
name
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Cm = ________
mg = ________
ML = ________
Gg = ________
ms = ________
km = _______
mL = _______
kJ = ________
mm= _______
Dg = _______
1. Centimeter
2. Milligram
3. Megaliter
4. Gigagram
5. Microsecond
6. Kilometer
7. Milliliter
8. Kilojoule
9. Millimeter
10. Decagram
Conversions
Kilo
Hecto Deca
UNIT
1.0
Deci
Centi
Milli
100
3.09
2594
73.60
0.000765
0.763
730638
84300
Conversions
G
M k
h
D
unit
d
c
m m
1.0
3.9
1x10-9
7x 106
403
7.62
27
848
626
n
p
Significant Figures
• Read the correct
number of
significant figures
Measure the following using
significant figures
Use Sig. Figs
to find this
measurement
Use the
correct
number of sig.
figs in this
measurement
What digit would be
estimated in using Ruler A?
A.
B.
C.
D.
E.
Ones
Tenths
Hundredths
Thousandths
Tens
What decimal place is
estimated when using Ruler
B?
A. Ones
B. Tenths
C. Hundredths
D. Thousandths
E. Tens
Measure the width of the
rectangle using the correct
number of sig figs.
A. 3. 75 cm
B. 3.6 cm
C. 2. 6 cm
D. 3.60 cm
E. 4.25 cm
Measure the length of the
rectangle using the correct
number of sig figs.
A. 12.55 cm
B. 12. 5 cm
C. 12.0 cm
D. 13. 5 cm
E. 13.50 cm
How many sig figs should be in
the correct measurement of the
length of this rectangle?
A. 2 sig figs
B. 3 sig figs
C. 4 sig figs
D. 5 sig figs
E. 1 sig fig
The width of this rectangle is 0.90
cm. How many significant figures
are in this measurement?
A. 3 sig figs
B. 2 sig figs
C. 1 sig fig
D. infinite sig figs
E. none of the above
Data Table 1.1
Dimension
Length (cm)
Longer side
Width (cm)
Shorter side
W
X
Y
Z
Data Table 1.2
Rectangle
W
X
Y
Z
Measured
Length
# of sig figs
in Length
Measured
Width
Number of
Sig Figs in
Width
Data Table 1.5
Dimension
Length (cm)
Longer side
Width (cm)
Shorter side
W
X
Y
Z
Data Table 1.6
Rectangle
W
X
Y
Z
Measured
Length
# of sig figs
in Length
Measured
Width
Number of
Sig Figs in
Width
Calculating with Uncertainty
CALCULATIONS WITH
SIGNIFICANT FIGURES
Multiplication and Division with
Sig Figs
• The least number of sig figs in the input
is equal to the number of sig figs in the
answer (output).
• Remember all conversion factors and
counted numbers have INFINITE sig
figs!
• Example: 8.03 g x 4.0 cm3/g = ?
Addition and Subtraction with
Sig Figs
• The least number of decimal places in
the input is the same number of
decimal places in the output (answer).
• 12.573 m + 3847.9 m - 378 m = ?
• 3482.473 (unrounded)
• 3482 rounded to the correct number of
decimal places
Calculations with Sig Figs
1. A rectangle has a width of 5.0 cm and
a length of 8.40 cm. What is the area
of this rectangle in cm2?
___________
2. A rectangular prism has the following
measurements: length 8.54 cm, width
7.80cm, and height 10.00 cm. What is
the volume in cm3? ______________
Data Table 1.2
Rectangle
Measured
Length (cm)
# of Sig Figs
in Length
Measured
Width (cm)
# of Sig Figs
in Width
W
12.2
3
10.1
3
X
13.6
3
0.7
1
Y
3.3
2
2.4
2
Z
20.2
3
2.5
2
Data Table 1.3
Rectangle
Justified # of
Sig Fig in Area
Unrounded Area Rounded Area
(cm2)
(cm2)
Data Table 1.6
Rectangle
Measured
Length (cm)
# of Sig Figs
in Length
Measured
Width (cm)
# of Sig Figs
in Width
W
12.19
4
10.09
4
X
13.51
4
.61
2
Y
3.27
3
2.49
3
Z
20.19
4
2.38
3
Data Table 1.7
Rectangle
Justified # of
Sig Fig in Area
Unrounded Area Rounded Area
(cm2)
(cm2)
2.3
VISUALIZING DATA
Linear Relationships
• Dependent (y) and
independent (x)
variables when
graphed form a
straight line
• Slope is positive if
they are DIRECTLY
PROPORTIONAL
• Slope is negative if
they are INVERSELY
PROPORTIONAL
4.5
4
3.5
3
Proportio
nal
Inverse
2.5
2
No
change
1.5
1
0.5
0
1
2
3
4
Nonlinear Relationships
• Quadratic
equations relate
parabolic
relationships
• Some inverse
relationships are
hyperbolic
Practice Problem
• Create a graph
from the data
chart given
• Describe the
relationship
(linear or non)
• If linear, find
slope (in/direct)
Volume of oil
(cc)
Concentration
(ppm)
0
0
30
1.3
40
1.5
48
2.0
49
3.0
51
11.0