Introductory Concepts
Units, dimensions, and
mathematics for problem solving
Unit Conversion
furlongs
S = 5 x10
fortnight
3
What is the value of S in cm per second?
Conversion Factor: number with 2 different units of measure
Solution requires convenient placement of conversion factors:
1 mile
furlongs
5280 feet 12inches 2.54cm 1 fortnight
S = 5 x10
x 8
x
x
x
x
fortnight furlong
mile
foot
inch
14days
3
1day
1hour
= 83.15 cm
x
x
sec
24hours 3600 sec
Scientific Notation
1. Format : N x 10x where 1≤ N< 10
and x = the location of the decimal place
in the original number
2. Indication of precision is by the number
of values after the decimal place.
3. Mathematics is algebraic:
a. Xa * Xb = Xa+b
b. Xa/ Xb = Xa-b
c. (Xa)b = Xa*b
d. b√Xa = Xa/b
Exponent Arithmetic
Multiplication: Xa * Xb = Xa+b
3.0 x 103 * 4.0 x 105 = (3.0*4.0) x 103+5
= 12 x 108 = 1.2 x 109
Division: Xa/ Xb = Xa-b
0.0008 / 0.016  8.0 x 10-4 / 1.6 x 10-2
8/1.6 = 5 ; 10-4 / 10-2 = 10-4-(-2) = 10-2
 = 5 x 10-2
Dimensional Analysis
Consider the equation:
y = x + a( w + z )
What do we know about the units on these
quantities?
(1) The units on x and y are the same.
(2) The units on w and z are the same.
(3) The units on aw and az are the same as on x and y.
Thus, if y has units of length (cm), x has units of cm. If the units on w
and z are grams, the units on a must be
cm
gram
Dimensional Analysis (Continued)
Consider the equation for the energy change occurring when an object is heated :
∆E = C[T 2 − T 1] + A[T 2 − T 1 ].
2
2
If ΔE has units of joules (J), What are the units of C?
K x (J/K) = J
What are the units on A?
K2 x (J/K2) = J
What can we say about the following equation?
∆E = C[T 2 − T 1 + T 2 − T 1]
2
2
Transcendental Functions
x
Let
Z=A+B (10)
y
How are the units of Z, A, and B related?
A and B units = Z units
How are the units of x and y related?
x units = y units
Let:
Z=A log[x/y] or Z=A ex/y
How are the units on Z and A related?
x units = y units
Z units = A units
If we write,
Z= A + log(
x
What are the units on A and Z?
Z units = A units
How are the units on x and y related?
y
)
Statistical Analysis
Let us assume that a quantity x is measured 6 times with the following results:
Set A
Set B
x1= 2.516
2.913
x2= 2.572
2.386
x3= 2.519
2.301
x4= 2.501
2.811
x5= 2.496
2.459
x6= 2.553
2.286
N
Define: Average Value of x =
Σ x /n
< x >=
i
i=1
< x > =(2.516+2.572+2.519+2.501+2.496+2.553)/6
= 2.526 for Set A
<X>
is also called the “first moment of the distribution”, or
the expectation value.
For Set B, the result is
<x>
=2.526
Spread of a Distribution
Although < x > is the same for both Set A and Set B, it is clear that the two values are
not equally precise. There is a lot more “spread” in the values for Set B. Hence, we say
these values are less precise.
Root Mean Square Deviation from the Mean
A measure of the spread of a distribution is given by σ, where we define σ to be
n
σ=
2
(
x
−
<
x
>
)
∑ i
i =1
[n(n − 1)]
For set A, we have:
σ ={[(2.516-2.526)2 + (2.572-2.526)2 +(2.519-2.526)2 + (2.501-2.526)2 + (2.4962.526)2 + (2.553-2.526)2]/(6)(5)}1/2
0.004708
= ±0.0125
σa =
σb = 0.1100
30
Significant Digits
Multiplication and Division:
The number of significant digits in the answer equals the number in the factor
containing the fewest number of significant digits.
x = (5.2)(11.621) = 60.4292 = 60 ± 1
that is, we expect x to be in range 59-61.
Addition and Subtraction:
1.All digits to the left of the decimal point are significant if all terms have only
significant digits to the left of the decimal point.
Example: 8231. + 1233. + 246. + 2088 =11,798.
5 significant digits in the answer.
2. If any # lacks significant digits to the left of the decimal point, the answer lacks
significance in positions where one or more terms in the sum do not have significant
digits.
8.23 × 10 + 1233. + 246. + 2088 = 11,797.
Example:
but the last digit is not significant:i.e. answer is
3
1.180 × 104
3.
The number of significant digits to the right of the decimal point for the
answer is equal to the number of significant digits to the right of the decimal point in
data with the least significant places.
Example:
9761.20+853.167+1.9832=10616.2702
But there are only 2 significant digits to the right. Hence, we report the answer as
10,616.27 [7 significant digits]
When subtraction occurs, significance can be easily lost:
Example:
116.532-116.531=0.001 (1 significant digit only)
and for:
116.5-116.531= -0.0031,
only the first digit to the right is significant:
answer = 0.0
which contains no significant digits.
The Number of Significant Figures in a Measurement
Depends Upon the Measuring Device
Common SI-English conversion factors
Quantity
Length
English to
SI Equivalent English Equivalent SI Equivalent
SI Unit
1 kilometer(km)
1 meter(m)
1000(103)m
0.62miles(mi)
1 mi = 1.61km
100(102)m
1.094yards(yd)
1 yd = 0.9144m
1000(103)mm
1 centimeter(cm)
0.01(10-2)m
1 kilometer(km)
1000(103)m
Volume 1 cubic meter(m3) 1,000,000(106)
39.37inches(in) 1 foot (ft) = 0.3048m
1 in = 2.54cm
0.3937in
(exactly!)
0.62mi
1 mi. = 5,280 ft.
35.2cubic feet (ft3) 1 ft3 = 0.0283m3
cubic centimeters
1 cubic decimeter
(dm3)
1 cubic
centimeter (cm3)
Mass
1000cm3
0.001 dm3
0.2642 gallon (gal) 1 gal = 3.785 dm3
1.057 quarts (qt) 1 qt = 0.9464 dm3
0.0338 fluid ounce 1 qt = 946.4 cm3
1 fluid ounce = 29.6 cm3
1 kilogram (kg) 1000 grams 2,205 pounds (lb) 1 (lb) = 0.4536 kg
1 gram (g) 1000 milligrams 0.03527 ounce(oz) 1 lb = 453.6 g
1 ounce = 28.35 g
The Freezing and Boiling Points of Water
Temperature Scales and Interconversions
Kelvin ( K ) - The “Absolute temperature scale” begins at
absolute zero and only has positive values.
Celsius ( oC ) - The temperature scale used by science,
formally called centigrade and most commonly used scale around
the world, water freezes at 0oC, and boils at 100oC.
Fahrenheit ( oF ) - Commonly used scale in the U.S. for our
weather reports; water freezes at 32oF,and boils at 212oF.
T (in K) = T (in oC) + 273.15
T (in oC) = T (in K) - 273.15
T (in oF) = 9/5 T (in oC) + 32
T (in oC) = [ T (in oF) - 32 ] 5/9
Converting Units of Temperature
PROBLEM: A child has a body temperature of 38.70C.
(a) If normal body temperature is 98.60F, does the child have a fever?
(b) What is the child’s temperature in kelvins?
PLAN:
We have to convert 0C to 0F to find out if the child has a fever
and we use the 0C to kelvin relationship to find the temperature
in kelvins.
SOLUTION:
(a) Converting from 0C to 0F
9
(38.70C) + 32 = 101.70F
5
(b) Converting from 0C to K
38.70C + 273.15 = 311.8K
SI - Base Units
Physical Quantity
Unit Name
Abbreviation
mass
kilogram
kg
length
meter
m
time
second
s
temperature
kelvin
K
electric current
ampere
A
amount of substance
mole
mol
luminous intensity
candela
cd
Common Decimal Prefixes Used with SI Units
Prefix
Prefix
Symbol
tera
giga
mega
kilo
hecto
deka
----deci
centi
milli
micro
nano
pico
femto
T
G
M
k
h
da
---d
c
m
µ
n
p
f
Number
Word
1,000,000,000,000
trillion
1,000,000,000
billion
1,000,000
million
1,000 thousand
100 hundred
10
ten
1
one
0.1
tenth
0.01 hundredth
0.001 thousandth
0.000001 millionth
0.000000001
billionth
0.000000000001
trillionth
0.000000000000001 quadrillionth
Exponential
Notation
1012
109
106
103
102
101
100
10-1
10-2
10-3
10-6
10-9
10-12
10-15
Determining the Number of Significant Figures
PROBLEM: For each of the following quantities, underline the zeros that are
significant figures(sf), and determine the number of significant
figures in each quantity. For (d) to (f) express each in
exponential notation first.
(a) 0.0030 L
(b) 0.1044 g
(c) 53.069 mL
(d) 0.00004715 m
(e) 57,600. s
(f) 0.0000007160 cm3
PLAN:
Determine the number of sf by counting digits and paying attention
to the placement of zeros.
SOLUTION:
(a) 0.0030 L 2sf
(b) 0.1044 g 4sf
(c) 53.069 mL 5sf
(d) 0.00004715 m
(e) 57,600. s
(f) 0.0000007160 cm3
(d) 4.715x10-5 m 4sf
(e) 5.7600x104 s 5sf
(f) 7.160x10-7 cm3
4sf
Rules for Significant Figures in Answers
1. For addition and subtraction. The answer has the
same number of decimal places as there are in the
measurement with the fewest decimal places.
Example: adding two volumes
83.5 mL
+ 23.28 mL
106.78 mL = 106.8 mL
Example: subtracting two volumes
865.9
mL
- 2.8121 mL
863.0879 mL = 863.1 mL
Rules for Significant Figures in Answers
2. For multiplication and division. The number with the least
certainty limits the certainty of the result. Therefore, the answer
contains the same number of significant figures as there are in the
measurement with the fewest significant
figures.
Multiply the following numbers:
9.2 cm x 6.8 cm x 0.3744 cm = 23.4225 cm3 = 23 cm3
Issues Concerning Significant Figures
Electronic Calculators
be sure to correlate with the problem
FIX function on some calculators
Choice of Measuring Device
graduated cylinder < buret ≤ pipet
Exact Numbers
60 min = 1 hr
numbers with no uncertainty
1000 mg = 1 g
These have as many significant digits as the calculation requires.
Rules for Rounding Off Numbers
1. If the digit removed is more than 5, the preceding number
increases by 1.
5.379 rounds to 5.38 if three significant figures are retained
and to 5.4 if two significant figures are retained.
2. If the digit removed is less than 5, the preceding number is
unchanged.
0.2413 rounds to 0.241 if three significant figures are retained
and to 0.24 if two significant figures are retained.
3.If the digit removed is 5, the preceding number increases by
1 if it is odd and remains unchanged if it is even.
17.75 rounds to 17.8, but 17.65 rounds to 17.6.
If the 5 is followed only by zeros, rule 3 is followed; if the 5 is
followed by nonzeros, rule 1 is followed:
17.6500 rounds to 17.6, but 17.6513 rounds to 17.7
4. Be sure to carry two or more additional significant figures
through a multistep calculation and round off only the final
answer.
Significant Figures and Rounding
PROBLEM: Perform the following calculations and round the answer to the
correct number of significant figures.
1g
(a)
4.80x104 mg
16.3521 cm2 - 1.448 cm2
(b)
11.55 cm3
7.085 cm
PLAN:
1000 mg
In (a) we subtract before we divide; for (b) we are using an exact
number.
SOLUTION:
(a)
16.3521 cm2 - 1.448 cm2
14.904 cm2
=
7.085 cm
7.085 cm
= 2.104 cm2
1g
4.80x104 mg
(b)
48.0 g
1000 mg
11.55 cm3
=
= 4.16 g/ cm3
11.55 cm3
Precision and Accuracy
Errors in Scientific Measurements
Precision Refers to reproducibility or how close the measurements are to each
other.
Accuracy Refers to how close a measurement is to the real value.
Systematic error Values that are either all higher or all lower than the actual value.
Random Error In the absence of systematic error, some values that are higher and
some that are lower than the actual value.
Error Analysis
Precise and
Accurate
Precise and
in-accurate
Lower precision
but average is
accurate
Systematic
error
Random
error
In-precise and
inaccurate