B-1
SIGNIFICANT FIGURES
Numbers, and their combinations by means of arithmetic, give us an exact way of speaking about
quantity. There are limits, however, to the accuracy of our measurements, and they, in turn, place limits
on our use of numbers to record our measurements.
Scientific notation enables us to show the limited accuracy of our measurements. For example, instead of
writing the radius of the earth as about 6,370,000 meters, we write it as
meters. Likewise, the
diameter of a hair is about 0.00003 meter, which we write as
meter. In this way of writing
numbers, we show the limited accuracy of our knowledge by omitting all digits about which we have no
m and not
m or
information. Thus, for the earth's radius, when we write
m, we are saying that we are reasonably sure of the third digit but have no idea of the value
of the fourth. The number of digits about which we do feel reasonably sure is called the number of
significant digits. In the example of the hair, we have indicated only one significant digit. This means that
we think 3 is a reasonable value, but we are not at all sure of the next digit (second significant digit). The
simplest way of expressing that reliability is by writing the proper number of significant digits.
To write additional digits that have no meaning is worse than a waste of time. It may mislead the people
who use those digits into believing them. This is particularly important to remember when we use
electronic calculators that display 8 or 20 digits whether they are significant or not. To keep track of the
inherent limitations of our measurements, we modify our ideas of arithmetic slightly so as to make sure
that we do not write meaningless digits in the results of our calculations.
Suppose we make the following time measurements: 27.8 h, 1.324 h, and 0.66 h. Now suppose we want
to find their sum. Paying no attention to significant digits, we might write
27.8 h
1.324 h
0.66 h
29.784 h
What is the meaning of this result? In any number obtained by measurement, all the digits following the
last significant one are unknown; for example, the hundredths and thousandths place in the first
measurement above. These unknown digits are not zero. Clearly, if you add an unknown quantity to a
known quantity, you get an unknown answer. Consequently, the last two digits in the sum above are in
fact unknown. In this case, then, we should round off all our measurements to the nearest tenth so that all
the digits in our answer will be significant. This gives
27.8
h
1.3 h
0.7 h
29.8
h
Since the first measurement is known only to the nearest tenth of an hour, we know the sum only to the
nearest tenth of an hour.
Subtraction of measured quantities works the same way. It makes no sense to subtract known and
unknown quantities. Particular care must be taken in subtracting two numbers of nearly equal magnitudes.
For example, suppose you wish to find the difference in length of two pieces of wire. One you have
measured to be 1.55 meters long and the other 1.57 meters long.
Notice that we do not write the answer as
, since we are somewhat uncertain about each
of the last digits in the original measurements. The difference certainly has only one significant figure,
and we would not be too surprised if the difference was either twice as large or zero, instead of 2 cm.
Subtraction of nearly equal quantities destroys accuracy. For this reason, you sometimes need
measurements which are much more accurate than the answers you want. To avoid the experimental
difficulties involved in making extremely accurate measurements, we would do well to put the two wires
side by side, if possible, and measure the difference directly with a micrometer screw rather than use the
difference between two large numbers.
Now what about multiplication? How do we modify it to take account of the limitations of measurement?
Suppose we wish to find the area of a long strip of tin. With a meter stick we measure its width to be 1.15
cm and its length to be 2.002 m. Here we have three-significant-figure accuracy in our width
measurement and four-significant-figure accuracy in our length measurement. To get the area, we
multiply length by width. If we pay no attention to significant figures, we get
But now think of the meaning of this answer. When we measured the width, we wrote 1.15 cm because
we were not sure that the real width might not be a bit bigger or a bit smaller by perhaps 0.01 cm. If in
fact the width is that much bigger, we have made a mistake in the area by the product of this extra width
times the length; that is,
Thus, we see that we have an uncertain number in the hundredths place, which means that our original
evaluation of the area may already be in error in the third significant figure. All the figures we write
, for
beyond the third have no significance. The proper way to express the answer is
when two numbers are multiplied together, their product cannot have more accuracy than the less accurate
of the two factors.
What has been said about multiplication applies equally well to division. Never carry a division out
beyond the number of significant figures in the least accurate measurement you are using.
It should be noted that numbers that are not the result of measurement may have unlimited accuracy and
may be taken to any degree of accuracy required by the nature of the problem. For example, if an area
was measured and found to be
, twice that area would be
.
The use of significant digits gives only a rough, though useful, guide to accuracy. A more reliable method
for expressing experimental errors is to state relative errors
percent.
in
EXERCISES
1. How many significant figures are there in each of the following measurements?
a)
mm km d)
b)
m e)
f)
cm c)
2. The diameter of a circle is 4.24 m. What is its area?
3. A stick has a length of 12.132 cm, a second stick a length of 12.4 cm.
a) If the two sticks are placed end to end, what is their total length?
b) If the two sticks in the last question are placed side by side, what is the difference in their lengths?
4. A student measures a block of wood and records the following results: length = 6.3 cm,
width = 12.1 cm, and height = 0.84 cm.
a) What is the volume of this block?
b) Assume the length and width measurements to be correct; however, you can see that the height
measurement may be off by 0.01 cm, either way. How would this change your answer for the
volume?
c) What fraction is this of the total volume?
5. A bus driver clocked the following times for portions of his route:
Station A to Station B
Station B to Station C
Station C to Station D
Station D to Station E
1.63 h
4.7 h
0.755 h
2.00 h
a) How long did it take him to drive from Station A to Station E?
b) What part of the whole traveling time does the time between Stations B and D represent?
c) The time to go from Station A to Station C is how much more than the time to go from Station C
to Station E?
B-2a
SCIENTIFIC NOTATION
Scientific notation, also sometimes known as standard form or as exponential notation, is a way of
writing numbers that accommodates values too large or small to be conveniently written in standard
decimal notation. Scientific notation has a number of useful properties and is often favored by scientists,
mathematicians and engineers, who work with such numbers. For example, the mass of an electron is
kg, and the mass of the earth is
kg. Such numbers in ordinary decimal form are clumsy to write
and to make calculations with, and it is hard to appreciate their precise magnitudes because of the sea of
zeros.
A better method for expressing numbers makes use of powers-of-ten notation. This method is based on
the fact that all numbers may be represented by a number between 1 and 10 multiplied by a power of 10.
In powers-of-ten notation the mass of an electron is written simply as
kg and the mass of
kg.
the earth is written as
In scientific notation, numbers are written in the form:
("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real
number. If the number is negative then a minus sign precedes the first of the decimal digits of a (as in
ordinary decimal notation). For example:
Ordinary decimal notation Scientific notation
Engineering Notation is a version of scientific notation in which the power of ten must be a multiple of
three (i.e, they are powers of a thousand, but written as, for example, 106 instead of 1,0002). As an
alternative to writing powers of 10, SI prefixes can be used, which also usually provide steps of a factor
of a thousand.
Example: In Scientific Notation the Earth is
miles from the sun, in Engineering Notation this
miles or 93 million miles.
would be written
Another example: the speed of light (defined as 299,792,458 m/s) is expressed in Scientific Notation as
3.00 × 108 m/s or 3.00 × 105 km/s , in Engineering Notation it is expressed as 300 × 106 m/s (300 million
m/s) , or 300 × 103 km/s (300 thousand km/s).
Table A-1: Powers of 10 from 10-10 to 1010.
Evidently positive powers of 10 (which cover numbers greater than 1) follow this pattern:
100
= 1.
=1
= 1 with decimal point moved 0 places,
101
= 1.0.
›
= 1.00.
››
= 1.000.
›››
= 1.000 0.
››››
= 1.000 00.
›››››
=10
= 1 with decimal point moved 1 place to the right,
=100
= 1 with decimal point moved 2 places to the right,
= 1,000
= 1 with decimal point moved 3 places to the right,
=10,000
= 1 with decimal point moved 4 places to the right,
=100,000
= 1 with decimal point moved 5 places to the right, and so on.
102
103
104
105
The exponent of the 10 indicates how many places the decimal point is moved to the right from 1.000 • • •
A similar pattern is followed by negative powers of 10, whose values always lie between 0 and 1:
100
= 1.
=1
= 1 with decimal point moved 0 places,
101
= 0.1.
‹
= 0.01.
‹‹
= 0.001.
‹‹‹
= 0.0 001.
‹‹‹‹
= 0.00.001.
‹‹‹‹‹
=0.1
=1 with decimal point moved 1 place to the left,
= 0.01
= 1 with decimal point moved 2 places to the left,
= 0.001
= 1 with decimal point moved 3 places to the left,
=0.000 1
= 1 with decimal point moved 4places to the left,
=0.000 01
= 1 with decimal point moved 5 places to the left, and so on.
102
103
104
105
The exponent of the 10 now indicates how many places the decimal point is moved to the left from 1.
Here are a few examples of powers-of-ten notation:
EXERCISES
6. Express the following numbers in decimal notation.
a)
d)
b)
e)
c)
f)
7. Express the following numbers in powers-of-ten notation.
(d) 0.007,890
(a) 70
(e) 3.81
(b) 1,000,000
(f) 351,600
(c) 0.14
B-2b
g)
h)
(g) 8400
(h) 31,415,192.654
USING SCIENTIFIC NOTATION: POWERS OF TEN
Let us see how to make calculations using numbers written in powers-of-ten notation. To add or subtract
numbers written in powers-of-ten notation, they must be expressed in terms of the same power of ten.
To multiply powers of ten together, add their exponents:
Be sure to take the sign of each exponent into account.
To multiply numbers written in powers-of-ten notation, multiply the decimal parts of the numbers together
and add the exponents to find the power of ten of the product:
If necessary, rewrite the result so the decimal part is a number between 1 and 10.
To divide one power of ten by another, subtract the exponent of the denominator from the exponent of the
numerator:
Be sure to take the sign of each exponent into account.
To divide a number written in powers-of-ten notation by another number written that way, divide the
decimal parts of the numbers in the usual way and use the above rule to find the exponent of the power of
ten of the quotient:
If necessary, rewrite the result so the decimal part is a number between 1 and 10:
To find the reciprocal of a power of ten, change the sign of the exponent:
Hence the prescription for finding the reciprocal of a number written in powers-of-ten notation is
For example,
The powers-of-ten method of writing large and small numbers makes arithmetic involving such numbers
relatively easy to carry out. Here is a calculation that would be very tedious if each number were kept in
decimal form.
EXERCISES
8. Perform the following additions and subtractions.
a
e
f)
h
b
g
c
d
9. Evaluate the following reciprocals.
a)
c)
b)
d)
10. Perform the following calculations.
a)
b)
e)
c)
d)
f)
g)
h)
B-3
FACTOR-LABEL (UNIT) CONVERSION
The factor-label method is the sequential application of conversion factors expressed as fractions and
arranged so that any dimensional unit appearing in both the numerator and denominator of any of the
fractions can be cancelled out until only the desired set of dimensional units is obtained. For example, 60
miles per hour can be converted to feet per second by using a sequence of conversion factors as shown
below:
First, since
,
.
So
Second, since
,
.
So, now
Third, since
,
.
And finally,
.
And so,
Usually all steps are done a the same time. For example, 10 miles per hour can be converted to meters per
second by using a sequence of conversion factors as shown below:
As can be seen, when the units mile and hour are cancelled out and the arithmetic is done, 10 miles per
hour converts to 4.47 meters per second.
As a more complex example, the concentration of nitrogen oxides (i.e., NOx) in the flue gas from an
industrial furnace can be converted to a mass flow rate expressed in grams per hour (i.e., g/h) of NOx by
using the following information as shown below:
NOx concentration
= 10 parts per million by volume = 10 ppmv = 10 volumes/106 volumes
NOx molar mass
= 46 kg/kgmol (sometimes also expressed as 46 kg/kmol)
Flow rate of flue gas
= 20 cubic meters per minute = 20 m³/min
The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kgmol.
After cancelling out any dimensional units that appear both in the numerators and denominators of the
fractions in the above equation, the NOx concentration of 10 ppmv converts to mass flow rate of 24.63
grams per hour.
Checking equations that involve dimensions
The factor-label method can also be used on any mathematical equation to check whether or not the
dimensional units on the left hand side of the equation are the same as the dimensional units on the right
hand side of the equation. Having the same units on both sides of an equation does not guarantee that the
equation is correct, but having different units on the two sides of an equation does guarantee that the
equation is wrong.
For example, check the Universal Gas Law equation of
•
•
•
•
•
, when:
the pressure P is in pascals (Pa)
the volume V is in cubic meters (m³)
the amount of substance n is in moles (mol)
the universal gas law constant R is 8.3145 Pa·m³/(mol·K)
the temperature T is in kelvins (K)
As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's
right hand side are cancelled out, both sides of the equation have the same dimensional units.
Limitations
The factor-label method can convert only unit quantities for which the units are in a linear relationship
intersecting at 0; i.e. when the magnitude of the quantity to be converted in terms of the initial units is
zero it must also be zero in terms of the final units. Most units fit this paradigm. An example for which it
cannot be used is the conversion between degrees Celsius and Kelvins. Between degrees Celsius and
Kelvins, there is a constant difference rather than a constant ratio. Instead of multiplying the given
quantity by a single conversion factor to obtain the converted quantity, it is more logical to think of the
original quantity being divided by its unit, being added or subtracted by the constant difference, and the
entire operation being multiplied by the new unit.
Since
we cannot do a straight unit conversion. We must need to know the difference, since
the difference is 32.
The unit conversion is based on the number of degrees between the freezing and boiling points in these
two systems;
To convert
and
. And so
and
.
one must first do a unit conversion and then add the difference,
.
To convert
one must first subtract the difference and then do a unit conversion,
.
Including the units makes these equations difficult to understand and therefore they are usually left out.
This results in the more common:
and
EXERCISES
11. On a separate sheet of paper, convert each of the following measurements to its equivalent, as ind
a)
60.0 mi/hr → km/hr
k)
60.0 mi/hr → ft/s
b)
6.14 g/cm3 → mg/mL
l)
60.0 mi/hr → m/s
c)
6.6 ft/s → cm/s
m)
60.0 mi/hr → mi/min
d)
283 L/s → gal/min
n)
108 km/hr → m/s
e)
486 km/hr → m/min
o)
108 km/hr → ft/s
f)
1.02 cm/day → m/yr
p)
108 km/hr → m/min
g)
44 ft/s → mi/hr
q)
1.86 x 105 mi/s → mi/hr
h)
0.432 kg/L → lb/ft3
r)
1.8 x 10-4 mm/hr → in/yr
i)
95.28 cm/min → in/s
s)
3.0 x 108 m/s → km/hr
j)
30 mile/gallon → mile/liter
t)
3.0 x 108 m/s → mi/hr
On a separate sheet of paper, determine the unit factors for each of the following conversions.
12.
a)
1 km/hr → m/s
f)
1 mile/hr → ft/s
b)
1 mile/hr → m/s
g)
1 lb/ft → kg/m
c)
1 mile/gallon → ft/cup
h)
1 lb/ft2 → kg/m
d)
1 ft3/min→ cm /sec
i)
1 lb/ft3 → kg/L
e)
1 mile/gallon → km/liter
j)
1 Dollar/gallon→Euro/liter
3
2
B-4
INTERNATIONAL SYSTEM of UNITS (SI)
The International System of Units (abbreviated SI from the French Le Système International d'Unités)
is the modern form of the metric system and is generally a system devised around the convenience of the
number 10. It is the world's most widely used system of units, both in everyday commerce and in science.
An extensive presentation of the SI units is maintained on line by NIST, including a diagram of the
interrelations between the derived units based upon the SI units. Definitions of the basic units can be
found on this site, as well as the CODATA report listing values for special constants such as the electric
constant, the magnetic constant and the speed of light, all of which have defined values as a result of the
definition of the meter and ampere. The older metric system included several groups of units. The SI was
developed in 1960 from the old meter-kilogram-second (mks) system, rather than the centimeter-gramsecond (cgs) system, which, in turn, had a few variants. Because the SI is not static units are created and
definitions are modified through international agreement among many nations as the technology of
measurement progresses, and as the precision of measurements improves.
The system is nearly universally employed, and most countries do not even maintain official definitions
of any other units. A notable exception is the United States, which still uses many old units in addition to
SI. The other two exceptions are Liberia and Myanmar. In the United Kingdom, conversion to metric
units is government policy, but the transition is not yet complete. Those countries that still recognize nonSI units (e.g. the US and UK) have redefined their traditional non-SI units in terms of SI units.
SI Base Units
SI base unit
Base quantity
Name
meter
kilogram
second
ampere
kelvin
mole
candela
length
mass
time
electric current
thermodynamic temperature
amount of substance
luminous intensity
Symbol
m
kg
s
A
K
mol
cd
SI Prefixes
Factor
Name
Symbol
Factor
Name
Symbol
10
24
yotta
Y
10
-1
deci
d
10
21
zetta
Z
10
-2
centi
c
10
18
exa
E
10
-3
milli
m
10
15
peta
P
10
-6
micro
µ
10
12
tera
T
10
-9
nano
n
10
9
giga
G
10
-12
pico
p
10
6
mega
M
10
-15
femto
f
10
3
kilo
k
10
-18
atto
a
2
hecto
deka
h
da
10
-24
10
-21
zepto
yocto
z
y
10
1
10
1 ångström (Å) = 10–10 meters
EXERCISES
On a separate sheet of paper , convert each of the following measurements to the unit indicated.
13.
a)
100 cm → m
k)
0.00467 kg → cg
b)
500 g → kg
l)
6.054 cm → hm
c)
0.01 km → cm
m)
83.59 Å → cm
d)
250 mL → L
n)
9.45 μm → mm
e)
35 kg → g
o)
10.4 cm → mm
f)
0.89 L → mL
p)
34.2 mm → cm
g)
3.484 cm → mm
q)
192.3 cg → mg
h)
15.93 mg → dg
r)
4.006 L → mL
i)
435.6 ms → s
s)
70.5 cm3 → mL
j)
89.05 mL → dL
t)
2.58 μg → cg
B-5
FORMULAS
Triangles:
Perimeter:
Semi-perimeter:
Area:
Hero’s Formula:
Law of Sines:
Law of Cosines:
Rectangle (squares):
Cube:
Perimeter:
Volume:
Area:
Surface Area:
Trapezoid:
Area:
Circle:
Sphere:
h
Volume:
Surface Area:
Circumference:
Area:
Cones & Pyramids:
Volume:
EXERCISES
B-6
PROPERTIES of MATTER
Density:
Linear Density: λ (lambda) Mass per Unit Length. The density of rope, steel cable, etc. might be
reported in this fashion.
As an example consider the following: A steel cable capable of supporting 1000 kg has a linear
density of 2 kg/m. The cable is to be used to lift ore from a mine. It must lift the ore 100 m in a basket
with a mass of 100 kg. What is the largest load this system could lift?
Solution: 100 m of cable (with a linear density of 2 kg/m) would have a mass of 200 kg. This
plus 100 kg basket means that just lifting the basket and cable requires 300 kg of the cables 1000 kg
rating. The cable must not only lift the load but also all of the cable above it! That leaves 700 kg of the
cable’s capability for the ore.
If instead of the linear density we were given the volume density and the cable’s diameter we
could still solve this problem, but it would not be as easy. The linear density offers a simple approach to
such problems.
Surface Density: σ (sigma) Mass per Unit Area. The density of sheet metal, floor tiles, road
surface, etc. might be reported in this fashion.
Volume Density: ρ (rho) Mass per Unit Volume. This is our friend, the “goode olde” regular
density!
EXERCISES
B-7
THE GREEK ALPHABET
χυστ φoρ φυν:
Capital
Lower Case
Greek Name
English
Α
α
Alpha
a
Β
β
Beta
b
Γ
γ
Gamma
g
∆
δ
Delta
d
Ε
ε
Epsilon
e
Ζ
ζ
Zeta
z
Η
η
Eta
h
Θ
θ
Theta
th
Ι
ι
Iota
i
Κ
κ
Kappa
k
Λ
λ
Lambda
l
Μ
µ
Mu
m
Ν
ν
Nu
n
Ξ
ξ
Xi
x
Ο
ο
Omicron
o
Π
π
Pi
p
Ρ
ρ
Rho
r
Σ
σ, ς
Sigma
s
Τ
τ
Tau
t
Υ
υ
Upsilon
u
Φ
φ
Phi
ph
Χ
χ
Chi
ch
Ψ
ψ
Psi
ps
Ω
ω
Omega
o
1) Ρ, ρ, ρ your boat
2) Φone
3) Χicken