Significant Digits and Units
EGR 120 – Introduction to Engineering
Significant Digits and Systems of Units
Reading Assignment
• Read Chapter 6 in Engineering Fundamentals – An Introduction
to Engineering, 5E by Moaveni
• Unit Conversion Tables - Inside front cover and inside back
cover of text (also on the course website)
Homework Assignments
• Homework Assignment #5 – Significant Digits and Units
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Significant Digits and Units EGR 120 – Introduction to Engineering
Significant Digits
The use of significant digits in expressing measured quantities gives us the ability
to indicate the intended degree of precision. A different degree of precision is
implied by 5 gallons versus 5.000 gallons.
Three basic rules for significant digits
1. All non-zero digits are significant (Example: 54.87 has 4 significant digits)
2. Leading zeros are not significant (Example: 0.003 has 1 significant digit)
3. Trailing zeros are significant (except with whole numbers where their
significance may be uncertain) (Example: 2.20 has 3 significant digits)
(Example: 100 has an unclear number of significant digits (1, 2, or 3))
Example: How many significant digits are in each number below?
1. 241.692
2. 10.000
3. 0.000173
4. 0.0440
5. 5250
2
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Significant Digits and Accuracy
When a measurement is recorded, the number should reflect
the accuracy of the measurement. A measurement of 15.3
implies a certain accuracy: that the true value of the
quantity being measured is between 15.25 and 15.35.
Illustration: Consider the three voltmeters shown to the
right. Why would the middle voltmeter read 15.3? It must
be that the actual value of the voltage is closer to 15.3 than
to 15.2 and is also closer to 15.3 than 15.4. Since the
midpoint between 15.2 and 15.3 is 15.25 and the midpoint
between 15.3 and 15.4 is 15.35, it follows that:
The reading 15.3 implies an accuracy of between 15.25
and 15.35.
1
5
.2
1
5
.3
1
5
.4
+
V
_
+
V
_
+
V
_
Example: Determine the accuracy for each measured quantity below.
1. 2.09
2. 4.80
3. 2.500
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Significant Digits and Whole Numbers
Whole numbers can include trailing zeros that may or may not be
significant. For example, the number the zeros in the number 4000 may
or may not be significant. It is unclear.
Examples: How many significant digits do you think are implied by
each of the following?
1) A person states that the outside temperature is 100 degrees.
Probably all 3 digits are intended to be significant as since expressing
temperatures to the nearest degree is common.
2) A person states that his swimming pool holds 10000 gallons of
water.
Probably all digits are not intended to be significant. If all 5 digits were
significant, this would imply an unusual accuracy.
So, how can the confusion over the number of significant digits be
eliminated? By using scientific notation.
Significant Digits and Units
EGR 120 – Introduction to Engineering
Significant Digits and Scientific Notation
Using scientific notation is a sure way to clear up any possible confusion
in the number of significant digits. Trailing zeros are always significant
when scientific notation is used. The number 4000 has an unclear number
of significant digits, but 4.000 x 103 clearly has 4 significant digits and 4.0
x 103 clearly has only 2 significant digits.
Example: Recall the case of the 10,000 gallon swimming pool. The
owner could have used scientific notation to make the accuracy of the
pool’s volume clear. Some possible choices are listed below.
Volume of water
in swimming pool
Number of
significant digits
Accuracy (or range)
1. x 104 gallons
1.0 x 104 gallons
1.00 x 104 gallons
1.000 x 104
1.0000 x 104 gallons
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Significant Digits and Units EGR 120 – Introduction to Engineering
Significant Digits and Percent Error
The number of significant digits used is related to the percent error.
Example: If 3 significant digits are used to express a whole number, then the
smallest whole number that can be expressed is 100 and the largest is 999. Note
that:
100 implies an accuracy of 99.5 - 100.5
999 implies an accuracy of 998.5 - 999.5.
max value - min value
x 100%
nominal value
100.5 - 99.5
Then when the display shows the value of 100 the error is
x 100% = 1%
100
999.5 - 998.5
And when the display shows the value of 999 the error is
x 100%  0.1%
999
If error is defined as error =
So a number expressed using 3
significant digits has a percent error
somewhere between 0.1% and 1% or a
maximum percent error of 1%. A
similar process could be used in other
cases, so complete the table shown.
# of Significant Digits
Max Error
1
2
3
4
5
1%
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Mathematical operations involving significant digits
Measured data is often used in calculations and the number of
significant digits used to express the result depends on the type of
calculations performed. There are 2 basic rules to consider:
Multiplication and Division: Express the result using the fewest
number of significant digits contained in any of the numbers.
Addition and Subtraction: Express the result using the smallest
number of digits to the right of any common(*) decimal point
contained in any of the numbers.
* Note that using scientific notation allows the user to move the
decimal point as desired
(for example: 0.123 x 105 = 1.23 x 104 = 12.3 x 103 , etc)
so be sure to use the same exponent when determining the number
of significant digits to the right of any common decimal point.
Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Express the result below using the proper number of
significant digits.
78
x 87
Example: Express the result below using the proper number of
significant digits.
3.729
x 1.6
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Express the result below using the proper number of
significant digits.
0.0064  28.2 =
Example: Express the result below using the proper number of
significant digits.
823.457
+ 438.9
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Lab Courses – The rules for significant digits are important in lab
courses since you are dealing with measured data.
Example: A student in lab uses a meter to measure voltage, V, and
current, I, as shown below. The student then needs to calculate
power using P = V*I (in watts, W).
P = VI = (11.6V)(1.30A)
P = 15.08 W
What value should the student record?
Discuss the rationale.
Voltage (V)
Current (A)
Power (W)
11.6
1.30
…
…
…
…
…
…
Image: www.dhgate.com
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Express the result below using the proper number of
significant digits.
1.67 x 103
+ 1.9 x 104
Example: Recall the example of the swimming pool that held
10000 gallons. Suppose that they measured the volume carefully
and can express the result with 5 significant digits. Now suppose
that someone spills a cup of water into the pool. What is the new
volume? (1 cup = 0.125 gallon)
10000.
+ 0.125
(Does the answer make sense?)
Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for significant digits - Are they perfect?
No. Using the rules for significant digits is simply an attempt to
express answers with a reasonable degree of accuracy. There are
other methods of error analysis which give better results, using
more complex techniques involving calculus and statistics (such as
worst-case analysis, Monte Carlo analysis, and sensitivity analysis).
Using exact quantities in problems
Error is not introduced until measurement is made. So if a
textbook problem states that a triangle has a height of 10 inches,
you can assume that it is an exact value and the rules for significant
digits are not an issue. If the problem states that a person measured
the height of the triangle to be 10.0 inches, then you can assume that
the quantity uses 3 significant digits.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Dimensions and Systems of Units
Dimension – a term used to describe a physical quantity, such as
length, temperature, time, etc.
Unit – a predetermined reference amount that is used to help us
understand the magnitude of a physical quantity
Dimensions can be described using various units.
Example: Length is a dimension. It can be expressed using a
variety of different units, such as in the example below:
Length = 0.8333 ft = 10 in = 25.4 cm
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Systems of Units
There are two systems of units:
1) SI Units (International System of Units)
2) US Customary Units*
SI Units have some significant advantages over US Customary
Units, but US Customary Units are still used heavily in the US, so
it is important to be very familiar with both sets of units.
* The text also discusses British Gravitational (BG) system (or the Imperial
system). They are very similar to the US Customary system, with some
differences in units for mass and force.
For example, the BG system uses the slug for mass and the US Customary
system uses the pound-mass (lbm). Both the slug and the lbm will be
introduced later.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Advantages of SI Units
The SI system is better than the US Customary system because:
1) The SI system is a decimal system.
The SI system uses a system of prefixes. There is only one unit for
each dimension (such as the meter for length), whereas US units
often have several units for each dimension (such as inches, feet,
yard, miles, etc., for length). Quantities are easily expressed in the
SI system with several different prefixes by simply moving the
decimal point. US Units, on the other hand, require the use of
conversion factors (such as 12 inches = 1 foot).
Example: Express the quantity below using other units (or
prefixes) in its system of units.
A) SI Units:
2500000 mm =
B) US Units:
2500000 in =
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Advantages of SI Units (continued)
2) The SI system is a non-gravitational system.
SI units are based on mass, whereas US units are based on force (or
weight). Mass is simply a quantity of matter and does not depend on
gravity. Force is the acceleration of mass. Recall Newton’s 2nd law of
physics:
F = ma
where a = acceleration
or if a = g = acceleration due to gravity, then force is called weight and
Newton’s law becomes
W = mg
The use of weight in the US system is awkward because weight varies
slightly over the surface of the earth (g is proportional to 1/R2, where R
is the radius of the Earth). Weight varies tremendously when we go
outside of the arena of the Earth. Acceleration due to gravity on the
moon is only about 1/6 of the value on Earth, so a person weighing 180
lb on Earth only weighs about 30 lb on the moon.
Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Suppose you are an engineer working for NASA and
you are providing specifications for a part on the space shuttle.
Which of the following two choices would you make?
A) specify its weight - If its weight is 100 lb on Earth then it
weighs only about 17 lb on the moon and its weight is
approximately zero in space. This is not a good choice!
B) specify its mass - If it has a mass of 100 kg on Earth then its
mass is also 100 kg on the moon or in space. This is a much
better choice!
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Advantages of SI Units (continued)
The SI system is better than the US Customary system because:
3) The SI system is a worldwide system.
The United States is the only major country still using US
Customary Units*. The rest of the world uses SI Units.
Obviously it would be better for the United States to switch to
the system used by everyone else.
* Burma and Liberia also use US Customary Units
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Will the United States ever switch to the SI system?
Efforts in the past
In the early 1970’s the United States attempted to switch to the SI
system. Speed limit signs on interstates were all changed to include
both mph and km/h. Speedometers on cars had to show speeds
using both types of units. Schools began teaching children the SI
system. Unfortunately, special interests such as the building and
manufacturing industries, derailed the efforts. The speed limit signs
were eventually changed back to only mph.
Many engineering textbooks currently
use 50% US units and 50% SI units.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Canada
In the early 1970’s Canada switched
to the SI system. They now use only
the SI system.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Efforts today
There is reason for hope.
• We have much more of a worldwide economy today.
• US Engineering companies often sell products and services to
other countries so they must work in their system of units.
• Many engineering companies report that a significant portion of
their work is now done in SI units.
• In 1992 the federal government began requiring engineering
work done on federal facilities to use SI units. This has had a
positive effect. For example, suppose a local a engineering
company designs a building for the Norfolk Naval base. Not
only is the design done in SI units, but the local subcontractors
that do the plumbing, electrical, HVAC, landscaping, etc., must
all deal with SI specifications.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Federal Agencies required to
use SI Units
The article shown is from the
local newspaper announcing
that the federal government
began requiring work in SI units
in 1992. It wasn’t exactly front
page news, but it will hopefully
bring us closer to someday
switching to the SI system.
Local engineering firms
performing design work on
military bases and contractors
doing construction all work with
SI units.
A10 THE VIRGINIAN-PILOT
THURSDAY OCTOBER 1, 1992
-----------------------------------------------COLORADO SPRINGS, Colo. –
All federal agencies, if possible, are
supposed to begin conducting business today like the rest of the world
does. By the metric system.
In reality, leaving behind a system in which Americans measure
their beer in pints and their binges
in pounds will probably happen at a
glacial pace.
Even Dr. Gary Carver, chief of
the Metric Program Office, a division of the Commerce Department,
realizes the government will have to
move as slowly as an inchworm –
or a centipede.
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The use of two different unit
systems was the cause of the
loss of the Mars Climate
Orbiter in 1998. NASA specified
metric units in the contract.
NASA and other organizations
applied metric units in their
work, but one subcontractor,
Lockheed Martin, provided
thruster performance data to the
team in pound force seconds
instead of newton seconds. The
spacecraft was intended to
orbit Mars at about 150
kilometers (93 mi) altitude, but
incorrect data probably caused it
to descend instead to about 57
kilometers (35 mi), burning up in
the Martian atmosphere.
http://en.wikipedia.org/wiki/Metrication_in_the_United_States
Significant Digits and Units
EGR 120 – Introduction to Engineering
The SI System Units
The SI system is built on 7 precisely defined base units. All other
units (called derived units) are built on these 7 units. Before
examining each unit in detail, a couple of points should be made.
1) The definitions of the units have changed over time. In recent
times attempts have been made to define the units in terms of
quantities in nature that are constant.
2) The units are man-made. They do not occur naturally. Any
advanced civilization would discover the value of  (the ratio of the
circumference to the diameter of a circle), e (the base of the natural
log), or many other constants. However, there is nothing in nature
that would lead them to discover the meter, second, kilogram, or
any other SI unit (or US unit).
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Significant Digits and Units
EGR 120 – Introduction to Engineering
SI Base Units
Quantity
Length
Mass
Time
Electric current
Thermodynamic temperature
Amount of substance
Luminous intensity
Name
meter
kilogram
second
ampere
kelvin
mole
candela
Symbol
m
kg
s
A
K
mol
cd
Suggestion:
Know the
SI Base
Units for
Test #1.
Note that the base unit for mass is the kilogram, not the gram. It is an odd
feature of the SI system that this unit includes a prefix. This sometimes
causes some confusion. Whenever performing calculations in the SI system
involving mass, be sure to express mass in kilograms, not grams. For
example,
Example: Calculate force using F = ma where m = 2 kg and a = 4 m/s2
F = (2 kg)(4 m/s2) = 8 N (not 8 kN)
Similarly, if m = 4 g and a = 3 m/s2
25
F = (4 g)(3 m/s2) = (0.004 kg)(3 m/s2) = 0.012 N = 12 mN (not 12 N)
This table from the
text may help to give
you a feel for each of
the SI base units.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Definitions of SI Base Units
Good news! You do not need
to know these definitions.
Length - The meter is a length equal to the distance traveled by
light in a vacuum during 1/299792458 s. The meter was defined
by the CGPM that met in 1983.
Time - The second is the duration of 9,192,631,770 periods of
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the cesium-133 atom. The
second was adopted by the Thirteenth CGPM in 1967.
Mass - The standard for the unit of mass, the kilogram, is
cylinder of platinum-iridium alloy kept by the International
Bureau of Weights and Measures in France. A duplicate copy is
maintained in the United States. The unit of mass is the only base
unit non-reproducible in a properly equipped lab. *
* A new definition for the kilogram is being developed.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Electric Current - The ampere is a constant current which, if maintained
in two straight parallel conductors of infinite length and of negligible
circular cross sections and placed one meter apart in vacuum, would
produce between these conductors a force equal to 2 x 10-7 newton per
meter of length. The ampere was adopted by the Ninth CGPM in 1948.
Temperature - The kelvin, a unit of thermodynamic temperature, is the
fraction 1/273.16 of the thermodynamic triple point of water. The kelvin
was adopted by the Thirteenth CGPM in 1967.
Amount of substance - The mole is the amount of substance of a system
that contains as many elementary entities as there are atoms in 0.012
kilogram of carbon-12. The mole was defined by the Fourteenth CGPM
in 1971.
Luminous intensity - The base unit candela is the luminous intensity in a
given direction of a source that emits monochromatic radiation of
frequency 540 x 1012 hertz and has a radiant intensity in that direction of
1/683 watts per steradian.
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Proposed redefinition of SI base units in 2018
A committee of the International Committee for Weights and Measures (CIPM) has
proposed revised formal definitions of the SI base units, which are being examined by
the CIPM and which will probably be introduced at the 26th 'General Conference on
Weights and Measures' in 2018. The metric system was originally conceived as a
system of measurement that was derivable from nature. When the metric system was
first introduced in France in 1799 technical problems necessitated the use of artifacts as
the prototype meter and kilogram. In 1960 the meter was redefined in terms of the
wavelength of light from a specified source, making it derivable from nature, but the
kilogram is still being defined by an artifact since its introduction. If the proposed
redefinition is accepted, the metric system (SI) will, for the first time, be wholly derivable
from nature.
The proposal can be summarized as follows:
"There will still be the same seven base units (second, meter, kilogram,
ampere, kelvin, mole, and candela). Of these, the kilogram, ampere, kelvin and
mole will be redefined by choosing exact numerical values for the Planck
constant, the elementary electric charge, the Boltzmann constant, and the
Avogadro constant, respectively. The second, meter and candela are already
defined by physical constants and it is only necessary to edit their present
definitions. The new definitions will improve the SI without changing the size of
any units, thus ensuring continuity with present measurements."[1]
http://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units
Significant Digits and Units
EGR 120 – Introduction to Engineering
Derived Units
There are hundreds of other units in the SI system; however, there
are no more definitions. All other SI units (called derived units) are
built on the 7 base units. Derived units fall into two categories:
1) Derived units that are given new names
Example: Force, F = ma = (1 kg)(1 m/s2) = 1 kg.m/s2 = 1 newton (1 N)
(so the derived unit name of newton was assigned to the kg.m/s2 )
Example: Work = (Force)(distance) = (1 N)(1 m) = 1 N.m = 1 joule (1 J)
(so the derived unit name of joule was assigned to the N.m = kg.m2/s2 )
So the newton and the joule don’t need to be scientifically defined.
They are based on the definitions of the kilogram, meter, and
second.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
2) Derived units that are NOT given new names
Example: Velocity = Distance/Time = (1 m)/(1 s) = 1 m/s
(but no new name was given to the m/s. Why not? Who knows?
Most new unit names are given in honor of a pioneer in the field,
such as Newton, Pascal, and Faraday (unit of farads). Perhaps the
m/s will some day be called the Andretti in honor of the race car
driver!)
Example: Acceleration = Velocity/Time = (1 m/s)/(1 s) = 1 m/s/s = 1 m/s2
(but again no new name was given to the m/s2)
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Note how
each derived
unit is built
from the 7
base units.
Some derived
units are
given new
names and
others are
not.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
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Significant Digits and Units
EGR 120 – Introduction to Engineering
SI Prefixes
Engineers should be familiar with
the SI prefixes shown in the table
to the right since they are
commonly used.
Suggestion: Know
this table for Test #1.
Multiplier Prefix name Symbol
10
+18
exa
E
10
+15
peta
P
10+12
tera
T
10+9
giga
G
10
+6
mega
M
10
+3
kilo
k
10+2
hecto
h
10+1
deka
da
10
-1
deci
d
10
-2
centi
c
10-3
milli
10-6
micro
m
m or u
-9
nano
n
10
-12
pico
p
10
-15
femto
f
atto
a
10
10-18
Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for SI Prefixes
1) Do not use scientific notation with prefixes.
Example: F = 2.75 x 106 N or F = 2.75 MN are acceptable, but F
= 2.75 x 103 kN is usually considered to be poor style.
2) Prefixes are not used with units of temperature.
Example: 4.00 (103) C, not 4.00 kC
1.25 (103) K, not 1.25 kK
Also note that the degree symbol () is not used with K.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for SI Prefixes
3) Choose the correct prefix such that the number lies in the
range 0.1 to 1000.
Example: Express each of the following quantities with the
correct prefix.
A) 6540.0 x 103 N
B) 0.000070 mm
C) 0.0139 x 10-8 g
D) 32752 GN
E) 0.0552 pg
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for SI Prefixes
4) For tables it is generally preferable to use a common prefix
(an exception to rule 3)
Example:
Example:
Poor style (mixed prefixes)
x
0.1
0.2
0.3
0.4
0.5
F
97 N
12.65 kN
2.95 MN
550 kN
80 N
Preferred style (common prefixes)
x
0.1
0.2
0.3
0.4
0.5
F (kN)
0.097
12.65
2950
550
0.08
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for Units Names and Symbols
Just as doctors and lawyers should know proper terminology and
notation within their profession, engineers should know the correct
way to write and use units. Several rules are presented below. You
will probably recognize how some of the rules are ignored in
everyday use.
1) Use proper unit symbols, not abbreviations.
Example: t = 10 s, not t = 10 sec.
(see the table of SI Base Units - the symbol for the unit second is s)
Example:
I = 15 A,
not
I = 15 Amps
(“Amps” is slang)
(I represents electric current)
2) Do not use periods with unit symbols.
Example: W = 10 lb, not
W = 10 lb.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Rules for Units Names and Symbols
3) Never make unit symbols plural.
Example: W = 10 lb, not W = 10 lbs
Example: Distance = x = 15 m, not
x = 15 ms
like milliseconds instead of meters!)
(it looks
Poorly
done!
Nicely
done!
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Significant Digits and Units
EGR 120 – Introduction to Engineering
4) Unit symbols use lower case letters unless they are proper
names and then only the first letter is in upper case.
Examples:
P = 10 kPa (not kPA or kpa)
(Pa is the unit symbol for pressure in pascals
from the proper name Pascal)
f = 10 kHz (not kHZ or khz)
(Hz is the unit symbol for frequency in hertz
from the proper name Hertz)
F = 10 kN (not kn)
(N is the unit symbol for force in newtons
from the proper name Newton)
t = 10 ms (not mS)
(s is the unit symbol for time in seconds – not a proper name)
G = 10 mS
(S is the unit symbol for conductance in
siemens from the proper name Siemen)
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Identify the unit/prefix in each case below. Case
obviously matters!
• ms
• Ms
• mS
• MS
5) Full symbol names can be made plural and should be in all
lower case.
Example: F = 10 N or F = ten newtons (not Newtons)
Example: P = 10 W or P = ten watts (not Watts)
6) Use a space between the number and the unit.
Example: W = 10 lb, not W = 10lb
Significant Digits and Units
EGR 120 – Introduction to Engineering
Performing calculations involving prefixes
The clearest method is:
1) Convert each quantity from using prefixes to using scientific
notation
2) Convert any units to base units if necessary (Note: the base
unit for mass in the kilogram, not the gram.)
2) Perform the calculation using scientific notation
3) Express the final result using a prefix.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Use Newton’s 2nd Law, F = ma, to calculate F in each
case below. Express the result with the correct SI prefix.
A) Find F if m = 6.00 Gg and a = 8.50 cm/s2
Step 1) F = ma = (6.00 Gg)(8.50 cm/s2)
= (6.00 x 109 g)(8.50 x 10-2 m/s2)
Step 2) F = (6.00 x 106 kg)(8.50 x 10-2 m/s2)
Step 3) F = 5.10 x 105 N (note that 1 kg.m/s2 = 1 N)
Step 4) F = 0.510 x 106 N = 0.510 MN
B) Find F if m = 35.0 Pg and a = 625 mm/s2
F = ma = (35.0 Pg)(625 mm/s2)
= (35.0 x 1015 g)(625 x 10-6 m/s2)
= (35.0 x 1012 kg)(625 x 10-6 m/s2)
F = 2.1875 x 1010 N = 21.875 x 109 N = 21.9 GN
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Examples: Use Newton’s 2nd Law, F = ma, to calculate F in each
case below. Express the result with the correct SI prefix.
1. m = 175 mg, a = 255 m/s2
2. m = 875.0 pg, a = 425.5 km/s2
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Examples: Use Newton’s 2nd Law, F = ma, to calculate F in each
case below. Express the result with the correct SI prefix.
3. m = 125 g, a = 7.00 m/s2
4. m = 0.625 Pg, a = 9.25 km/min2
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Significant Digits and Units
EGR 120 – Introduction to Engineering
US Customary Units
The following US Customary Units are three of the key base units:
Quantity
Length
Time
Force
Unit name
feet
second
pound
Unit symbol
ft
s
lb
Unfortunately, US Customary units also rely on numerous other
units.
Examples:
• Length might be expressed in terms of inches, feet, yards, or miles.
• Time might be expressed in terms of seconds, minutes, or hours.
• Force might be expressed in terms of ounces, pounds, or tons.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Mass versus Force
Note that the kilogram, a unit of mass, is a base unit in the SI system,
whereas the pound, a unit of force, is a base unit in the Imperial (BG)
system. Using F = ma it can be seen that the newton, a unit of force, is a
derived unit in the SI system and there is a derived unit for mass in the
Imperial system - called the slug. These relationships are shown below.
Using SI Units
If m = 1 kg and a = 1 m/s2 (all base units),
then
F = ma = (1 kg)(1 m/s2 )
F = 1 kg.m/s2 =
F = 1 newton = 1 N (a derived unit)
Using Imperial (BG) Units
If F = 1 lb and a = 1 ft/s2
(all base units), then
m = F/a = (1 lb)/(1 ft/s2 )
m = 1 slug (a derived unit)
So the slug is the derived unit for mass in the BG system and
1 slug = (1 lb)/(1 ft/s2 ) = 1 lb.s2/ft
lb  s 2
1 slug  1
ft
Also note that lb and lbf (pound-force) are used interchangeably, so:
1 lb  1 lbf  1 lb f
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Calculating Weight
A special case of
F = ma
(Newton’s 2nd Law of Physics)
occurs when a = g = acceleration due to gravity.
F = ma is then often written as W = mg.
Note that g varies slightly over the surface of the area, but the
following average values are commonly used:
g = 32.174  32.2 ft/s2 = acceleration due to gravity (using the
average radius of the Earth)
g = 9.806650 m/s2  9.81 m/s2
It is recommended that Engineering students memorize the
following constants:
g = 32.2 ft/s2 = 9.81 m/s2
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: A man weighs 180 lb. Determine his mass in slugs.
Example: A piece of equipment has a mass of 12.5 slugs.
Determine its weight in lb.
Example: A truck tire has a mass of 85 kg. Determine its weight in
N.
Example: A crate weighs 1.5 kN. Determine its mass in kg.
Significant Digits and Units
EGR 120 – Introduction to Engineering
Alternate unit for mass in the US system: pound-mass (lbm)
• The unit slug is unfamiliar to the general public in the US.
• The US Customary system uses an alternate unit for mass: the
pound-mass (lbm).
• The mass in pounds-mass is equal to the weight in pounds (lb) or
pounds-force (lbf) only when g = 32.2 ft/s2 (exactly).
• If g is not 32.2 ft/s2, then lbm should be converted to an alternate
unit (slug or kg) before performing calculations.
So,
1 lbm = 1 lbf = 1 lb if g = 32.2 ft/s2
Otherwise covert lbm to kg or slugs using:
1 slug = 32.2 lbm
1 lbm = 0.453592 kg
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: A man weighs 180 lb on Earth (assume that g = 32.2
ft/s2). Determine his mass in lbm.
Solution: Since g = 32.2 ft/s2 , his mass is 180 lbm
(This essentially means that he has a mass such that when g = 32.2
ft/s2 his weight will be 180 lbf.)
Example: A man has a mass of 180 lbm and travels to the moon
where g = 5.37 ft/s2. Determine his weight in lbf. (Hint: convert
mass to slugs)
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Other alternate (informal or non-standard) units
The kilogram is the base unit for mass in the SI system.
One kilogram could also be expressed as one kilogram-mass (so 1
kg = 1 kgm).
The kilogram force (kgf) is an informal unit of force equal to the
force exerted on one kilogram of mass by g = 9.80665 m/s2
(exactly). Since the official unit for force in the SI system is N, care
should be taken to convert from kgf to N when doing calculations.
The following conversion factor can be used:
1 kgf = 9.80665 N
Other alternate units could also be defined.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Unit conversions
Engineers must be able to work with a wide variety of units in both
the SI and the US systems. A systematic approach to unit
conversions will help the engineer to avoid errors. Unit conversions
can be accomplished by the following methods:
1) Dimension analysis
2) Calculators
3) Software programs (such as MATLAB, MathCAD, or Excel)
4) Online conversion programs (such as DigitalDutch)
www.digitaldutch.com/unitconverter/
Note: Conversion factors are available on the inside front and back
covers of the text and on the course website.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Dimension Analysis
Dimension analysis is a method of balancing units during the unit
conversion process. When someone is converting from inches to
feet and knows that 12 in = 1 ft, sometimes the person mistakenly
divides by 12 when he or she should have multiplied by 12, or vice
versa. Dimension analysis will eliminate this type of error.
Example: Convert a length
of 3.95 ft to inches.
Example: Convert a length
of 123.5 in to feet.
12 in 
Length  3.95 ft 
 47.4 in

 1 ft 
 1 ft 
Length  123.5 in 
 10.29 ft

12 in 
multiply
by 1
multiply
by 1
Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Convert a velocity V = 47.5 in/s to miles per hour (mi/h
or mph). Use dimension analysis. Use the conversion factors
provided in the textbook.
Example: Convert a moment (or torque) of M = 875 Ncm to inlb.
Use dimension analysis. Use the conversion factors provided in the
textbook.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Dimension Analysis with units raised to powers
Conversion factors sometimes need to be squared or cubed when
converting units raised to powers, such as when working with areas
or volumes.
Example: Convert an area of 6.2500 in2 to mm2. Suppose that the
following conversion factors are available:
1 in2 = 645.16 mm2
1 in = 25.4 mm
Two possible solutions are shown below:
 645.16 mm 2 
2
Area  6.2500 in 

4032.2
mm

2
1
in


2
2
 25.4 mm 
2
Area  6.2500 in 

4032.2
mm

 1 in 
2
Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Convert the pressure P = 32.5 lb/in2 to N/m2. Use
dimension analysis. Use the conversion factors provided in the
textbook.
Example: Convert a flow rate of 500 in3/min to m3/s. Use
dimension analysis. Use the conversion factors provided in the
textbook.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Convert the mass density,  = 1.60 slugs/ft3 to g/cm3.
Use dimension analysis. Use the conversion factors provided in the
textbook.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Temperature Conversions
Recall that the base unit for temperature in the SI system is the
kelvin. The base unit for temperature in the US system is degrees
Fahrenheit.
The following conversion formulas are
available if:
F = temperature in degrees Fahrenheit
C = temperature in degrees Celsius
R = temperature in degrees Rankine
K = temperature in Kelvin*
* Note that “degrees” is not used with Kelvin
9
F  C  32
5
5
C  F - 32 
9
R  F  459.67
9
R K
5
K  C  273.16
Significant Digits and Units
EGR 120 – Introduction to Engineering
Example: Convert the temperature 72F to Celcius, Kelvin, and
Rankine.
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Significant Digits and Units
EGR 120 – Introduction to Engineering
Conversions using online conversion programs
You can easily search
online for a unit
conversion program
such as Digital Dutch
shown below. Online
programs may not
include all possible
dimensions or
combinations of
units. For example,
the flow rate
conversion example
recently covered is
http://www.digitaldutch.com/unitconverter
not easily performed
using Digital Dutch.
Significant Digits and Units
EGR 120 – Introduction to Engineering
Conversions using calculators
• Engineering calculators typically have built in unit conversion
capabilities.
• Some calculators, such as the TI-89 or TI-Nspire CAS, attach an
assigned unit to a number and use it in subsequent calculations,
generating derived or compound units if necessary. For example,
multiplying 10_kg by 20_m/s2 will yield a result of 200_N.
• Check with your instructor before purchasing a calculator.
Powerful calculators that can work with units, vectors, complex
numbers, and symbolic variables will be very useful in later
courses.
• Some good choices for engineering calculators include:
• TI-89
• TI-Nspire CAS
63
• HP Prime Graphing Calculator
Significant Digits and Units
EGR 120 – Introduction to Engineering
Microsoft Equation Editor
• Microsoft Office products include Microsoft Equation 3.0, a
powerful equation editor. It doesn’t perform calculations, but
will neatly display them.
• Select Insert – Object – Microsoft Equation 3.0
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Significant Digits and Units
EGR 120 – Introduction to Engineering
• Select View – Toolbar if the toolbar isn’t displayed
Toolbar
• Enter the desired equation. Use various tools on the toolbar.
• Select File – Exit to return to the document
Velocity  225
ft 12 in   2.54 cm  1 min 
 115.7 cm/s
min  1 ft   1 in   60 s 
See the PowerPoint
presentation
“Equation Editor and
Drawing Tools” on the
course website.