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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.7 §1.7 SciNotatSciNotat
Using UnitsUsing Units
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Review §Review §
Any QUESTIONS About• §1.6 → Exponent Properties
Any QUESTIONS About HomeWork• §1.6 → HW-02
1.6 MTH 55
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Scientific NotationScientific Notation
Scientific notation for a number is an expression of the type
N × 10m
• Where: N is at least 1 but less than 10 (that is, 1 ≤ N < 10),
• N is expressed in decimal notation
• m is an integer.
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Scientific NotationScientific Notation
Scientific notation for a number is an expression of the type
N × 10m
Note that when • m is positive the decimal point moves
right m places in decimal notation
• m is negative, the decimal pointmoves left |m| places.
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Example Example Scientific Notation Scientific Notation Example - Convert to decimal notation:
a) 3.842 106 b) 5.3 10−7
Solution:
a) Since the exponent is positive, the decimal point moves right 6 places.
3.842000 → 3.842106 = 3,842,000
b) Since the exponent is negative, the decimal point moves left 7 places.
0.0000005.3 → 5.310−7 = 0.00000053
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Example Example Scientific Notation Scientific Notation
Write in scientific notation:
a) 94,000 b) 0.0423
Solution a) We need to find m such that 94,000 = 9.4 10m.
This requires moving the decimal point 4 places to the right.
94,000 = 9.4 104
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Example Example Scientific Notation Scientific Notation
Write in scientific notation:
a) 94,000 b) 0.0423
Solution b) To change 0.0423 to 4.23 we move the decimal point 2 places to the left.
0.0423 = 4.23 10–2
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Multiplying and Dividing Using Multiplying and Dividing Using Scientific NotationScientific Notation Products and quotients of numbers
written in scientific notation are found using the rules for exponents.
Example - Simplify: (1.7108)(2.210−5)
Solution (1.7 108)(2.2 10−5)
= (1.7 2.2) (108 10−5)
= 3.74 108 +(−5)
= 3.74 103
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Example Example Divide Divide
Simplify (6.2 10−9) (8.0 108) Solution
(6.210−9) (8.0108) = 9 9
8 8
6.2 10 6.2 10
8.08.0 10 10
170.775 10
1717.75 10 10
187.75 10
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Multiply & Divide SummaryMultiply & Divide Summary
If Multiplying in Scientific Notation, then• MULTIPLY Decimal Numbers
• ADD Exponent Numbers
If Dividing in Scientific Notation, then• DIVIDE Decimal Numbers
• SUBTRACT Exponent Numbers
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Scientific Notation ProcedureScientific Notation Procedure Move the decimal point to the right or left
until you have a number that is greater than or equal to 1, but less than 10.
Count how many places you moved the decimal point. This number will become the absolute value of the exponent.
If you moved the decimal point to the left, the exponent will be positive.
If you moved the decimal point to the right, make the exponent negative.
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LeftLeft↔Right? Top↨Bottom? What???↔Right? Top↨Bottom? What???
When deciding on the SIGN for the Exponent in Scientific Notation
If the Number is ≥10, then theExponent is POSTIVEPOSTIVE
If the Number is <1, then the Exponent is NEGATIVENEGATIVE
If the Number is ≥1 & <10, then the Exponent is ZEROZERO• i.e., NO “x10n” needed
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More ExamplesMore Examples Write in Scientific
Notation:a) 1043
b) 2.5
c) 0.000495
Scientific Notation Solutionsa) 1.043103
b) 2.5100 = 2.51 = 2.5
c) 4.9510−4
a) The decimal is to the right of the 3. Move it LEFT 3 places.
b) This number is already greater than or equal to one and less than 10. Therefore, the decimal does NOT have to be moved and the exponent will be 0
c) Move the decimal RIGHT 4 places.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Mass of an Atom Mass of an Atom As you will learn
when you take CHEM1A & ENGR45 all matter is made of VERY small particles called ATOMS
Atoms are, in turn, composed of SUB-atomic Particles
The Primary SubAtomic Particles and their masses• Protons →
1.6710−27 kg
• Neutrons → 1.6710−27 kg
• Electrons → 9.1110−31 kg
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Example Example Mass of Mass of 107107AgAg
Now take the Metal Silver (Chem Symbol Ag).
The 107Ag atom “Isotope” contains• 47 Protons
• 47 Electrons
• 60 Neutrons
Find the Mass of a 107Ag atom
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Example Example Mass of Mass of 107107AgAg
1. Find Total PROTON Mass
kg 108497Proton 47Proton
kg 10671 2627
..
2. Find Total ELECTRON Mass
kg 102824Electron 47Electron
kg 10119 2931
..
3. Find Total NEUTRON Mass
kg 10002.1Neutron 60Neutron
kg 1067.1 2527
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Bruce Mayer, PE Chabot College Mathematics
Example Example Mass of Mass of 107107AgAg
Add the Total Masses of the all the SubAtomic particles
25-
25-
25-
25-
101.784TOTAL
100.0004282Electron
100.7848Proton
101.002Neutron
(kg) MassParticle
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Chp1 ExtraChp1 Extra
Using Using UnitsUnits
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Bruce Mayer, PE Chabot College Mathematics
Physical QuantitiesPhysical Quantities
Anything that we can “Feel” or “See” or “Sense” can be MEASURED. These Things are PHYSICAL Quantities• e.g.; Time, Temperature, Length, Angle
To “Measure” a physical quantity We need a “Ruler” that describes the “Size” of the Quantity. This “Sizing” leads to the concept of UNITS
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Units IntroductionUnits Introduction People MEASURE quantities through
COMPARISONS with STANDARDS. Every measured quantity has an associated
“UNIT” Which is the NAME of the Standard. Need to define SENSIBLE and PRACTICAL
"units" and "standards" that People everywhere can AGREE upon
Even though there exist an almost INFINITE number of different physical quantities, we need no more than a handful of “BASE” standards.
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SI System of UnitsSI System of Units
Système International d'Unités (International System of Units)
A Completely Consistent Set of Basic Units• Requires NO Conversion
factors– e.g., 18 inches = 1.5 feet
• Defined by UNCHANGING Physical Phenomena– Except for one... http://www.bipm.org/en/si/
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SI Base UnitsSI Base UnitsSI Base Units
Base quantity Name Symbol
length meter m
mass kilogram kg
time second s
electric current ampere A
thermodynamic temperature
kelvin K
amount of substance mole mol
luminous intensity candela cd
All But the kg are defined by Physical Phenomena• Examine the Defs
From this List Observe• Very common Units
– Mass (kg)
– Length (m)
– Time (s)
• Some Not so Common Units– Current (A)
– Temperature (K)
• Some Uncommon units– Substance amt (mol)
– Luminous Int (cd)
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Second DefinedSecond Defined Time
(Second)
The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom• This is the Definition of an
“Atomic” Clock– more than 200 atomic clocks are
located in metrology institutes and observatories in more than 30 countries around the world
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Meter DefinedMeter Defined Length or
Distance (meter)
Laser
1 meter
1/299792458 sphoton
“The path traveled by light in vacuum during a time interval of 1/299792458 of a second.”
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Kilogram DefinedKilogram Defined Mass
(Kilogram)
If The ProtoType Were Cubic, its Edge Length would be About 36.2
mm (1.42”); quite small
a cylinder of PLATINUM-IRIDIUM alloy maintained under vacuum conditions by the International Bureau of Weights and Measures in Paris
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Amp DefinedAmp Defined Electric
Current (ampere)
That constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 x 10−7 Newton per metre of length.• What’s a Newton?→ 1kg-m/(s2)
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Kelvin (Temperature) DefinedKelvin (Temperature) Defined Thermo-
dynamic temperature (Kelvin)
273.16K = 0.0098 °C Room Temperature
(72 °F) is about 295.5 Kelvins
NO “Degree” Sign Used with the Kelvin Unit
The unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
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mole (amt of Substance) Definedmole (amt of Substance) Defined Amount of
Substance (mole)
1 mol = 6.023x1023 entities
• entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.
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Luminous Intensity DefinedLuminous Intensity Defined Light
Brightness (candela)
The are 4 (12.57) Steradians in a sphere• 1 Str = 7.96% of the
Sphere Surface
The luminous intensity, in a given direction, of a source that emits monochromatic radiation (one-color light) of frequency 540 x 1012 Hertz (λ = 555.5 nm) and that has a radiant intensity in that direction of 1/683 watt per steradian
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Units Have EvolvedUnits Have Evolved
Candela Predecessor based on a Flame• Hence the Name
Temperature Based on Freezing points• Water
• Platinum
Second Based on the Sidereal (standard) day
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Units Have EvolvedUnits Have Evolved History of the Meter (or Metre)
• One ten millionth of the distance from the North pole to the equator.
• The distance between two fine lines engraved near the ends of a platinum-iridium bar
• 1 650 763.73 wavelengths of a particular orange-red light emitted by atoms of krypton-86 (86Kr).
• The length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second.
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SI Derived UnitsSI Derived Units The Seven Base Units May be
Arithmetically Combined to Produce “Derived Units”• e.g.:
m/sseconds
meters timeof units
distance of units velocityof Units
Several DerivedUnits have SpecialUsefulness AndGiven their OWNNames
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Some Derived UnitsSome Derived Units
Derived quantity Name Symbol
Expression in terms of
other SI units
Expressionin terms of
SI base units
plane angle radian (a) rad - m·m-1 = 1 (b)
solid angle steradian (a) sr (c) - m2·m-2 = 1 (b)
frequency hertz Hz - s-1
force newton N - m·kg·s-2
pressure, stress pascal Pa N/m2 m-1·kg·s-2
energy, work, quantity of heat
joule J N·m m2·kg·s-2
power, radiant flux
watt W J/s m2·kg·s-3
electric charge, quantity of electricity
coulomb C - s·A
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Some (more) Derived UnitsSome (more) Derived UnitsDerived quantity Name Symbol
Expression in terms of
other SI units
Expressionin terms of
SI base units
electric potential difference,electromotive force
volt V W/A m2·kg·s-3·A-1
capacitance farad F C/V m-2·kg-1·s4·A2
electric resistance
ohm V/A m2·kg·s-3·A-2
electric conductance
siemens S A/V m-2·kg-1·s3·A2
magnetic flux Weber Wb V·s m2·kg·s-2·A-1
magnetic flux density
tesla T Wb/m2 kg·s-2·A-1
inductance henry H Wb/A m2·kg·s-2·A-2
Celsius temperature
degree Celsius °C - K
luminous flux lumen lm cd·sr (c) m2·m-2·cd = cd
illuminance lux lx lm/m2 m2·m-4·cd = m-
2·cd
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Old (and Tired) Unit SetsOld (and Tired) Unit Sets
MKS• Stands for Meter-Kilogram-Second in the
Most Common Units– Predecessor to The SI Units
CGS• Means Centimeter-Gram-Second
– Still Widely Used
IPS, FPM, FPH• Inch-Pound-Sec, Foot-Lb-Min, Ft-Lb-Hour
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American Engineering System, American Engineering System, AES – Still in (declining) UseAES – Still in (declining) UseFundamental Dimension Base Unit
length
mass
force
time
electric charge [Q]
absolute temperature
luminous intensity
amount of substance
foot (ft)
pound (lbm)
pound (lbf)
second (sec)
coulomb (C)
degree Rankine (oR)
candela (cd)
mole (mol)
Some Are the
SAME SI
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Conservation of UnitsConservation of Units
Principle of conservation of units:• Units on the LEFT side of an equation
MUST be the SAME as those on the RIGHT side of an Equation
Then Have Dimensional homogeneity• Needed to Prevent “Apples & Oranges”
Confusion– e.g., I Buy 100 ft of Wire at One Store and 50 m
at another; how much total Wire do I have? It’s NOT “150”
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Unit Conversion by Chain-LinkUnit Conversion by Chain-Link
To Determine the Amount of Wire I have Need to Convert to Consistent (Homogeneous) Units
Start by Thinking About the Definition of “1”• Now Consider a “minute” minute 1 Seconds60
1 mintherefore 1
60 sec
60 secor 1
1 min
Read as “60 Seconds per minute”
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Chain-Link Unit ConversionChain-Link Unit Conversion
Also Units can be Multiplied and Divided in a manner similar to Numbers• This how we get, say, “Square Feet”
– e.g.; Consider an 8ft x 10ft Engineer’s Cubicle in Dilbert-Land. How Much WorkSpace Does the Engineer Have?
2ft 80 ftxft8x10 ft 108ft x rkSpc W
Now Back to the Wire• Want to Know how many FEET of Wire
I have in Total
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Chain-Link Unit Conversion cont.Chain-Link Unit Conversion cont.
Check on the Internet and Find that there are 3.2808ft in one meter• Multiply the 50m by this special Value of 1
feet 164.04meter 1
feet 3.2808meter 50 1meter 50
Can “Cancel” The Units by Division
So then the Total Wire = 100ft + 164ft = 264 ft
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Chain Link ExamplesChain Link Examples A World-Class Sprinter can Run 100m in 10s.
• How Fast is this in MPH?
hr
miles 37.22
hr 1
min 60
min 1
s 60
ft 5280
mile 1
m 1
ft 3.2808
s 10
m 100
Gasoline In Hamburg Germany Costs 1.10 € for one Liter of Regular Unleaded• How Much is this in $ per Gallon
– Find Currency Exchange Rate → $1 = 0.787 €
Gal
$ 29.5
Gal 7.48
ft 1
ft 1
Liter 28.317
€ 0.787
$ 1
liter 1
€ 1.10 3
3
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Some Unit ConversionsSome Unit Conversions
See Also http://www.onlineconversion.com/
1 mile = 5280 feet 1 Btu = 1054.4 Joule 1 hour = 60 min
1 meter = 3.281 feet 1 Watt = 1 Joule/sec 1 min = 60 sec
1 foot = 12 inches 1 HorsePower = 2545 Btu/hr 1 gallon = 3.785 liters
1 yard = 3 feet 1 km = 1000 meters 1 Cubic Foot = 7.4805 gallons
1 lb = 4.448 Newtons 1 furlong = 220 yards 1 Pascal = 1 Newton/m2
1 m2 = 1973.5 Circular inches °F = 1.8x°C +32 1 HorsePower = 550 ft-lb/s
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WhiteBoard WorkWhiteBoard Work
Problems From §1.7 Exercise Set• 64, 66, 68
“Seward’s Folly” ≡ 1868 Purchase of Alaska from RussiaFor $7.2M
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White Board ExampleWhite Board Example
The USA FDA recommends that Adults consume 2200 Calories per Day
• What then is the “Power Rating” of a Grown Human Being?
– Note that are TWO types of “Calories”1. The Amount of Heat Required to Raise the
Temperature of 1 GRAM of water by 1 °C (or 1 Kelvin) Often Called the Gram-CAL; used in the CGS system
2. The Amount of Heat Required to Raise the Temperature of 1 KILOgram of water by 1 °C Often Called the KgCAL; This is what you read
on the side of Food Packaging
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White Board ExamplesWhite Board Examples
A 1966 Dodge Hemi Engine• 426 Cubic Inch V8
– What is the Engine Displacement in Litres?
• Develops 425 HP– What is the Power in Watts?
The 2006 Toyota Prius Hybrid Synergy Drive System has a Torque rating of 400 Newton-Meters (Nm)• What is this Torque in Ft-Lbs?
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All Done for TodayAll Done for Today
More Infoon SilverIsotopes
IsotopeAtomicMass
(g/mol)
Relative Abundanc
e(at %)
107Ag 106.905 51.839
109Ag 108.905 48.161
http://www.webelements.com/webelements/elements/text/Ag/isot.html
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All Done for TodayAll Done for Today
CookingConversions
• 16 tablespoons = 1 cup • 12 tablespoons = 3/4 cup • 10 tablespoons + 2 teaspoons =
2/3 cup • 8 tablespoons = 1/2 cup • 6 tablespoons = 3/8 cup • 5 tablespoons + 1 teaspoon =
1/3 cup • 4 tablespoons = 1/4 cup • 2 tablespoons = 1/8 cup • 1 tablespoon = 1/16 cup • 2 cups = 1 pint • 2 pints = 1 quart • 3 teaspoons = 1 tablespoon • 48 teaspoons = 1 cup
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22