Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH55_Lec-03_sec_1-7_SciNot_Units_Rules.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.7 §1.7 SciNotatSciNotat

Using UnitsUsing Units



Review §Review §

Any QUESTIONS About• §1.6 → Exponent Properties

Any QUESTIONS About HomeWork• §1.6 → HW-02

1.6 MTH 55



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.



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.



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



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



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



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



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



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



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.



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



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.



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



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




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




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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Chabot Mathematics

Chp1 ExtraChp1 Extra

Using Using UnitsUnits



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



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.



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/



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)



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



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.”



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



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)



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.



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.



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



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



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.



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



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



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



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



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



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”



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”



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



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



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



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



WhiteBoard WorkWhiteBoard Work

Problems From §1.7 Exercise Set• 64, 66, 68

“Seward’s Folly” ≡ 1868 Purchase of Alaska from RussiaFor $7.2M



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



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?



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



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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Chabot Mathematics

AppendiAppendixx

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srsrsr 22

Documents

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege