Transcript Exponent Notation_02sep13

Math a Challenge?
• Don’t Blame Nature!
– Nature establishes it’s own rules
• No need for calculations of any kind
– Humans create models to understand nature
• We developed math models as analogies
• We’re the ones with 10 fingers
– But not all people or animals have 10
– Digital Computers work with powers of 2
Origin of formulas
• Mathematicians create math models
– Geometry, trigonometry, algebra
– No objects required, a thought process
• Physicists utilize the math models
– Usually not materials oriented
– Newton’s Laws of motion, energy, velocity
– Are we running out of useful math models?
• Brian Green says so, proposes “string theory”
• Chemists apply the models to materials
– Gas laws, temperature, reactions
– Use models to explain how & why of materials
Powers of 10 is arbitrary
• We find it convenient to count with fingers
– 10 is our “base” number
– Counting is 0,1,2,3,4,5,6,7,8,9...10,11
• Dogs & cats have 8 fingers/toes on front paws
– 8 would be their base number
– Cat counting is 0,1,2,3,4,5,6,7…10,11.
• Horses have 2 hooves in front
– 2 would be their base number
– Horse counting would be 0,1…10,11,100, 101, …
– Computer counting is based on powers of 2
• Horses would find computer math natural
12 finger math?
Take-away messages
• Don’t be intimidated by the math
– It’s just a way of explaining things
– WE created the system, not nature
• No square roots, logs, or imaginary #’s in nature
– Models are analogies, and analogies fail
• Most models don’t work in all situations
– Newton’s laws fail for the very large and very small
– Some are probably too complex to be correct (Strings)
• No “theory of everything” exists (yet)
• Use the simplest model which solves the problem
– Minimize complexity, remember “it’s just a model”
English system a mess!
• Length based on a King’s foot
– What happens when we change Kings? (save the foot!)
– The King’s foot might change with age …
– Definition is arbitrary, but now standardized
• Mass depended on natural objects (e.g. grains of wheat)
– Inconsistent by location, time, plant variety, humidity …
• Nonsensical multiples evolved over time
– 4 quarts/gallon, 32 ounces/quart,
– 6 feet per fathom
– 12 eggs per dozen (13 donuts in baker’s dozen)
– 42 gallons per barrel of oil
– 12 inch/foot, 3 feet/yard, 5280 ft/mile, leagues, furlongs …
– 7000 grains/pound, 14 pounds/stone
– 20 schillings per currency “pound”,
– 144 items per gross (a dozen dozen)
• France attacked the problem
– Defined new measurements (no plants or people)
– Based values using powers of 10, became the “metric” system
SI or “metric” system of units
(SI = System International)
• Employ a Decimal System, of powers of 10
– Defined kilometer, meter, centimeter, millimeter, nanometer
• Replacing feet, fathoms, knots, cubits, furlongs, etc.
• Volume defined as 1 liter = 10 x 10 x 10 cm = 1000cm^3
– Kilogram, gram, metric ton (1000 Kg)
• Replacing pounds, stones, grains, ounces, drams
• Related to water (1 liter = 1000 cm^3 = 1 kilogram)
– Second, millisecond, microsecond
• Preserved historical units, impractical to change all clocks
• Tied old units to more precise standards
Basic CGS metric scheme
Preceded SI / ISO system of units (cm vs meter)
1 cm^3 = 1 milliliter = 1 gram H2O
Why use Exponents?
• Huge range of values in nature
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299,792,458 meters/sec speed of light
602,214,200,000,000,000,000,000 atoms/mole
0.000000625 meters is wavelength of red light
0.0000000000000000001602 electron charge
• Much simpler to utilize powers of 10
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3.00*108 meters/sec speed of light
6.02*1023 atoms/mole
6.25*10-7 meters for wavelength red light
1.60*10-19 Coulombs for electron’s charge
Parts of a Value
Setup of a scientific number
this is Avogadro’s number, atoms in a mole
Exponent Conventions
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1000 = 103 exponent as a superscript
1000 = 10E3 used in Excel, “E” means exponent
1000 = 10^3 also in Excel, “carat ^” is exponent
1000 = 10exp3 used by some calculators
“EE” key used on TI-30XII
5 EE 3 yields 5,000 (EE is 2nd function)
• 100 = 102 =10E2 =10^2 all mean the same
• 10 = 101 =10E1 = 10^1 all the same
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1 = 100 = 10E0 =10^0, by definition
– Anything raised to zero power is one
Negative Exponents are handy for
very small numbers
Decimal vs Scientific
“normalized” refers to small number of leading digits
Exponential Notation
• Scientific Notation
– Powers of 10
• Applications
– Measuring mass of atoms versus stars
– Length of viruses versus interstellar travel (light year)
– Volume of cells versus oceans (cubic miles)
• Measurement systems
– English is current system in USA
• One of last countries to use it
– Rest of world is Metric, using exponents
• We’re getting there slowly (2 liter sodas, 750mL wine)
People like small numbers
• Tend to think in 3’s
– good, better, best (Sears appliances)
– Small, medium, large (T-shirts, coffee serving)
• 1-3 digit numbers easier to remember
– Temperature, weight, volume
– Modifiers turn big back into small numbers
• 2000 lb  1 ton, 5280 feet  1 mile
• Kilograms, Megabytes, Gigahertz, picoliters (ink jet)
Exponential Notation
• Notation method
– Single digit (typically) before decimal point
– Significant digits (2-3 typical) after decimal
– Power of 10 after the significant digits
• More Examples
– 1,234 = 1.234 x 103 = 1.234E3 (Excel)
– 0.0001234 = 1.234 x 10-4 = 1.234E-4
• 6-7/8 inch hat size, in decimal notation
– 6+7/8 = 6+0.875 = 6.875 inch decimal equivalent
– 6.875, also OK is 0.6875E1 = 6.875E0 = 68.75E-1
Exponential Notation
• 3100 x 210 = 651,000
• In Scientific Notation: 3.100E3 x 2.10E2
• Coefficients handled as usual numbers
– 3. 100 x 2.10  6.51 with 3 significant digits
• Exponents add when values multiplied
– E3 (1,000) * E2 (100) = E5 (100,000)
– Asterisk (*) indicates multiplication in Excel
• Final answer is 6.51E5 = 6.51*10^5
– NO ambiguity of result or accuracy
Exponential Math
• Exponents subtract in division
– E3 (1,000) / E2 (100) = E1 (10)
– Forward slash (/) indicates division
• Computers multiply & divide FIRST
– Example 1+2*3= 7, not 9
– Example (1+2)*3 = 9
– Work inside parenthesis always done first
– Use (extra) parenthesis to avoid errors
How to decide number of digits
Examples
A few more examples
Another Example
Positive and negative exponents
A few more examples
Kahn Acadamy
• http://www.khanacademy.org/
• Huge number of short You-Tube lectures
• Math is a specialty, free tutoring
• Try it out, a GREAT on line resource!
Manipulation of Exponents
• Multiplication
– Exponents add, 103 * 102 = 105 = 100,000
• Division
– Exponents subtract, 103 / 102 = 101 = 10
• Addition, Subtraction
– Normal addition, must use SAME exponents
– 1.23E2 + 1.23E3  (1.23+12.3)E2 = 1,353
– More detailed example later
Multiplication – exponents add
Division – exponents subtract
Exponents are very useful for
manipulating large & small values
• 0.000000123 * 62300000000 = ?
Rewriting in exponential notation is easier
• 1.23*10-7 * 6.23*1010 = 7.663*103 = 7,663
– OK for manual calculations
• 1.23E-7 * 6.23E10 = 7.663E3 = 7,663
– Simplest for Excel, calculators
• 1.23*10^-7 * 6.23*10^10 = 7.663*10^3
– Also handy for Excel, computers
Final slide
• End of Exponents