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
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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.
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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 schemePreceded 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
– 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– 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 numberthis is Avogadro’s number, atoms in a mole
Exponent Conventions
• 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• 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
• 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
• 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 ExamplePositive and negative exponents
A few more examples
Kahn Acadamy
• http://www.khanacademy.org/ • Huge number of short You-Tube lectures