1. 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

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

3. 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

4. 12 finger math?

5. 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”

6. 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

7. 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

8. Basic CGS metric scheme Preceded SI / ISO system of units (cm vs meter) 1 cm^3 = 1 milliliter = 1 gram H 2 O

9. 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*10 8 meters/sec speed of light –6.02*10 23 atoms/mole –6.25*10 -7 meters for wavelength red light –1.60*10 -19 Coulombs for electron’s charge

10. Parts of a Value

11. Setup of a scientific number this is Avogadro’s number, atoms in a mole

12. Exponent Conventions 1000 = 10 3 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 2 nd function) 100 = 10 2 =10E2 =10^2 all mean the same 10 = 10 1 =10E1 = 10^1 all the same 1 = 10 0 = 10E0 =10^0, by definition –Anything raised to zero power is one

13. Negative Exponents are handy for very small numbers

14. Decimal vs Scientific “normalized” refers to small number of leading digits

15. 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)

16. 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)

17. 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 10 3 = 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

18. 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

19. 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

20. How to decide number of digits

21. Examples

22. A few more examples

23. Another Example Positive and negative exponents

24. A few more examples

25. 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!

26. Manipulation of Exponents Multiplication –Exponents add, 10 3 * 10 2 = 10 5 = 100,000 Division –Exponents subtract, 10 3 / 10 2 = 10 1 = 10 Addition, Subtraction –Normal addition, must use SAME exponents –1.23E2 + 1.23E3  (1.23+12.3)E2 = 1,353 –More detailed example later

27. Multiplication – exponents add Division – exponents subtract

28. Exponents are very useful for manipulating large & small values 0.000000123 * 62300000000 = ? Rewriting in exponential notation is easier 1.23*10 -7 * 6.23*10 10 = 7.663*10 3 = 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

29. Final slide End of Exponents