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

• 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 ExamplePositive 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 addDivision – 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