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## Digital Values

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**Digital Values**• Digital Measurements • Integers only, “0” & “1” for computers • On or Off, Yes or No, In or Out, up or down … • Dozen eggs is exactly 12, not 12 +/-1 • Biped has exactly 2 legs, tripod has 3 • NO fractions or partial values, just integers • Relatively error free transcription • Can apply automatic corrections, parity, ECC • NO uncertainty, values are exact • Nature modeled digitally at atomic levels • Quantum numbers, energy levels, spin direction**Analog Values**• Analog measurements, everyday norm • Variable quantities, any value allowed • Intensity of light and sound, level of pain • Everyday life is continuously variable • What we weigh, sense of smell & hearing • Values experienced are NOT fixed • If any value is OK, how to prevent errors? • Precision & accuracy become important**Number Notation**• Common symbols in text books • 102 = 100, • √25 = 5 • Calculators and computers (e.g. Excel) use other conventional symbols • 100 = 10^2 = 10E2 (Excel) = 10exp2 (Casio) • 25^0.5 = 25E0.5 = 25^(1/2) for square roots • yx also does ANY powers & roots**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, 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**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 or Scientific Notationkeeps numbers relatively**simple • Decimal number identifying significant digits • Example: 5,050,520 • Exponent of 10 identifies overall magnitude • Example: 10^6 or E6 (denoting 1 million) • Combined expression gives entire value • 5.05052 x 106 (usual text book notation) • 5.05052*10^6 (computers, Excel) • 5.05052*10exp6 (some calculators) • 5.05052E6 (alternative in Excel)**Exponential Notation**Notation method Leading digit (typically) before decimal point Significant digits (2-3 typical) after decimal Power of 10 after all 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, could also write 0.6875E1 = 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 Notation**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**Significant Figures**• Precision must be tailored for the situation • Result cannot be more precise than input data • Data has certain + uncertain aspects • Certain digits are known for sure • Final (missing) digit is the uncertain one • 2/3 cups of flour (intent is not 0.66666666667) • Fraction is exact, but unlimited precision not intended • Context says the most certain part is 0.6 • Uncertain part is probably the 2nd digit • Recipe probably works with 0.6 to 0.7 cups • How to get rid of ambiguity?**Significant Figures**• “Sig Figs” = establish values of realistic influence • 1cup sugar to 3 flour does not require exact ratio of 0.3333333 • Unintended accuracy termed “superfluous precision” • Need to define actual measurement precision intended • “Cup of flour” in recipe could be +/- 10% or 0.9 to 1.1 cup • Can’t be more Sig-Figs than least accurate measure • Final “Sig Fig” is “Uncertainty Digit” … least accurately known • adding .000001 gram sugar to 1.1 gram flour = 1.1 gram mixture**How to Interpret Sig-Figs(mostly common sense)**• All nonzero digits are significant • 1.234 g has 4 significant figures, • 1.2 g has 2 significant figures. • “0” between nonzero digits significant: • 3.07 Liters has 3 significant figures. • 1002 kilograms has 4 significant figures**Handling zeros in Sig-Figs**• Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point (overall magnitude): • 0.001 oC has only 1 significant figure • 0.012 g has 2 significant figures • 1.51 nanometers (or 0.00000000151 meter), 3 sig figs • Trailing zeroes that are to theright of a decimal point with numerical values are always significant: • 0.0230 mL has 3 significant figures • 0.20 g has 2 significant figures • 1.510 nanometers (0.000000001510 meters), 3 sig figs**More examples with zeros**• Leading zeros don’t count • Often just a scale factor (0.000001 = microgram) • Middle zeros between numbers always count • 1.001 measurement has 4 decades of accuracy • Trailing zeros MIGHT count • YES if part of measured or defined value, 1.0010 • YES if placed intentionally, 7000 grains = 1 pound • NO if zeros to right of non-decimal point • 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs • NO if only to demonstrate scale • Carl Sagan’s “BILLIONS and BILLIONS of stars” • Does NOT mean “BILLIONS” + 1 = 1,000,000,001**More Sig-Fig ExamplesClass interaction: how many sig figs**below? • Zeros between • 60.8 has __ significant figures • 39008 has __ sig-figs • Zeros in front • 0.093827 has __ sig-figs • 0.0008 has __ sig-fig • 0.012 has __ sig-figs • Zeros at end • 35.00 has __ sig-figs • 8,000.000 has __ sig-figs • 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4**More Sig-Fig Examples**Zeros between 60.8 has 3 significant figures 39008 has 5 sig-figs Zeros in front 0.093827 has 5 sig-figs 0.0008 has 1 sig-fig 0.012 has 2 sig-figs Zeros at end 35.00 has 4 sig-figs 8,000.000 has 7 sig-figs 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4**Sig-Fig Exponential Notation**• A number ending with zeroes NOT to right of decimal point are not necessarily significant: • 190 miles could be 2 or 3 significant figures • 50,600 calories could be 3, 4, or 5 sig-figs • Ambiguity is avoided using exponentialnotation to exactly define significant figures of 3, 4, or 5 by writing 50,600 calories as: • 5.06 × 10E4 calories (3 significant figures) or • 5.060 × 10E4 calories (4 significant figures), or • 5.0600 × 10E4 calories (5 significant figures). • Remember values right of decimal ARE significant**Exact Values**• Some numbers are exact because they are known with complete certainty, or are defined by exact values: • Many exact numbers are simple integers: • 12 inches per foot, 12 eggs per dozen, 3 legs to a tripod • Exact numbers are considered to have an infinite number of significant figures. • Apparent significant figures in any exact number can be ignored when determining the number of significant figures in the result of a calculation • 2.54 cm per inch (exact) • 5/9 Centigrade/Fahrenheit degree (exact) • 5280 feet per mile (exact, based on definitions) • The challenge is to remember which numbers are exact !**more Sig-Fig Accounting**• Addition & Subtraction • Least Significant Figure determines outcome • 1.01 + 1.00000001 = 2.01 (limited by 1.01) • Multiplication & Division • Least Significant Figure determines outcome • 1.01 x 1.0000001 = 1.01 • Round-Off • Calculators yield more sig-figs than justified • Must reduce answer to lowest sig-fig component**Sig-Fig Multiply & Divide**Good first step to use scientific notation Multiply 0.113 * 5280 1.13E-1 * 5.280E3 Multiply the leading values, add the exponents Becomes 5.96640E2 Sig.Fig. set by least precise input 5.96E2 Divide 4995 by .0012 4.995E3 / 1.2E-3 Divide leading values, subtract the exponents Becomes 4.1625E6 Sig.Fig. set by least precise input 4.2E6**Sig-Fig Addition & Subtraction**First get the decimals (blue) to align Take 1.0234E3 same as 1,023.4 Then add 1.0E-4 same as + 0.0001 Then subtract 15.22 same as - 15.22 Do the math 1,008.1803 Round to least decimal sig fig 1,008.2 “spitting in the ocean” analogy … if you measure ocean volume by cubic meters or miles, adding a teaspoon is undetectable !**Partial Values**• Averages, fractions, yields • 2/3 cups flour = 0.66666666666666 …cups? • >2 digit precision inappropriate for cookies • See Mrs. Fields Cookie Recipe • “superfluous accuracy” • unjustified or unwarranted level of detail • Precision needs to fit the situation • “Rounding Off” to appropriate accuracy • Need rules to set the values**more Sig-Fig Accounting**• Round-Off • Calculations can yield more sig-figs than justified • Must reduce result to lowest sig-fig component • Methodology (usual & customary rules) • If value beyond last sig-fig is ≥5, round UP • For 3 sig-fig accuracy, 5.255123 becomes 5.26 • If value beyond last sig-fig is <5, round OFF • For 3 sig-figs accuracy, 5.254459 becomes 5.25**Rounding Rules … Traditional Rule is Simplest**• When trailing digit is <5 round off • 1.244 rounded to 3 digits 1.24 • 1.2449999 rounded to 3 digits 1.24 • When trailing digit is ≥5 round up • 1.246 rounded to 3 digits 1.25 • 1.2460111 rounded to 3 digits 1.25 • Note lack of symmetry at “5” • 5 is in the middle, but rounds up • Unintended bias is towards larger values**Rounding Rules … “Banker’s Rule” addresses bias**• When trailing digit is < 5 round off • 1.244 rounded to 3 digits 1.24 • When trailing digit is > 5 round up • 1.246 rounded to 3 digits 1.25 • What to do with a trailing “5” ? • Aim is equal opportunity, round up or down • Try to avoid statistical bias in large data sets • “rule” is to look at digit preceding rounding • Equal probability of odd or even value • Arbitrary rule to round up if odd, down if even • 17.75 17.8 also 17.85 17.8**Guidelines for using calculators**• Don’t round off too soon, do it at end of calculation • (5.00 / 1.235) + 3.000 + (6.35 / 4.0) • 4.04858 + 3.000 + 1.5875 = 8.630829 • 1st division results in 3 sig-figs, last division results in 2 sig-figs. • 3 numbers added should result in 1 digit after the decimal. Thus, the correct rounded final result should be 8.6. This final result has been limited by the accuracy in the last division. • Warning: carrying all digits through to the final result before rounding is critical for many mathematical operations in statistics. Rounding intermediate results when calculating sums of squares can seriously compromise the accuracy of the result.**Rounding & Sig-Figs NOT exact**• Several papers illustrate the issues • Wikipedia article • Rounding issues tend to be academic • Prof. Mulliss, Univ. of Toledo Ohio • Tried millions of calculations to test the rules • Add-Subtract simple rule ≈ 100% accurate • Multiply-Divide standard rule ≈ 46% accurate • Multiply-Divide (Std Rule+1) ≈ 59% accurate • Mult-Divide best-case rules ≈ 90% accurate**Metric-English Conversions**• Convert 10.0 inches to centimeters • 10.0 inch * 2.54 cm/inch = 25.4 cm • Precision is 3 sig figs, input & output • But …. Inches are bigger units of measure • 3rd significant figure for inches is 2-½ x larger ! • Inches not the same size as centimeters! • A tolerance setting problem for international companies • Often add one more sig-fig to inches when converting**Take Away Message**• Rounding & Sig-Figs not infallible • It’s a math model, numbers on a page • Reality may be different (hopefully not by much) • Units of measure may not have same magnitude • Utility is to make results more rational • Avoids a conclusion not justified by the input • Numerical methods fail when pushed too far • Nature is not the problem • Our use of numbers and rules are the issue • Walt Kelly in “Pogo” had it right, “we’ve met the enemy … and it’s us”**Dimensional Analysis**• Making the units come out right • Useful strategy to avoid calculation errors • Relies on “cancellation of dimensions” • If sec^2 instead of sec/sec cancel, something got inverted • Should always put dimensions on initial formulas • Good News • Easy to do • Avoids silly answers with wrong dimensions. • Bad News • Does not insure right physical relationships • No guarantee of right answer … but units OK**Dimensional Analysis**Speed Limit 100 km/hr vs. miles/hr (100 km/hr *1000 m/km *100 cm/m) / (2.54 cm/inch*12 inch/foot*5280 foot/mile) = 62.13711922 mph If 100 km/hr limit is exact (e.g. 100.00000 …) An exact value leads to infinite precision 62.13711922 … Mathematically correct, but impractical for speedometers If 100 km/hr limit is NOT exact (e.g. 99.5 - 100.4) 3 sig fig limit sets speed at 62.1 mph 2 sig fig limit sets speed at 62 mph 1 sig fig sets speed limit at 60 mph**Dimensional Analysis**Human Body Temperature Accepted healthy value in USA is 98.6oF Convert to Celsius: (98.6– 32) oF * (5oC/9oF) = 37.0oC Accepted (customary) value in Europe is 37oC Convert to Fahrenheit (37oC * 9oF/5oC) + 32oF = 99oF Result is 2 sig-figs, and an apparent temperature rise What happened… are Europeans bodies hotter? 2 digit sig-fig on a larger unit of measure (oC), vs 3 sig figs on smaller degree (oF) is inconsistent. Europeans might argue that variability between health people negates need for higher sig fig.**From Chem. 15 Lab ManualExercises Page 2, # 4J**(0.0048965 – 0.00347) x (3.248E4 – 4.58983E3) • Solve what’s inside parenthesis FIRST • Initial value 1st parenthesis 0.0048965 4.8965 E-3 • Subtract 2nd value 0.00347 3.47 E-3 • Result after subtraction 0.0014265 1.4265 E-3 • Round to least accurate 0.00143 1.43 E-3 • Second Parenthesis Calculation • 3.248E4 same as 32,480 32.48 E3 • Subtract 4.58983E3 same as 4,589.83 - 4.58983 E3 • Result after subtraction 27,890.17 27.89017 E3 • Round to low of 4 sig fig 27,890 27.89 E3 • Multiply results from parenthesis calculations • 0.00143 * 27,890 = 39.88270 39.9 • Multiplication accuracy limited to least sig figs = 3 in this case**Accuracy and Precision**• Accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value. • Precision, also called reproducibility or repeatability, is the degree to which further measurements or calculations show the same or similar results. • A measurement can be accurate but not precise; precise but not accurate; neither; or both. • A result is valid if it is both accurate and precise • Related terms are error (random variability) and bias (non-random or directed effects) caused by a consistent and possibly unrelated factor. • Show water slide video … is he accurate or precise?**Accuracy**• Degree of error in achieving the established measurement goal • The Cubit average value has not changed much since biblical times at about 18 inches so it has remained relatively accurate over hundreds of years.**Good accuracy This example shows good accuracy, but low**precision**Precision**• How well multiple measurements agree with one another to provide a consistent value. (e.g. tight grouping, low dispersion, “all together” series of events). • The “cubit” is not a very precise measure of distance, since it varies between observers using the same definition. No two people are the same, so length data is dispersed. (e.g. inconsistent individual measurements).**Target analogyThis example has high precision, but poor**accuracy**Standard Deviation, why bother?**Range a poor indicator of accuracy One bad measurement controls the range Averaging scheme redefines error RMS (root mean squared) is common tool Moves error to an average value basis Suppresses random error contribution**Non-Linear representations**Exponential Growth (or decline) Changes associated with exponent of 10 value Example: exp of 2100x, exp of 31000x Moore’s Law, Chain Reaction of Uranium Logarithmic Scale Some differences too large to put on a linear scale Hearing, visual acuity, earthquakes, concentration of ions Logarithm scale “compresses” scales “decibel” for sound, “pH” for acid concentration Richter scale for earthquakes Richter 9 (SF 1906) is 1000x that of Richter 6 (mild shake)**Gordon Moore’s Lawtransistors in a CPU doubles every 18-24**months**Summary, Exponential Notation**Number represented by decimal + exponent Example: 1,234 = 1.234*10^3 or 1.234E3 Multiplication: multiply decimals, add exponents 6*10^6 x 2*10^2 = 12*10^8 = 1.2*10^9 (or 1.2E9) Division: Divide decimals, subtract denominator exponent from numerator 6*10^6 / 2*10^2 = 3*10^3 (or 3E3) Addition & Subtraction Line up numbers by decimal (and same exponent) before adding Add 1,234 or 1.234 E3 Add to 5.678 or .005678 E3 Sum = 1,239.678 or 1.23978 E3**Summary, Significant Figures**All nonzero digits are significant 1.234 has 4 significant figures, “0” between nonzero digits significant: 1002 has 4 significant figures “0” after decimals always significant 0.12300 has 5 significant figures “0” in front of decimal NOT significant 0.00000000123 has 3 significant figures “0” after non-zero digit MAY be significant 1,000 could be 1, 2, 3, or 4 significant figures 4 if an exact number, e.g. 1000 grams per kilogram Depends on context, better to write in exponentials**Summary, Rounding Rules**When trailing digit is <5 round off (truncate) 1.244 rounded to 3 digits 1.24 When trailing digit is ≥5 round up 1.2460111 rounded to 3 digits 1.25 Lack of symmetry at “5” Unintended bias is towards larger values “Banker’s rule”: look at digit preceding rounding Equal probability of odd or even value Arbitrary rule to round up if odd, down if even 17.75 17.8 also 17.85 17.8**Summary, Dimensional Analysis**Relies on “cancellation of dimensions” 7 days/week * 52 week/year = 364 days/year Always put dimensions on initial formulas List starting and ending (desired) dimensions Conversion dimensions between start and end Multiply or divide to eliminate unwanted dimensions Writing dimensions avoids squared vs cancelled Use exact values when practical Avoids sig-fig confusion Round off answer only after all calculations Rounding too soon can multiply uncertainty error