The Grammar and Etiquette of Scientific Math By T. Webb HHS You can be a mathematician without a lot of science, however, you cannot be a scientist without math Part 1 - Terminology in Basic Data Analysis Quantitative analysis is expressing data in numerical form. The data is measured within a given degree of accuracy and precision. This is an objective form of data. (Quantity how much answered with a number) Qualitative analysis involves descriptive terms that rely on the senses, such as sight, touch, sound, smell and taste. It is a subjective form of data that may be biased by personal experiences. (Quality a value answered by a description) Applications and terms used in quantitative analysis: (A) Relative Error: the magnitude, degree or size of an error; the deviation from the true or accepted value as compared to the derived value (your answer). The derived value is subtracted from the accepted/true value, and then divided by the accepted/true value; multiplied by 100 to get the percent error. (B) Precision: the reproducibility of results (data and measurements). A caliper may have the precision of measuring to 0.01 cm every time. Automated assembly lines have a high degree of precision they can do the same thing every time. (C) Accuracy: the correctness of a measurement to the desired result. The closer the result is to the true, standard or accepted value, the higher its degree of accuracy. Ideally, one should strive for a high degree of BOTH accuracy and precision! (D) Average or mean: obtained by adding together all results and dividing by the sum of the number of results. (E) Tolerance: the amount of accepted variation in the precision and accuracy in reference to a measuring instrument the instrument s limitations. For example, the average bathroom scale will be accurate up to 350 lb, and precise to 0.10 lb if used properly. It would not be suitable if you weighed more than 350 lb, or you wanted to get the mass of a toothpick. (A) Is bowling a strike once in 4 turns accurate? Precise? (B) Would it be better to have a precise soccer player, or an accurate one occasionally? 1
Part 2 - Digital Integrity 101 aka Significant Figures ( sig fig or sf) All digits (as in numbers, not just your fingers and toes) count as significant when obtained from a properly taken measurement. The last reasonably measured digit is uncertain, since we do not know the next number. The value of 56.9 cm has 3 sf, with the 9 being the uncertain digit. For another example, we would not record a measurement of 34.9384633 cm taken by a common ruler as it is unreasonable and beyond the ruler s limitation of measure. The last reasonably measured digit would be the 3 a good website resource is http://dbhs.wvusd.k12.ca.us/webdocs/sigfigs/ A) Count all reasonably measured digits from 1 9 as significant. B) Exact numbers are not uncertain, and have an infinite number of sig figs. These are defined numbers such as 1000 m in 1 km, or 100 cm in 1 m. They also include numbers of counting objects that you cannot reasonably break down further 4 people or 23 pennies. Zeroes Do NOT count zeroes in front of a number, as they are only placeholders. Example: 0.0224 cm has only 3 sig figs the 224 part. We can convert this to 224 um and still have the same value. Do NOT count zeroes following a number unless there is a decimal in the measured value. Example: You ran 3 000 m (1 sf) but unless you measured it exactly on a track at 3 000.0 m (5 sf), you may have gone more or less than 3 000 m Zeroes between numbers count as significant. Example: 204 m has 3 sf; 3 007 km has 4 sf; 0.02030 m has sf Trailing zeroes do not count, and a bar indicates the last sig fig of uncertainty. Example: 1 200 000 km has 2 sf; (draw bar over the 2 nd zero after the 4) 1 400 000 000 m has 4 sf (up to the bar) Quick Practice How many significant digits are in these measurements? 1. 44.5 6. 902.3 2. 0.0034 7. 5200 3. 90.00 8. 0.00 4. 10300 9. 200.20 5. 0.020 10. 40 000 000 2
Digital Integrity Part 2 - How Many Digits Can I Have in My Answer? Specific SI Rules - The International System of Units has accepted values for calculations based on reasonable measurements. It is expected that all data and numerical information is expressed using these rules. http://dbhs.wvusd.k12.ca.us/webdocs/metric/metric-units.html 1. Addition and Subtraction Rule: When adding or subtracting, calculate the answer and then round off to the LEAST number of DECIMAL PLACES contained in the question. 34.66 g 0.033 g 2.005 g 1.0288 g 10 g * no decimal place 4.54 g 46.665 g = 47 g 5.6018 g = Please note: if more than one size of unit is given, generally convert to the larger unit and then do the calculation.) 12.04 km 12.04 km 609 m 0.609 km 989 m 0.989 km???? = 13.638 km = 13. km 2. Multiplication and Division Rule: Do the math, and then round off to the LEAST number of DIGITS (Sig Figs) contained in the question. Please note - If doing a series of calculations, round off the final answer only. You will end up with a very inaccurate answer if you round off after every step! 13.52 m x 2.1 m = 28.392 = 28 m 2 389.81 km / 4.2 h = 92.8119 = If using an exact number, do the math and then use the least number of decimal places in the other values of the question. Example: 42 horses x 1.25 bales/horse/day = 52.5 = 52.50 bales/day 3. Rounding Off Rule If the number following the last one you can keep is a 6 or more, round it up. If the number following the last one you can keep is a 4 or less, leave it as is. If the number following the last one you can keep is a 5, use the ODD/EVEN rule. Statistically, the number 5 is exactly in the middle. There are 5 even digits (0,2,4,6,8) and 5 odd digits (1,3,5,7,9). The reason of the odd/even rule is that if you always round digits from 5 9 up, then there is only 1 4 times that you don t round at all. Therefore, if the number preceding the 5 is odd, round up; if it is even, leave it. Example keep only two digits for the following: (A) 12.74 + 3.599 + 7.2 = 23.5 = 24 (B) 12.68 0.1 = 12.5 = 12 3
4. Scientific Notation Writing very large or small numbers can be awkward, and difficult to manage in calculations. Besides using the preferred metric system of prefixes, scientific notation can also make values much easier to work with. For example, the average wavelength of gamma rays is 0.000 000 000 064 m. This is 64 trillionth of a metre try doing calculations with that! Scientific notation would change this to 6.4 x 10-11 m. The coefficient number (11) is how many placeholders (for a power of ten each) are present, and if it is negative, you have a decimal value; a positive is a whole number. To put into scientific notation, move the decimal until only ONE digit is in front of it. Count how many spaces you moved, and that is your coefficient for the power of ten. 23 000 cm = 2.3 x 10 4 cm 0.000 000 388 m = 3.88 x 10-7 m To expand a number, multiply it by the power of 10 4.3 x 10 3 m = 4 300 m 5.08 x 10-2 cm = 0.0508 cm Notate: (A) The human body contains ~ 5 340 ml of blood. (B) Astronomers predict that our sun will last another 7 500 000 000 a. (C) The mass of a proton is ~ 0.000 000 000 000 000 000 000 001 68 g. Expand: (A) The area of Canada is ~ 1.0 x 10 7 km 2. (B) X-rays have a 4.84 x 10-10 m wavelength. 5. Metric Rules and Symbols (A) Symbols are always printed, and lowercase letters used. The exceptions are units derived from a proper name, such as Joule, Watt, Newton, Litre (B) Symbols are never pluralized 12 g, not 12 gs (C) Do not put a period after a symbol, unless it is at the end of a sentence. (D) A full space is between the number and the symbol: 25 m, not 25m (E) Use decimals, not fractions, and put a zero before the decimal: 0.87, not.87 (F) Generally, we use the term mass for the weight of something. (G) The Celsius scale indicates temperatures. (H) Numbers and symbols are used together, not numbers and names. 10 km, not 10 kilometres Find the errors and correct them: (A) Mike ran in the 400-metre relay when it was 32 degrees outside! (B) The recipe called for 1 ½ cups of sugar and 45 ml. of vanilla. (C) The joules to heat one gram of water 1 degree Celsius is 4.2 j. 4
Practice 1. Give an example of an instrument s tolerance and explain what it means. 2. Describe how a person could be accurate, but not precise, using an example of your choice. 3. List three quantitative observations about this room. (a) (b) (c) 4. List three qualitative observations about yourself. (a) (b) (c) 5. Why would scientists, mathematicians, engineers etc be interested in relative error? 6. What value is considered a reasonable margin of error? Would this depend on what was being measured, or the situation involved? Explain and give reasons with examples. 7. Calculate, and give your answer according to the SI rules for significant figures (A) 2.1 g (B) 0.005 m (C) 10.92 L (D) 16.02 m 10.41 g + 1.04 m - 8.1 L x 1.4 m + 3 g (E) 0.034 km (F) 19.3 cm (G) 91.2 g (H) 76.4 L 1.01 h x 4.5 cm 4.2 g/doz 9 people = = = 5