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Calculation Practice 01/05/2016 Cal-1 Math Calculations Practice Exercises Introduction The International System of Measurement (SI) is used worldwide and has been adopted as the official system of measurement by most countries. It is commonly called the metric system. Our traditional American/English system of measurement (e.g., miles, quarts, pounds) requires many conversion factors. Take length, for example – there are inches, feet, yards, rods, chains, and miles! The metric system is much different. It is based on standard units that can be easily converted by multiplying or dividing by factors of ten. Engineers and scientists most often use these standard metric units: the meter, for length; the gram, for mass (or weight); the liter, for volume; and the degree Celsius (or less often Kelvin) for temperature. Estimating and Uncertainty Whenever you take a scientific measurement, you are making a quantitative observation. When you use a device to make a measurement, you obtain a measured number. When you report your data, you usually are estimating the last significant figure. In other words, you will report the digits in the measurement that you are certain about plus one additional digit that are you allowed to estimate. We will discuss significant figures later in this exercise. You can also look in your lecture text or Appendix F of the lab manual if you need help with significant figures. Here are typical uncertainties of common measuring devices: Measuring Device 12 cm ruler triple-beam balance analytical balance 10 ml graduated cylinder 100 ml graduated cylinder 50 ml buret 25 ml volumetric flask 25 ml transfer pipet Uncertainty ± 0.05 cm ± 0.05 g ± 0.0001 g ± 0.05 ml ± 0.5 ml ± 0.02 ml ± 0.02 ml ± 0.02 ml The last digit in your measured number is the digit that is considered uncertain because it is estimated. For example, if you are measuring the length of a piece of metal with a ruler that has marks down to units of 0.1 cm, you could estimate the length down to the 0.05 cm, in other words, between tenths of a cm. The number of significant figures gives us an idea of the accuracy of a measuring device. In this and future labs in this class, you will be expected to keep track of significant figures for performing calculations and reporting results. Appendix F provides rules for rounding off the results of calculations with significant figures. When you use a conversion factor or a definition, those would be examples of an exact number – in other words, a number for which you do not need to use a measuring device. Can you think of number relationships that would fit into this category of number? Accuracy and Precision In common English, we often use the terms “accuracy” and “precision” interchangeably, to indicate how “correct” an answer is. However, in science the two terms have different meanings. Accuracy is a measure of how closely an observation is to the “true” or “accepted” value. Precision is a measure of how closely a group of observations are to one another. If you think about a dartboard, “accurate” would be hitting the bulls-eye or center of the target; “precise” would mean that all of your darts hit the target close to one another, without reference to whether or not you hit the bulls-eye. So, it is possible to be precise (all the darts close together) but not accurate (missing the bulls-eye). Of course, we would like to be both accurate and precise in our laboratory measurements. Appendix D (Useful Math Relationships) gives you some mathematical ways of reporting accuracy and precision. Calculation Practice 01/05/2016 Cal-2 Random and Systematic Errors As we can see from the information above, each measurement has a certain amount of uncertainty associated with it, which means that each measurement has a certain amount of error. Errors refer to the calculated difference between a measured value and the “true” value. There are actually two kinds of error: random error and systematic error. Random errors result from the uncertainty of your measurement device. They are not caused by a mistake in your technique, but are caused by unpredictable or imperceptible factors that are beyond your control as an experimenter. An example would be two different experimenters determining the mass of an object using two kinds of balances that have different sensitivities for mass. Errors that have definite causes are called systematic errors. In general, systematic errors are generally reproducible and will result in values that are always higher than the true value or lower than the true value. An example would be a thermometer that is not calibrated correctly and always gives a reading that is lower than the actual or true value. Random errors are always present – but you want to eliminate or minimize the systematic errors in carrying out your experiments. A. Measured and Exact Numbers Example 1 Describe each of the following as a measured or exact number a. 14 inches ______________________________________ b. Four chairs at the dining table _____________________ c. 60 minutes in 1 hour _____________________________ d. 7.5 kg _____________________________________ e. 2.54 cm in 1 inch ____________________________ f. 1000 mg in 1 g ______________________________ g. 12 inches in 1 ft ______________________________ h. 2.20 lb in 1 kg ________________________________ B. Scientific Notation In scientific work, small numbers such as 0.000000025 m and large numbers such as 4,000,000 g are often expressed using powers of 10, such as 2.5 x 10 -8 m and 4 x 10 6 g. The values of 2.5 and 4 are coefficients; the values 10 -8 and 10 6 are powers of ten. (See Table 1 and Example 2). The rules for converting standard numbers or scientific notation are given below. Calculation Practice 01/05/2016 Table 1 Standard number 10,000 1000 100 10 = = = = Cal-3 Some Powers of Ten Power of ten 10 4 10 3 10 2 10 1 Standard number 0.1 0.01 0.001 0.0001 = = = = Power of ten 10 -1 10 -2 10 -3 10 -4 Rules for Writing Numbers in Scientific Notation For numbers larger than 10: a. Move the decimal point to the left until it follows the first non-zero digit in the number b. Write a power of ten that is equal to the number of places the decimal was move to the left For numbers smaller than 1: a. Move the decimal point to the right until it is located after the first digit in the number b. Write a negative power of ten that is equal to the number of places the decimal point was moved to the right Example 2 Write the following standard numbers in scientific notation: a. 35,000 km _______________________________ b. 608 g _______________________________________ c. 0.0000815 m _________________________________ C. Significant Figures In measured numbers, all the reported figures are called significant figures. The first significant figure is the first nonzero digit. The last significant figure is always the estimated digit. Zeros between other digits or at the end of the decimal part of a number (i.e., trailing zeros in a number with a decimal part) are counted as significant figures. However, leading zeros are not significant; they are placeholders. Zeros are not significant in numbers equal to, or greater than 10, with no decimal part (there may or may not be a decimal point); they are placeholders needed to express the magnitude of the number. In a number written in scientific notation, all the figures in the coefficient are significant. Examples of counting significant figures in measured numbers are given in Table 2 and Example 3. Table 2 Measurement 455.2 cm 0.800 m 50.2 L 0.0005 lb 25,000 ft 3.20 x 10 4 g Examples of Counting Significant Figures Number of Significant Figures 4 3 3 1 2 e Reason All nonzero digits are significant The trailing zeros in the decimal part are significant A zero between nonzero digits is significant Leading zeros are not significant Placeholder zeros are not significant All the digits in a coefficient are significant Calculation Practice 01/05/2016 Cal-4 Example 3 State the number of significant figures in each of the following measured numbers a. 588 m ____________________ c. 1.5 x 10 3 mL _________________ b. 0.0700 g d. 0.00045 L ___________________ e. 45.0 kg ________________________ f. g. 2.20 lbs h. 1000 mm _____________________ ___________________ 500 lb ______________________ ___________________ D. Rounding Off Often, you will use a measurement in a mathematical operation such as multiplication, division, addition or subtraction. When the calculator display shows more numbers than the measurements support, it is necessary to round of the calculated answer. If the numbers to be dropped begin with a number less than 5, they are simply dropped. However, if the numbers dropped begin with a 5 or greater, the value of the last retained digit is increased by 1. For a calculator display already in scientific notation, round off the coefficient to the correct number of significant figures. On your calculator, an answer may appear in scientific notation, which means that a coefficient and a power of ten are shown. In scientific notation, the correct number of significant figures is shown in the coefficient. Often the numbers shown in a calculator display before the power of ten must be rounded. Be sure to keep the power of ten! See Table 3 and Example 4. Table 3 Examples of Scientific Notation from Calculator Results Calculator display 2.512 4.1585 8.775 05 12 -08 Number of significant figures to be shown in the coefficient 2 3 2 Rounded and written in scientific notation 2.5 x 10 3 4.16 x 10 12 8.8 x 10 -8 Example 4 Round off each of the following calculator displays to report answers with three significant figures and two significant figures. a. 75.6243 km _______________ _____________ b. 0.528392 g _______________ _____________ c. 387.600 miles _______________ _____________ d. 0.007000 m _______________ _____________ E. Multiplication and Division If a calculated answer is obtained from multiplication and/or division, it is rounded off to the same number of significant figures are the measured number with the fewest significant figures. See Examples 5 and 6. Calculation Practice 01/05/2016 Cal-5 Example 5 Solve: (0.025 𝑔)(4.62 𝑔) 3.44 𝑔 = Solution: On the calculator, the steps are: Enter keys 0.025 X 4.62 = ÷ 3.22 = Display reads 0.025 0.025 4.62 0.1155 0.1155 3.44 0.033575581 0.03 Two significant figures Three significant figures Three significant figures Calculator display to 9 decimal places Final answer rounded to two sig figs Example 6 Solve 3.4 𝑥 10−4 𝑚 2.75 𝑥 10 8 𝑚 = Solution: On the calculator, the steps are: Enter keys 3.4 EXP ( or EE) 4 ± ÷ 2.75 EXP 8 = Display reads 3.4 3.4 00 3.4 04 3.4 -04 3.4 -04 2.75 2.75 00 2.75 08 1.2363636 -12 1.2 x 10 -12 Two significant figures Three significant figures Calculator display Coefficient rounded to two sig figs F. Addition and subtraction After you have added or subtracted measured numbers, you may need to round off the result. An answer from + or – has the last significant figure in the column where all the numbers added or subtracted also have significant figures. See Examples 7 and 8. Example 7 Add 42.11 g + 4.056 g + 30.1 g Solution: + + 42.11 4.056 30.1 __________ 76.266 76.3 digits in this number end at the tenth’s place; all other numbers go further calculator display final answer rounded to give digit in tenth’s place Calculation Practice 01/05/2016 Example 8 Subtract: 14.621 mL – 3.39 mL Solution: + 14.621 3.39 __________ 11.231 11.23 digits end at hundredth’s place; all other numbers go further calculator display final rounded to give a digit in hundredth’s place Cal-6 Calculation Practice 01/05/2016 Cal-7 Student Name: _____________________________________ Section: ______________ Pre-lab Activity Questions 1. Give one example of a measured number. 2. Give one example of exact numbers. 3. What is meant by “degree of uncertainty”? 4. Give 2 reasons why a zero would be considered NOT significant. Exercises A. Measured and Exact Numbers Circle “M” or “E” to indicate whether each of the following numbers is measured or exact: Counted 5 books M E 2.54 cm in 1 inch M E 5 lb M E 12 inches in 1 foot M E 9.25 g M E 361 miles M E 0.035 kg M E 100 cm in 1 m M E B. Scientific Notation Write the following numbers in scientific notation: 4,450,000 ________________________ 0.00032 ____________________ 38,000 ________________________ 25.2 ____________________ 0.0505 ____________________ 0.0000000021 ________________________ Write the following as standard numbers: 4 x 10 2 ________________________ 3 x 10 -4 ____________________ 5 x 10 3 ________________________ 8.2 x 10 -3 ____________________ 3.15 x 10 5 ________________________ 2.46 x 10 -6 ____________________ Calculation Practice 01/05/2016 Cal-8 C. Significant Figures State the number of significant figures in each of the following measured quantities: 4.5 m _____________________________ 204.52 g ______________________ 0.0004 L _____________________________ 625,000 mm ______________________ 805 lb _____________________________ 34.80 km ______________________ D. Rounding Off Round off each of the following to the number of significant figures indicated. Don’t forget placeholder zeros when necessary! 0.4108 143.63212 532,800 5.448 x 10 2 8,950,000 Three significant figures Two significant figures E. Multiplication and Division Do the following multiplication and division calculations. Give a final answer with the correct number of significant figures. 4.51 ft x 0.28 ft ___________________________ 0.184 cm x 8.00 cm x 0.034 cm ___________________________ 2.51 x 10 4 ms -1 x 5.0 x 10 -7 m ___________________________ (42.4 𝑎𝑎𝑎)(1.45 𝐿) ÷ (4.8 𝐾)(0.0821 𝐿 𝑎𝑎𝑎 mol-1 K-1 ___________________________ F. Addition and Subtraction Do the following addition and subtraction calculations. Give a final answer with the correct number of significant figures. 13.4 mL + 0.4552 mL ____________________________ 145.5 m + 86.5 m + 1045 m ____________________________ 245.625 g - 80.2 g ____________________________ (11.2 g – 2.11 g)/(10.2 cm x 2.3 cm x 1.0 cm) ____________________________