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Clinical Calculation 5th Edition Appendix B from the book – Pages 314 - 315 Appendix E from the book – Pages 319 - 321 Scientific Notation and Dilutions Significant Digits Graphs Appendix B – Conversion between Celsius and Fahrenheit Temperatures Although digital thermometers are replacing the old fashion thermometers these days, but as health care provider you should be able to convert between the Celsius and Fahrenheit and vise versa. Comparing different thermometers The ones we are concern are Celsius ( C ) and Fahrenheit ( F ) http://asp.usatoday.com/we ather/CityForecast.aspx?txt SearchCriteria=Oklahoma& sc=N http://weather.msn.com/ Converting Fahrenheit to Celsius 32F = _________________C F 32 C 1.8 32 32 0 C 0C 1.8 1.8 Converting Fahrenheit to Celsius 212F = _________________C F 32 C 1.8 212 32 180 C 100C 1.8 1.8 Converting Fahrenheit to Celsius 100F= _________________C F 32 C 1.8 100 32 68 C 37.8C 1.8 1.8 Converting Fahrenheit to Celsius 28F = ________________C F 32 C 1.8 28 32 4 C 2.2C 1.8 1.8 Converting Celsius to Fahrenheit 50 C = _________________ F F 1.8C 32 F 1.85 32 41F Converting Celsius to Fahrenheit 500 C = _________________ F F 1.8C 32 F 1.850 32 122F Converting Celsius to Fahrenheit 250 C = _________________ F F 1.8C 32 F 1.825 32 77F Appendix E – Twenty-four hour clock Twenty-four hour clock is for documenting medication administration, specially with use of computerized MARs. Rules: To convert from traditional to 24-hours: 1:00am and 12:00noon – delete the colon and proceed single digit number with a zero Between 12noon and 12 midnight – add 12hours to the traditional time. To convert from 24-fours clock to traditional: Between 0100and 1200-replace colon and drop zero proceeding single digit numbers Between 1300 and 2400-subtract 1200 (12 hours) and replace the colon. 01 00 - 1:00 am 02 00 - 2:00 am 03 00 - 3:00 am 04 00 - 4:00 am 05 00 - 5:00 am 06 00 - 6:00 am 07 00 - 7:00 am 08 00 - 8:00 am 09 00 - 9:00 am 10 00 - 10:00 am 11 00 - 11:00 am 12 00 - 12 noon 13 00 - 1:00 pm 14 00 - 2:00 pm 15 00 - 3:00 pm 16 00 - 4:00 pm 17 00 - 5:00 pm 18 00 - 6:00 pm 19 00 - 7:00 pm 20 00 - 8:00 pm 21 00 - 9:00 pm 22 00 - 10:00 pm 23 00 - 11:00 pm 24 00 - midnight 24-Hour Clock Conversion Table Example: on the hour 24 Hour Clock AM / PM Example: 10 minutes past 24 Hour Clock AM / PM 12hr Time 12 am (midnight) 1 am 2 am 3 am 4 am 5 am 6 am 7 am 8 am 9 am 10 am 11 am 12 pm (noon) 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm 7 pm 8 pm 9 pm 10 pm 11 pm 24hr Time 0000hrs 0100hrs 0200hrs 0300hrs 0400hrs 0500hrs 0600hrs 0700hrs 0800hrs 0900hrs 1000hrs 1100hrs 1200hrs 1300hrs 1400hrs 1500hrs 1600hrs 1700hrs 1800hrs 1900hrs 2000hrs 2100hrs 2200hrs 2300hrs 0100 1:00 AM 0010 12:10 AM 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2:00 AM 3:00 AM 4:00 AM 5:00 AM 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12 Noon 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00 PM 11:00 PM 12:00 PM 0110 0210 0310 0410 0510 0610 0710 0810 0910 1010 1110 1210 1310 1410 1510 1610 1710 1810 1910 2010 2110 2210 2310 1:10 AM 2:10 AM 3:10 AM 4:10 AM 5:10 AM 6:10 AM 7:10 AM 8:10 AM 9:10 AM 10:10 AM 11:10 AM 12:10 PM 1:10 PM 2:10 PM 3:10 PM 4:10 PM 5:10 PM 6:10 PM 7:10 PM 8:10 PM 9:10 PM 10:10 PM 11:10 PM Converting traditional clock to 24-hour clock Examples: 12 Midnight = 12:00 AM = 0000 = 2400 12:35 AM = 0035 11:20 AM = 1120 12:00PM = 12:00 Noon = 1200 12:30 PM = 1230 4:45 PM = 1645 11:50 PM = 2350 Midnight and Noon "12 AM" and "12 PM" can cause confusion, so we prefer "12 Midnight" and "12 Noon". Converting 24 Hour Clock to AM/PM traditional Examples: 0010 = 12:10 AM 0040 = 12:40 AM 0115 = 1:15 AM 1125 = 11:25 AM 1210 = 12:10 PM 1255 = 12:55 PM 1455 = 2:55 PM 2330 = 11:30 PM Scientific Notation Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10. The number 123,000,000,000 in scientific notation is written as : 1.23 10 The first number 1.23 is called the coefficient. 11 It must be greater than or equal to 1 and less than 10. The second number is called the base . It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 1011 the number 11 is referred to as the exponent or power of ten. To write a number in scientific notation: To write 123,000,000,000 in scientific notation: Put the decimal after the first non-zero digit and drop the zeroes. 1.23 In the number 123,000,000,000 The coefficient will be 1.23 To find the exponent count the number of places from the decimal to the end of the number. 1011 In 123,000,000,000 there are 11 places. Therefore we write 123,000,000,000 as: 1.23 X 1011 Exponents are often expressed using other notations. The number 123,000,000,000 can also be written as: 1.23 E+11 or as 1.23 X 10^11 Scientific Notation For small numbers we use a similar approach. Numbers less smaller than 1 will have a negative exponent. A millionth of a second (0.000001 sec) is: Put the decimal after the first non-zero digit and drop the zeroes 1.0 (in this problem zero after decimal is place holder) To find the exponent count the number of places from the decimal to the end of the number 0.000001 has 6 places 0.000001 in scientific notation is written as: Exponents are often expressed using other notations. The number 0.000001 can also be written as: 1.0 E-6 or as 1.0^-6 Fun Do you know this number, 300,000,000 m/sec.? It's the Speed of light ! 3.0 10 8 Do you recognize this number, 0.000 000 000 753 kg. ? This is the mass of a dust particle! 7.53 10 10 Now it is your turn. Express the following numbers in their equivalent scientific notational form: 1. 2. 3. 4. 5. 6. 123,876.3 1,236,840. 4.22 0.000000000000211 0.000238 9.10 1. 1.23876310 6 2. 1.236840 10 5 3. 4.22 10 4. 2.1110 -13 0 5. 2.38 10 -4 0 6. 9.110 Now it is your turn. Express the following numbers in their equivalent standard notational form: 1. 5.66310 5 2. 1.2310 3. 7.002 10 -1 4. 9.18 10 5. 7.18 10 6. 8.0 10 4 0 2 1. 2. 3. 4. 5. 6. 7 566.3 123,000. 70,020,000 0.918 7.18 80,000 Dilutions Understanding how to make dilutions is an essential skill for any scientist. This skill is used, for example, in making solutions, diluting bacteria, diluting antibodies, etc. It is important to understand the following: - how to do the calculations to set up the dilution - how to do the dilution optimally - how to calculate the final dilution Volume to volume dilutions describes the ratio of the solute to the final volume of the diluted solution. To make a 1:10 dilution of a solution, you would mix one "part" of the solution with nine "parts" of solvent (probably water), for a total of ten "parts." Therefore, 1:10 dilution means 1 part + 9 parts of water (or other diluent). Serial dilutions http://www.wellesley.edu/Biology/Concepts/Animations/d ilution.mov Serial dilutions - 1 mL 1 mL 1 mL 1 mL Original solution 9mL 9mL 9mL 9mL 1 10 1 10 1 10 1 10 1 4 10 Serial dilutions - 0.1 mL 1 mL 1 mL 1 mL Original solution 9.9 mL 9mL 9mL 9mL 0.1 1 1 2 10 100 10 1 10 1 10 1 10 4 1 10 1 10 3 1 10 5 Serial dilutions - 1 mL 1 mL 1 mL 1 mL Original solution 2 mL 2mL 2mL 2mL 1 3 1 3 1 1 2 3 9 1 3 1 1 3 3 27 1 3 1 1 4 3 81 Build Dilution ratio of 1:16 using 4 water blanks provided 3 mL 3 mL 3 mL 3 mL 3 mL Original solution 3 mL 3 mL 3 mL 1 2 1 2 1 1 2 2 4 1 2 1 1 3 2 8 1 2 1 1 4 2 16 Build Dilution ratio of 1:104 using 4 water blanks 1 mL 1 mL 1 mL 1 mL 9 mL Original solution 9 mL 9 mL 9 mL 1 10 1 10 1 10 2 1 10 1 10 3 1 10 1 10 4 Build Dilution ratio of 1:104 using 3 water blanks 0.1 mL 1 mL 1 mL 9.9 mL Original solution 9 mL 9 mL 0.1 1 10 100 1 10 1 1 103 1000 1 10 1 1 104 10000 Build Dilution ratio of 1:104 using 2 water blanks provided 0.1 mL 0.1 mL 9.9 mL Original solution 9.9 mL 0.1 1 10 100 0.1 1 10 100 1 4 10 Build Dilution ratio of 1:27 using water blanks provided 5 mL 5 mL 5 mL 10 mL Original solution 10 mL 10 mL 1 3 1 3 1 1 2 3 9 1 3 1 1 3 3 27 Serial dilutions 1 mL 1 mL 1 mL 1 mL 1 mL 3.25 103 # of bacteria found Original solution 9mL 9mL 9mL 9mL N 1 10 1 10 1 10 1 10 1 1 1 1 3.25 103 N 10 10 10 10 1mL 1 3.25 103 N 4 10 1mL EXPECTED NUMBER OF BACTERIA IN ORIGINAL SOLUTION N 3.25 103 104 3.25 107 Serial dilutions 0.1 mL 1 mL 1 mL 1 mL 5 mL 1.15 10 4 # of bacteria found Original solution 9mL 9mL 9mL 9mL N 1 100 1 10 1 10 1 10 1 1 1 1 1.15 104 N 100 10 10 10 5mL 1.15 104 105 N 0.23 109 2.3 108 5ml 1 1.15 104 N 5 10 5mL EXPECTED NUMBER OF BACTERIA IN ORIGINAL SOLUTION Serial dilutions 0.1 mL 0.1 mL 0.1 mL 0.1 mL 2 mL 5.12 101 # of bacteria found Original solution 9.9mL 9.9mL 9.9mL 9.9mL N 1 100 1 100 1 100 1 100 1 1 1 1 5.12 101 N 100 100 100 100 2mL 5.12 101 108 N 2.56 109 2ml EXPECTED NUMBER OF BACTERIA IN ORIGINAL SOLUTION Significant Digits The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data When are Digits Significant? Non-zero digits are always significant. Thus, 22 has two significant digits, and 22.3 has three significant digits. With zeroes, the situation is more complicated: Zeroes placed before other digits are not significant; 0.046 has two significant digits. Zeroes placed between other digits are always significant; 4009 kg has four significant digits. Zeroes placed after other digits but behind a decimal point are significant; 7.90 has three significant digits. Zeroes at the end of a number are significant only if it is followed by a decimal point or underlined emphasized on the precision: 8300 has two significant digits 8300. has four significant digits 8300 has three significant digits Example: Identify number of significant digits 27.4 18.045 7600 7600. 7600 0.4003 4003 0.40030 40030 400.30 0.00403 40300 3 significant digits 5 significant digits 2 significant digits 4 significant digits 3 significant digits 4 significant digits 4 significant digits 5 significant digits 4 significant digits 5 significant digits 3 significant digits 3 significant digits Operation using significant digits Adding and subtracting – add and subtract as you normally do. For the final solution the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted. . Add the following problem 5.67 (two decimal places) 1.1 (one decimal place) 0.9378 (four decimal place) 7.7078 7.7 (one decimal place) Example - How precise can the answers to the following be expressed to? 17.142 + 2.0013 + 24.11 17.142 has 3 numbers after the decimal points 2.0013 has 4 numbers after the decimal points 24.11 has 2 numbers after the decimal points The answer could have two positions to the right of the decimal since the least precise term, 24.11, has only two positions to the right. Example: Add / Subtract Subtract: Add: 10.003 17.034 – 4.57 12.464 Final answer is 12.46 + 173.1 4 8.00003 195.00303 Final solution is 195. Subtract: 76 – 5.839 70.161 Final answer is 70. Add: 18.123 3.1 4.76 + 1.00 26.983 Final solution is 27.0 Operation using significant digits Multiplying and dividing – do the operation as you normally do. For the final solution use the least significant digits between all the numbers involved. For example: 0.000170 X 100.40 The product could be expressed with no more than three significant digits since 0.000170 has only three significant digits, and 100.40 has five. So according to the rule the product answer could only be expressed with three significant digits. Example - Indicate the number of significant digits the answer to the following would have. (I don't want the actual answer but only the number of significant digits the answer should be expressed as having.) (20.04) ( 16.0) (4.0 X 102) (20.04) has 4 significant digits ( 16.0) has 3 significant digits (4.0 X 102) has 2 significant digits Final answer will have 2 significant digits Sample problems on significant figures 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 37.76 + 3.907 + 226.4 = 319.15 - 32.614 = 104.630 + 27.08362 + 0.61 = 125 - 0.23 + 4.109 = 2.02 × 2.5 = 600.0 / 5.2302 = 0.0032 × 273 = (5.5)3 = 0.556 × (40 - 32.5) = 45 × 3.00 = 1. 268.1 (4 significant) 2. 286.54 (5 significant) 3. 132.32 (5 significant) 4. 129 (3 significant) 5. 5.0 (2 significant) 6. 114.7 (4 significant) 7. 0.87 (2 significant) 8. 1.7 x 102=170 (2 significant) 9. 4 (1 significant) 10. 1.4 x 102 (2 significant) Rounding or Precision significant digits Rules for rounding off numbers If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example, 12.6 is rounded to 13. If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example, 12.4 is rounded to 12. If the digit to be dropped is 5, and if any digit following it is not zero, the last remaining digit is increased by one. For example, 12.51 is rounded to 13. If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit is increased by one if it is odd, but left as it is if even. For example, 11.5 is rounded to 12, 12.5 is rounded to 12. This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is always rounded to the even digit. The rationale is to avoid bias in rounding: half of the time we round up, half the time we round down. Graphs – Plotting Points on the Graph – how? y x 1 3 4 6 0 -1 -2 -4 y 2 7 3 6 -2 -3 5 -6 x Decide the scale and follow within that scale setting (1=1) Graphs – Plotting Points on the Graph y x 1 3 4 6 0 -1 -2 -4 y 2 1 3 4 -2 -3 5 -1 x Decide the scale and follow within that scale setting (2=1) Graphs – Plotting Points on the Graph y x 1 3 4 0 -1 y 1 7 10 -2 -5 x 1=1 Drawing Straight Line y x 1 3 4 0 -1 y 1 7 10 -2 -5 x 1=1 Points written as ordered pair: (1, 1), (3, 7), (4, 10), (0, -2), (-1, -5) Drawing Straight Line y = 2x - 3 x 0 1 2 -1 y -3 -1 1 -5 (2, 1) (1, -1) (0, -3) (-1, -5) 1=1 Drawing Straight Line y = 5x + 7 x 0 1 2 -1 y 7 12 17 2 (0, 7) (-1, 2) 1=1 (2, 17) (1, 12) 1=2 Drawing Straight Line y = -30x + 50 x 0 1 2 -1 y 50 20 -10 +80 (1, 20) (0, 50) (-1, 80) 1=1 (2, -10) 1=10 Slope of the line Positive and negative slope POSITIVE SLOPE Slope of the line Positive and negative slope NEGATIVE SLOPE Finding slope from the known points rise slope run Rise Run Rise Run rise 3 slope run 5 Finding slope from the known points rise slope run Rise Run Rise Run rise 3 1 slope run 9 3 Finding slope using the 2 ordered pair (x1, y1) and (x2, y2) rise y2 y1 slope run x2 x1 Finding slope using the 2 ordered pair (-1, -1) and (3, 6) rise 6 (1) 6 1 7 slope run 3 (1) 4 4 RISE = 7 RUN = 4 Finding slope using the 2 ordered pair (1, -1) and (3, 6) rise 6 (1) 6 1 7 slope run 3 1 2 2 Finding slope using the 2 ordered pair (-2, -3) and (-1, 5) rise 5 (3) 53 8 slope 8 run 1 (2) 1 2 1 Finding slope using the 2 ordered pair (-1, 0) and (1, 2) rise 2 (0) 2 2 slope 1 run 1 (1) 1 1 2 Finding slope using the 2 ordered pair (0, -3) and (1, -5) rise 5 (3) 5 3 2 slope 2 run 1 (0) 1 1 Equation of straight line y = mx + b Identifying slope and y-intercepts y= mx +b x and y represents points on the graph m = Slope b = y-intercepts (0, b) ordered pair Drawing Straight Line y = 2x - 3 For this problem: Slope = 2 and y-intercept = -3 [if written as ordered pair (0, -3)] Drawing Straight Line y = 5x + 7 For this problem: Slope = 5 and y-intercept = 7 [if written as ordered pair (0, 7)] Drawing Straight Line y = -30x + 50 For this problem: Slope = -30 and y-intercept = 50 [if written as ordered pair (0, 50)] Collecting data and plotting the points Height (inches) of a child at different age (year) x 0.5 1 2 3 4 5.5 y 16 21 28 40 35 50 Year Height (1=5) (2=1) What is the child height at the age 5? What is the child height at the age 6? It is about 46 inches It is about 55 inches Interpolation and Extrapolation Definition Interpolation – When the value for dependent variable is estimated from independent variable within the data set range Extrapolation – When the value for dependent variable is estimated from independent variable out side the data set range Collecting data and plotting the points Height (inches) of a child at different age (year) x 0.5 1 2 3 4 5.5 y 16 21 28 40 35 50 From last problem! X = 5 is within the data range (0.5 – 5.5) X = 6 is outside the data range (0.5 – 5.5) Year Height (1=5) (2=1) What is the child height at the age 5? What is the child height at the age 6? It is about 46 inches - Interpolation It is about 55 inches - Extrapolation Find the equation of the line for this graph. What is the y-intercept? What is the slope of this line? Use the equation of the line y=mx+b Then write the equation of the line Find the equation of the line for this graph. What is the y-intercept? What is the slope of this line? Use the equation of the line y=mx+b Then write the equation of the line EBay, 2007 - http://moneycentral.msn.com/investor/charts/chartdl.aspx? EBay, 2007 - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 EBay, 2007 - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 Google, 2007 - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 Microsoft, 2007 - - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 Dell, 2007 - - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 Yahoo, 2007 - http://quotes.nasdaq.com/quote.dll?page=nasdaq100 Bar graph http://www.shodor.org/interactivate/activities/BarGraph/?version=1.6.0_02&b rowser=MSIE&vendor=Sun_Microsystems_Inc. Bar graph http://www.shodor.org/interactivate/activities/BarGraph/?version=1.6.0_02&b rowser=MSIE&vendor=Sun_Microsystems_Inc. Pie graph http://www.shodor.org/interactivate/activities/BarGraph/?version=1.6.0_02&b rowser=MSIE&vendor=Sun_Microsystems_Inc.

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