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Clinical Calculation5th 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/weather/CityForecast.aspx?txtSearchCriteria=Oklahoma&sc=N
http://weather.msn.com/
Converting Fahrenheit to Celsius
CC
08.1
0
8.1
3232
32F = _________________C
8.1
32F
C
Converting Fahrenheit to Celsius
CC
1008.1
180
8.1
32212
212F = _________________C
8.1
32F
C
Converting Fahrenheit to Celsius
CC
8.378.1
68
8.1
32100
100F= _________________C
8.1
32F
C
Converting Fahrenheit to Celsius
CC
2.28.1
4
8.1
3228
28F = ________________C
8.1
32F
C
Converting Celsius to Fahrenheit
328.1 CF
50 C = _________________ F
FF 413258.1
Converting Celsius to Fahrenheit
328.1 CF
500 C = _________________ F
FF 12232508.1
Converting Celsius to Fahrenheit
328.1 CF
250 C = _________________ F
FF 7732258.1
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 am02 00 - 2:00 am03 00 - 3:00 am04 00 - 4:00 am05 00 - 5:00 am06 00 - 6:00 am07 00 - 7:00 am08 00 - 8:00 am09 00 - 9:00 am10 00 - 10:00 am11 00 - 11:00 am12 00 - 12 noon
13 00 - 1:00 pm14 00 - 2:00 pm15 00 - 3:00 pm16 00 - 4:00 pm17 00 - 5:00 pm18 00 - 6:00 pm19 00 - 7:00 pm20 00 - 8:00 pm21 00 - 9:00 pm22 00 - 10:00 pm23 00 - 11:00 pm24 00 - midnight
Example: on the hour Example: 10 minutes past
24 Hour Clock AM / PM 24 Hour Clock AM / PM
0100 1:00 AM 0010 12:10 AM
0200 2:00 AM 0110 1:10 AM
0300 3:00 AM 0210 2:10 AM
0400 4:00 AM 0310 3:10 AM
0500 5:00 AM 0410 4:10 AM
0600 6:00 AM 0510 5:10 AM
0700 7:00 AM 0610 6:10 AM
0800 8:00 AM 0710 7:10 AM
0900 9:00 AM 0810 8:10 AM
1000 10:00 AM 0910 9:10 AM
1100 11:00 AM 1010 10:10 AM
1200 12 Noon 1110 11:10 AM
1300 1:00 PM 1210 12:10 PM
1400 2:00 PM 1310 1:10 PM
1500 3:00 PM 1410 2:10 PM
1600 4:00 PM 1510 3:10 PM
1700 5:00 PM 1610 4:10 PM
1800 6:00 PM 1710 5:10 PM
1900 7:00 PM 1810 6:10 PM
2000 8:00 PM 1910 7:10 PM
2100 9:00 PM 2010 8:10 PM
2200 10:00 PM 2110 9:10 PM
2300 11:00 PM 2210 10:10 PM
2400 12:00 PM 2310 11:10 PM
24-Hour Clock Conversion Table
12hr Time 24hr Time 12 am (midnight) 0000hrs 1 am 0100hrs 2 am 0200hrs3 am 0300hrs 4 am 0400hrs 5 am 0500hrs 6 am 0600hrs 7 am 0700hrs 8 am 0800hrs 9 am 0900hrs 10 am 1000hrs 11 am 1100hrs 12 pm (noon) 1200hrs 1 pm 1300hrs 2 pm 1400hrs 3 pm 1500hrs 4 pm 1600hrs 5 pm 1700hrs 6 pm 1800hrs 7 pm 1900hrs 8 pm 2000hrs 9 pm 2100hrs 10 pm 2200hrs 11 pm 2300hrs
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 :
The first number 1.23 is called the coefficient.
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.
111023.1
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 !
Do you recognize this number, 0.000 000 000 753 kg. ? This is the mass of a dust particle!
8100.3
101053.7
Now it is your turn. Express the following numbers in their equivalent scientific notational form:
1. 123,876.3
2. 1,236,840.
3. 4.22
4. 0.000000000000211
5. 0.000238
6. 9.10
5101.238763 .1 6101.236840 .2
0104.22 .3 -13102.11 .4
-4102.38 .5 0109.1 .6
Now it is your turn. Express the following numbers in their equivalent standard notational form:
1. 566.3
2. 123,000.
3. 70,020,000
4. 0.918
5. 7.18
6. 80,000
2105.663 .1 5101.23 .2
7107.002 .3 -1109.18 .4 0107.18 .5
4108.0 .6
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/dilution.mov
Serial dilutions -
410
1Original solution
9mL 9mL 9mL 9mL
1 mL 1 mL 1 mL 1 mL
10
1
10
1
10
110
1
Serial dilutions -
510
1
Original solution
9.9 mL 9mL 9mL 9mL
0.1 mL 1 mL 1 mL 1 mL
210
1
100
1
10
1.0
10
1
10
110
1
310
1410
1
Serial dilutions -
81
1
3
14
Original solution
2 mL 2mL 2mL 2mL
1 mL 1 mL 1 mL 1 mL
3
13
1
3
13
1
9
1
3
12
27
1
3
13
Build Dilution ratio of 1:16 using 4 water blanks provided
16
1
2
14
Original solution
3 mL 3 mL 3 mL 3 mL
3 mL 3 mL 3 mL 3 mL
2
12
1
2
12
1
4
1
2
12
8
1
2
13
Build Dilution ratio of 1:104 using 4 water blanks
410
1
Original solution
9 mL 9 mL 9 mL 9 mL
1 mL 1 mL 1 mL 1 mL
10
1
10
110
110
1
210
1310
1
Build Dilution ratio of 1:104 using 3 water blanks
Original solution
9.9 mL 9 mL 9 mL
0.1 mL 1 mL 1 mL
100
1
10
1.0
10
110
1
1000
1
10
13
10000
1
10
14
Build Dilution ratio of 1:104 using 2 water blanks provided
Original solution
9.9 mL 9.9 mL
0.1 mL 0.1 mL
100
1
10
1.0
410
1100
1
10
1.0
Build Dilution ratio of 1:27 using water blanks provided
Original solution
10 mL 10 mL 10 mL
5 mL 5 mL 5 mL
3
13
1
3
1
9
1
3
12
27
1
3
13
Serial dilutions -
mLN
1
1025.3
10
1
10
1
10
1
10
1 3
743 1025.3101025.3 N
Original solution
9mL 9mL 9mL 9mL
1 mL 1 mL 1 mL 1 mL1 mL
N
31025.3
10
1
10
1
10
1
10
1
EXPECTED NUMBER OF BACTERIA IN ORIGINAL SOLUTION
# of bacteria found
mLN
1
1025.3
10
1 3
4
Serial dilutions -
mLN
5
1015.1
10
1
10
1
10
1
100
1 4
8954
103.21023.05
101015.1
mlN
Original solution
9mL 9mL 9mL 9mL
0.1 mL 1 mL 1 mL 1 mL5 mL
N
41015.1
100
1
10
1
10
1
10
1
EXPECTED NUMBER OF BACTERIA IN ORIGINAL SOLUTION
# of bacteria found
mLN
5
1015.1
10
1 4
5
Serial dilutions -
mLN
2
1012.5
100
1
100
1
100
1
100
1 1
Original solution
9.9mL 9.9mL 9.9mL 9.9mL
0.1 mL 0.1 mL 0.1 mL 0.1 mL2 mL
N
11012.5
100
1
100
1
100
1
100
1
981
1056.22
101012.5
mlN EXPECTED NUMBER OF BACTERIA
IN ORIGINAL SOLUTION
# of bacteria found
Significant DigitsThe 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 problem5.67 (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:
17.034
– 4.57
12.464
Add:
10.003
173.1
4
+ 8.00003
195.00303
Subtract:
76
– 5.839
70.161
Add:
18.123
3.1
4.76
+ 1.00
26.983
Final answer is 12.46Final solution is 195.
Final answer is 70.
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. 37.76 + 3.907 + 226.4 =
2. 319.15 - 32.614 =
3. 104.630 + 27.08362 + 0.61 =
4. 125 - 0.23 + 4.109 =
5. 2.02 × 2.5 =
6. 600.0 / 5.2302 =
7. 0.0032 × 273 =
8. (5.5)3 =
9. 0.556 × (40 - 32.5) =
10. 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?
1
3
4
6
0
-1
-2
-4
x y
2
7
3
6
-2
-3
5
-6
x
y
Decide the scale and follow within that scale setting (1=1)
Graphs – Plotting Points on the Graph
1
3
4
6
0
-1
-2
-4
x y
2
1
3
4
-2
-3
5
-1
x
y
Decide the scale and follow within that scale setting (2=1)
Graphs – Plotting Points on the Graph
1
3
4
0
-1
x y
1
7
10
-2
-5
x
y
1=1
Drawing Straight Line
1
3
4
0
-1
x y
1
7
10
-2
-5
x
y
1=1
Points written as ordered pair: (1, 1), (3, 7), (4, 10), (0, -2), (-1, -5)
Drawing Straight Line y = 2x - 3
0
1
2
-1
x y
-3
-1
1
-5
1=1
(2, 1)
(1, -1)
(-1, -5)
(0, -3)
Drawing Straight Line y = 5x + 7
0
1
2
-1
x y
7
12
17
2
1=1
1=2
(-1, 2)
(0, 7)
(1, 12)
(2, 17)
Drawing Straight Line y = -30x + 50
0
1
2
-1
x y
50
20
-10
+80
1=10
1=1
(-1, 80)
(0, 50)
(1, 20)
(2, -10)
Slope of the linePositive and negative slope
POSITIVE SLOPE
Slope of the linePositive and negative slope
NEGATIVE SLOPE
Finding slope from the known points
run
riseslope
Rise
Run
5
3
run
riseslope
RiseRun
Finding slope from the known points
run
riseslope
RiseRun
3
1
9
3
run
riseslope
RiseRun
Finding slope using the 2 ordered pair (x1, y1) and (x2, y2)
12
12
xx
yy
run
riseslope
Finding slope using the 2 ordered pair (-1, -1) and (3, 6)
4
7
4
16
)1(3
)1(6
run
riseslope
RISE = 7
RUN = 4
Finding slope using the 2 ordered pair (1, -1) and (3, 6)
2
7
2
16
13
)1(6
run
riseslope
Finding slope using the 2 ordered pair (-2, -3) and (-1, 5)
81
8
21
35
)2(1
)3(5
run
riseslope
Finding slope using the 2 ordered pair (-1, 0) and (1, 2)
12
2
11
2
)1(1
)0(2
run
riseslope
Finding slope using the 2 ordered pair (0, -3) and (1, -5)
21
2
1
35
)0(1
)3(5
run
riseslope
Equation of straight line y = mx + bIdentifying slope and y-intercepts
y = m x + bx 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 + 50For this problem:
Slope = -30 and y-intercept = 50 [if written as ordered pair (0, 50)]
Collecting data and plotting the pointsHeight (inches) of a child at different age (year)
0.5
1
2
3
4
5.5
x y
16
21
28
40
35
50
Year
(2=1)Height
(1=5)
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 pointsHeight (inches) of a child at different age (year)
0.5
1
2
3
4
5.5
x y
16
21
28
40
35
50
Year
(2=1)Height
(1=5)
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
X = 5 is within the data range (0.5 – 5.5)
X = 6 is outside the data range (0.5 – 5.5)
From last problem!
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
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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&browser=MSIE&vendor=Sun_Microsystems_Inc.
Bar graph - http://www.shodor.org/interactivate/activities/BarGraph/?version=1.6.0_02&browser=MSIE&vendor=Sun_Microsystems_Inc.
Pie graph http://www.shodor.org/interactivate/activities/BarGraph/?version=1.6.0_02&browser=MSIE&vendor=Sun_Microsystems_Inc.
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