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

 0C
1.8
1.8
Converting Fahrenheit to Celsius
212F = _________________C
F  32
C
1. 8
212  32 180
C

 100C
1.8
1.8
Converting Fahrenheit to Celsius
100F= _________________C
F  32
C
1. 8
100  32 68
C

 37.8C
1.8
1.8
Converting Fahrenheit to Celsius
28F = ________________C
F  32
C
1. 8
28  32  4
C

 2.2C
1.8
1.8
Converting Celsius to Fahrenheit
50 C = _________________ F
F  1.8C  32
F  1.85  32  41F
Converting Celsius to Fahrenheit
500 C = _________________ F
F  1.8C  32
F  1.850  32  122F
Converting Celsius to Fahrenheit
250 C = _________________ F
F  1.8C  32
F  1.825  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
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
Example: on the hour
24 Hour Clock
Example: 10 minutes past
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
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
11
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.
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. 1.238763 10
6
2. 1.236840 10
5
1.
2.
3.
4.
5.
6.
123,876.3
1,236,840.
4.22
0.000000000000211
0.000238
9.10
3. 4.22 10
4. 2.1110
0
-13
5. 2.38 10
-4
6. 9.110
0
Now it is your turn. Express the following numbers in their equivalent standard
notational form:
1. 5.663 10
5
2. 1.23 10
3. 7.002 10
-1
4. 9.18 10
5. 7.18 10
6. 8.0  10
4
2
1.
2.
3.
7
4.
5.
6.
0
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/di
lution.mov
Serial dilutions -
1 mL
Original
solution
1 mL
9mL
1
10
1 mL
9mL
1
10
1 mL
9mL
1
10
9mL
1
10
1
4
10
Serial dilutions -
0.1 mL
Original
solution
1 mL
9.9 mL
0 .1
1
1

 2
10 100 10
1 mL
1 mL
9mL
9mL
9mL
1
10
1
10
1
10
1
10 3
1
10 4
1
10 5
Serial dilutions -
1 mL
Original
solution
1 mL
1 mL
1 mL
2 mL
2mL
2mL
1
3
1
3
1
3
1 1

2
3
9
1
1

3
3
27
2mL
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
1
2
1
2
3 mL
3 mL
3 mL
Original
solution
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
Original
solution
1 mL
1 mL
1 mL
9 mL
9 mL
9 mL
9 mL
1
10
1
10
1
10
1
10
1
10 2
1
10 3
1
10 4
Build Dilution ratio of 1:104 using 3 water blanks
0.1 mL
1 mL
9.9 mL
Original
solution
0 .1
1

10 100
1 mL
9 mL
9 mL
1
10
1
10
1
1

103 1000
1
1

10 4 10000
Build Dilution ratio of 1:104 using 2 water blanks provided
0.1 mL
0.1 mL
9.9 mL
Original
solution
0 .1
1

10 100
9.9 mL
0 .1
1

10 100
1
4
10
Build Dilution ratio of 1:27 using water blanks provided
5 mL
5 mL
10 mL
5 mL
10 mL
10 mL
Original
solution
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
N
1
10
1
10
1
10
1 1 1 1 3.25 103
N    
10 10 10 10
1mL
N  3.25 103 10 4  3.25 107
9mL
1
10
1
3.25 103
N 4 
10
1mL
EXPECTED NUMBER OF BACTERIA
IN ORIGINAL SOLUTION
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 1.15 104
N 5 
10
5mL
1.15 10 4 105
N
 0.23 109  2.3 108
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
N
9.9mL
9.9mL
9.9mL
1
100
1
100
1
100
9.9mL
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
5.67
1.1
0.9378
7.7078
problem
(two decimal places)
(one decimal place)
(four decimal place)
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
173.1
– 4.57
4
12.464
+
8.00003
195.00303
Final answer is 12.46
Final solution is 195.
Subtract:
Add:
76
18.123
– 5.839
3.1
70.161
Final answer is 70.
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
y
1
2
3
7
4
3
6
6
0
-2
-1
-3
-2
5
-4
-6
Decide the scale and follow within that scale setting (1=1)
x
Graphs – Plotting Points on the Graph
y
x
y
1
2
3
1
4
3
6
4
0
-2
-1
-3
-2
5
-4
-1
Decide the scale and follow within that scale setting (2=1)
x
Graphs – Plotting Points on the Graph
y
x
y
1
1
3
7
4
10
0
-2
-1
-5
x
1=1
Drawing Straight Line
y
x
y
1
1
3
7
4
10
0
-2
-1
-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
y
0
-3
1
-1
2
1
-1
-5
(2, 1)
(1, -1)
(0, -3)
(-1, -5)
1=1
Drawing Straight Line y = 5x + 7
x
y
0
7
1
12
2
17
-1
2
(2, 17)
(1, 12)
(0, 7)
(-1, 2)
1=1
1=2
Drawing Straight Line y = -30x + 50
x
y
0
50
1
20
2
-10
-1
+80
(-1, 80)
(0, 50)
(1, 20)
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)
53 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
y
0.5
16
1
21
2
28
3
40
4
35
5.5
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 points
Height (inches) of a child at different age (year)
x
From last problem!
y
0.5
16
1
21
2
28
3
40
4
35
5.5
50
X = 5 is within the data range
(0.5 – 5.5)
X = 6 is outside the data range
(0.5 – 5.5)
Year
Height
(2=1)
(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
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
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Microsoft, 2007 - - http://quotes.nasdaq.com/quote.dll?page=nasdaq100
Dell, 2007 - - http://quotes.nasdaq.com/quote.dll?page=nasdaq100
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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.