1. Chapter 1 Basic Math Skills For Nuclear Medicine

2. Objectives
Explain how to do significant figures and rounding of numbers
Determining powers and exponents
How to do square roots
Explanation on how to do significant figures
Definition of mathematical operations using exponential numbers as well as direct and inverse proportions and converting within the metric system
Explanation on how to convert curie to becquerel, Rad to gray, Rem to sievert, pound to kilogram, and fahrenheit to centigrade
Determining Logs, natural logs and antilogs
Discuss how to solve equations with an unknown in the exponent
Explain how to graph on linear and semi log papers as well as slope calculations

3. Significant figures Integers from 1-9 are always significant
If 0 is a space holder it is not a sig fig
When 0 is sandwiched between other integers than it is a sig fig
Actual Numbers Number of Sig Figs
25,
20,

4. Rounding Numbers Select digits based on potential for accuracy
Examine the first digit to be dropped . This is the first number to the right of the retained number
If the first digit to be dropped is less than 5, the retained number is left unchanged.
If the first number to be dropped is less greater than 5, increase the retained by a number of 1 unit
If the first digit to be dropped is a 5, round down where the retained number is even and round up if the retained number is odd
If a decimal place is rounded up to a 0, drop the zero

5. Rounding Numbers Examples Round 1.9582 to four sig figs
The answer is because the 2 that was dropped is less than 5
Round to two sig figs
The answer is 9.4 because the first digit dropped is a 5 and the retained number is odd. The retained number is therefore increased by 1 unit
Round to two sig figs
The answer is 8.0 because the first digit dropped was a 5 and the retained number is even. The retained number therefore remains unchanged.

6. Significant figures and Mathematical Operations
Each rounding operation adds a small element of error or accuracy.
You must therefore decide if rounding off numbers as you proceed through multiple mathematical operations will add significant error to your results.
The more steps in the operation , the more error rounding is used at every step.

7. Significant figures and Mathematical Operations
Example
Perform the following operation without rounding until the final answer
(123.7)(35) = 4,329.5 = 0.038
(1395)(82) 114,390

8. Significant Figures In Product or Quotient
The product or quotient of an operation can only have as many sig figs as the least accurate number in the equation. If any number in the equation has only one sig fig, so then must be the answer. If the least accurate number contains three sig figs, then so must the answer.

9. Significant Figures In Product or Quotient
Example:
Determine the product of 6.31 x to the appropriate significant figure
6.31 x =
Since 6.31 is the least accurate number, with only 3 sig figs, than the product of the operation can only assure accuracy to three sig figs. The answer should therefore be recorded as In this case the 0 is a sig fig
Therefore 6.31 x = 14.0

10. Significant Figures in Sum or Difference
The sum or difference of a mathematical operation must have the same number of significant decimal places as the least accurate number in the operation
If there are no decimal places the answer is rounded to the number of sig figs in the least accurate number

11. Significant Figures in Sum or Difference
Example
Add and record the sum with the appropriate number of significant decimal places.
The sum is The number with the fewest decimal places is 1.2, so the sum must be expressed as 20.4

12. Powers and Exponents
When a number is multiplied by itself one or more times, it is said to be raised to a given power, If 4 is multiplied by 4, than 4 has been raised to the second power or squared. For 4x4x4, 4 has been raised to the 3rd power, or cubed.

13. Powers and Exponents Examples: Convert 62 to a whole number
Answer: 36
Convert 54 to a whole number
Answer : 625
Convert 39 to a whole number
Answer : 19683

14. Roots
The square root of a value is the number that was raised to the power of two to create the value.
The cube root gives the value that was raised to the third power to produce the number that appears within the sign

15. Roots Examples: Calculate the square root of 64
Answer = 8, because 82 = 64
Convert to a simple number
Answer: = 22.5 because = 506

16. Scientific Notation
Scientific notation uses exponents to make the recording and manipulation of very large and very small numbers more convenient. Typically, the number is expressed as a single digit to the left of the decimal and 1 or more digits to the right, followed by 10 raised to the appropriate power.
Example:
A. 8,210,000 = 8.21 x 106
B ,000 = 82.1 x 105 or 821 x 104

17. Scientific Notation (Whole number to scientific notation)
Move the decimal point to the left until a single digit lies to the left of the decimal point
Count the number of places the decimal was moved
Use the number of places as the exponent of 10
Re write the number
Example:
Convert 8,210,000 to standard scientific notation
The decimal is moved 6 places to the left
The exponential notation becomes 106
Rewrite the number as 8.21x 106

18. Scientific Notation (scientific notation to whole number)
Move the decimal to the right the number of places to which 10 has been raised
Add zeros if necessary
Examples:
Convert 4.37 x 1012 to a whole number
The decimal must be moved 12 places to the right, which is the opposite direction the decimal is moved in order to create the scientific notation. A sufficient number of zeros is added to the number to accomplish this
so 4.37 x 1012 = 4,370,000,000,000

19. Scientific Notation (decimal to scientific notation)
Move the decimal point to the right until a single digit lies to the left of the decimal point
Count the number of places the decimal was moved
Use the number of places as the negative exponent of 10. The negative denotes that the number was created by moving the decimal to the right .
Rewrite the number
Example:
Convert to scientific notation
The decimal point is moved 7 places to the right, so equals 7.5 x 10-7

20. Scientific Notation (scientific notation to simple decimal form)
Move the decimal to the left the number of places to which ten has been raised
Add zeros between the decimal point and the first digit if necessary
Example:
Convert 5.9 x 10-6 to a simple decimal
The decimal must be moved 6 places to the left, which is the opposite direction the decimal was moved in order to create the scientific notation. A sufficient number of zeros is added to the number to accomplish this
Therefore 5.9 x 10-6 equals

21. Mathematical Operations using Exponential Numbers (multiplication)
Apply the rules of exponents ( am)(an)= a m+n
Group the whole numbers together and the exponents together
Multiply whole numbers by whole numbers
Add the exponents
Round off to the appropriate sig figs
Reformat the answer into standard scientific notation.
Example:
Multiply 3.75 x 102 by x 105 by 1.67 x 10-4
Regroup as (3.75 x x 1.67)(102 x 105 x 10 -4)
Multiply whole numbers and add exponents: ( ) ( (-4)) = x 103
Round to sig figs and reformat into standard scientific notation:
55.8 x 103 = 5.58 x 104

22. Mathematical Operations using Exponential Numbers (division)
Apply the rules of exponents : am = a m-n an
Group the whole numbers together and exponents together
Divide whole numbers by whole numberss
Subtract the exponents
Round off to the appropriate sig fig
Reformat the answer into scientific notation
Example:
Divide
8.63 x 107 = 8.63 x = x 102
4.61 x
The answer is rounded to 3 sig figs: 1.87 x 102

23. Mathematical Operations using Exponential Numbers (adding and subtracting)
Reformat the numbers so the exponents are identical
Add or subtract
Round off to the appropriate sig fig
Reformat the answer into standard scientific notation
Example
Add x 105 and x 104
Reformat one of the numbers: x 105 = x 104
Add: x 104
x 104
x 104

24. Direct and Inverse Proportions
Direct proportions: When a ratio X1 equals a Y2
ratio of X2 the relationship is called a direct Y1
proportion.
Using X1 = X2
Y Y2
Cross multiply to give: x1y2 = x2y1
Isolate the unknown (x) which can be any element in the equation
Solve for X.

25. Direct and Inverse Proportions
Example:
If a 200uCi point source produces 6,000 cps, how many cps will be produced by a 500uCi source?
200uci = 500uci
6,000cps X cps
Cross-Multiply: (200uCi)(X cps) = (500uci)(6,000cps)
Isolate and solve for X: X= (500uci)(6,000cps)
200uci
=15,000cps

26. Inverse Proportions
Inverse proportions: proportions do not always vary directly. In some instances, one element decreases as the other increases and vice versa. This is an inverse proportion.
Using X1Y1 = X2Y2
Isolate the unknown (X), which can be any element in the equation
Solve for X
Example:
Two ml of solution of 20% HCL are placed in a volumetric flask and diluted to 100 ml. What is the percent concentration of the diluted solution?
(20%)(2ml) = (X%)(100ml)
Isolate and solve for X: X = (20%)(2ml) = 0.4%
100ml

27. Converting within the Metric System
Nuclear medicine utilizes the metric system for all measurements. The technologist needs to be able to convert numbers within the metric system.
A large number with small units, such as 40,000uci, is better expressed and more easily manipulated as a smaller number with larger units such as 40mci.
You can also convert from large units to smaller ones when appropriate,

28. Converting within the Metric System
Identify the powers of the units of measurement
Apply the inverse proportion equation
X1Y1 = X2Y2 or X1 = Y2
X2 Y1
Isolate and solve for X
Reformat the answer into standard scientific notation

29. Converting within the Metric System
Example:
Convert proportional method:
Identify the powers of the units of measurement
Apply the inverse proportion and solve
(3.6 x 1011)(10-2) = (x)(10-6)
X = (3.6 x 1011)(10-2) = 3.6 x 1011+(-2) = 3.6 x 109-(-6)
X = 3.6 x 1015um
The answer is already in standard scientific notation

30. Converting Ci to mCi to uCi or GBq to MBq to kBq
Increase or decrease the number of decimal places to the right, or the number of zeros to the left by three or six, depending on the change in units.
Examples:
Convert 500uCi to mCi
When converting from a small unit to a larger one, the number decreases, so the decimal is moved to the left 3 places.
5 0 0 uCi = mCi
Convert mCi = 85 uCi
When converting from a large unit to a smaller one, the number increases, so the decimal is moved to the right 3 places.
mCi = 85uCi
Convert 423,000 uCi to Ci
The number must become smaller, so the decimal is moved left. In this case the decimal is moved 6 places as you are converting from uCi to Ci , which are 6 decimal places apart
uCi = Ci

31. Converting Curie and Becquerel
According to the international system (SI) of units the we accepted as standard measurements by the international Commission on Radiological Units (ICRU) in 1975, the Becquerel (Bq) is the preferred unit of measurement for radioactivity. It is equivalent to 1 disintegration per second (dps)
The conventional curie units are usually converted to Becquerel as follows.
Curies (Ci) to gigabecquerel (GBq)
Millicurie (mCi) to megabecquerel (MBq
Microcurie (uCi) to kilobecquerel (kBq)

32. Converting Curie and Becquerel
Multiply the number of curies by the equivalent number of becquerels
1 Ci = 37 GBq
1mCi = 37 MBq
1 uCi = 37 kBq
Multiply the number of becquerels by the equivalent number of curies.
1 Bq = 2.7 x Ci
1GBq = Ci or27 mCi
1 MBq = mCi or 27uCi
1 kBq = uCi

33. Converting Curie and Becquerel
Examples:
Convert 100 uCi to kBq
If 1 uCi = 37 kBq, then (100 uCi)(37 kBq/uCi) = 3,700 kBq or 3.70 x 103 kBq
Convert 50 mCi to MBq
(50 mCi)(37MBq/mCi) = 1,850 or 1.85 x 103 MBq

34. Converting between rad and gray
The traditional unit for absorbed dose is the rad (radiation absorbed dose). The international system (SI) unit is the gray (Gy)
Rad to Gray:
Multiply the number of rad by the equivalent number of Gy.
1 rad = 0.01 Gy
1 mrad = 0.01 mGy
Gray to Rad
Multiply the number of Gy by the equivalent number of Rad.
1 Gy = 100 rad
1 mGy = 100 mrad

35. Converting between rad and gray
Examples:
Convert 5 rad to grays
(5 rad)(0.01 Gy/rad) = 0.05 Gy or 50 mGy
Convert 0.2 mGy to mrad
(0.2 mGy)(100mrad/mGy) = 20 mrad

36. Converting between Rem and Sievert
The traditional unit for dose equivalent is the rem (roentgen equivalent man). The international system (SI) unit is the sievert (SV)
Multiply the number of rem by the equivalent number of Sv
1 rem = 0.01 Sv
1 mrem = 0.01 mSv
Multiply the number of Sv by the equivalent number of rem
1 Sv = 100 rem
1 mSv = 100 mrem

37. Converting between Rem and Sievert
Examples:
Convert 5 rem to sieverts
(5mrem)(0.01 Sv/rem) = 0.05 Sv or 50 mSv
Convert 35 mrem to mSv
(35mrem)(0.01 mSv/mrem) = 0.35 mSv
Convert 0.3 Sv to rem
(0.3 Sv)(100 rem/Sv) = 30 rem

38. Converting between Pounds and kilogram
The traditional US unit for human weight is the pound (lb), while the international system (SI) unit is the kilogram
1 lb = 0.45 kg
1 kg = 2.2 lb
Pound to Kilogram:
Kg = (lb)(0.45kg/lb)
Kilogram to pound
Lb = (kg)(2.2 lb/kg)

39. Converting between Pounds and kilogram
Examples:
Convert 155 lb to kg
(155 lb)(0.45 kg/lb) = 70 kg
If a child weighs 12 kg, how much does he weigh in pounds?
(12kg)(2.2 lb/kg) = 26lb

40. Converting between Fahrenheit and centigrade
On the fahrenheit (f) scale, water freezes at 320 and boils at 100 degrees. To convert between the scales use the calculations listed below.
Fahrenheit to centigrade:
0C = (oF – 32)(5/9)
Centigrade to fahrenhiet:
0F= (0C x 9/5) + 32

41. Converting between Fahrenheit and centigrade
Examples:
Convert F to 0C
(98.80F – 32)(5/9) = 0C (66.80)(5/9) = 0C 334/9= 37.10C
Convert 450C to 0F
(450 C x 9/5) + 32 = 0F = 1130F

42. Logs, Natural Logs and Antilogs
The logarithm (log) of a number is the power to which a base must be raised to produce that number.
Refer to the instruction booklet that came with your scientific calculator to determine how to utilize the (LOG), natural log (LN), antilog (10x), and natural antilog (ex), functions on your instrument

43. Logs, Natural Logs and Antilogs
Examples:
Find the log of 10,000
Answer: log 10,000 = 4 Note that 10,000 is written in exponential form as 104
Find the Ln of 25.
Answer: Ln 25 = 3.219
Find the antilog of 4(104) using the 10x function key
Answer: 104 = 10,000
Find the natural antilog of (e3.219) using the ex function
Answer: e = 25

44. Graphing on linear and semi-log papers
The horizontal axis of a graph is called the abscissa and is designated the x-axis. The vertical axis is called the ordinate and is refered to as the y-axis
When drawn on linear paper, a graph showing elapsed time (x) versus the percent of radioactivity remaining (y) will produce a curvier rather than straight line. This is because radioactive decay is not a linear process.
If it were, the same amount of radioactivity would be lost in each half life
In reality , the same percent of radioactivity decays within each half life, Ex 50% of the amount present at the beginning of each half life. Therefore if you started with 100mCi there would be 50mCi at the end of one half life, 25 mCi at the end of two half lives, 12.5 and the end of three and so forth

45. Determining Values between the discrete data point on a graph
Read from the point of interest on one axis to the graphed line
Now read directly across to the opposing axis to obtain the corresponding data, in this case 10% of the activity remaining

46. Slope Calculations
The slope of a line is the mathematical expression of its steepness. A positive value for the slope indicates the line is rising. A negative value indicates a downward curve.
The following equation is used to determine the slope of a line.
Slope = Y1 – Y2
X1 – X2
How to calculate the slope of a line:
Select two points on the line
Determine the coordinates for each point: X1, Y1 and X2, Y2
Apply the coordinates to the equation: Y1 – Y2

47. Slope Calculations Example:
Find the slope of a line having the following coordinates.
(250, 62) (350, 84)
62 – 84 = -22 = 0.22