Chapter 1 Basic Math Skills For Nuclear Medicine

Chapter 1 Basic Math Skills For Nuclear Medicine

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

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, 600 3 56. 89 4 0. 28946 5 0. 0009538 4 20, 456 5 �

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

Rounding Numbers � Examples › Round 1. 9582 to four sig figs �The answer is 1. 958 because the 2 that was dropped is less than 5 › Round 9. 351 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 8. 057 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.

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.

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

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.

Significant Figures In Product or Quotient � Example: › Determine the product of 6. 31 x 2. 223 to the appropriate significant figure � 6. 31 x 2. 223 = 14. 02713 � 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 14. 0. In this case the 0 is a sig fig › Therefore 6. 31 x 2. 223 = 14. 0

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

Significant Figures in Sum or Difference � Example › Add 9. 392 + 1. 2 + 4. 001 + 5. 8240 and record the sum with the appropriate number of significant decimal places. › The sum is 20. 417. The number with the fewest decimal places is 1. 2, so the sum must be expressed as 20. 4

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 4 x 4 x 4, 4 has been raised to the 3 rd power, or cubed.

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

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

Roots � Examples: › Calculate the square root of 64 �Answer 64 = 8, because 82 = 64 › Convert �Answer: 506 to a simple number 506 = 22. 5 because 22. 52 = 506

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. 8. 210, 000 = 82. 1 x 105 or 821 x 104

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 � 8 2 1 0 0. �The decimal is moved 6 places to the left �The exponential notation becomes 106 �Rewrite the number as 8. 21 x 106

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 � 4. 3 7 0 0 0 0 0 so 4. 37 x 1012 = 4, 370, 000, 000

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 0. 00000075 to scientific notation � 0. 0 0 0 7 5 �The decimal point is moved 7 places to the right, so 0. 00000075 equals 7. 5 x 10 -7

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 � 0 0 0 5. 9 �Therefore 5. 9 x 10 -6 equals 0. 0000059

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 8. 904 x 105 by 1. 67 x 10 -4 › Regroup as (3. 75 x 8. 904 x 1. 67)(102 x 105 x 10 -4) › Multiply whole numbers and add exponents: (55. 7613) (10 2+5+(-4)) = 55. 7613 x 103 › Round to sig figs and reformat into standard scientific notation: 55. 8 x 103 = 5. 58 x 104

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 107 -5 = 1. 872 x 102 4. 61 x 105 4. 61 105 The answer is rounded to 3 sig figs: 1. 87 x 102

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 8. 273 x 105 and 5. 821 x 104 › Reformat one of the numbers: 8. 273 x 105 = 82. 73 x 104 Add: 82. 73 x 104 + 5. 821 x 104 88. 551 x 104

Direct and Inverse Proportions � Direct proportions: When a ratio X 1 equals a Y 2 ratio of X 2 the relationship is called a direct Y 1 proportion. � Using X 1 = X 2 Y 1 Y 2 Cross multiply to give: x 1 y 2 = x 2 y 1 � Isolate the unknown (x) which can be any element in the equation � Solve for X. �

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

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 X 1 Y 1 = X 2 Y 2 � 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%)(2 ml) = (X%)(100 ml) › Isolate and solve for X: X = (20%)(2 ml) = 0. 4% 100 ml

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, 000 uci, is better expressed and more easily manipulated as a smaller number with larger units such as 40 mci. � You can also convert from large units to smaller ones when appropriate,

Converting within the Metric System � Identify the powers of the units of measurement � Apply the inverse proportion equation X 1 Y 1 = X 2 Y 2 or X 1 = Y 2 X 2 Y 1 � Isolate and solve for X � Reformat the answer into standard scientific notation

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) 10 -6 X = 3. 6 x 1015 um The answer is already in standard scientific notation

Converting Ci to m. Ci to u. Ci or GBq to MBq to k. Bq 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 500 u. Ci to m. Ci �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 u. Ci = 0. 500 m. Ci › Convert 0. 085 m. Ci = 85 u. Ci �When converting from a large unit to a smaller one, the number increases, so the decimal is moved to the right 3 places. � 0. 0 8 5 m. Ci = 85 u. Ci › Convert 423, 000 u. Ci 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 u. Ci to Ci , which are 6 decimal places apart � 4 2 3 0 0 0 u. Ci = 0. 423 Ci

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 (m. Ci) to megabecquerel (MBq › Microcurie (u. Ci) to kilobecquerel (k. Bq)

Converting Curie and Becquerel Multiply the number of curies by the equivalent number of becquerels 1 Ci = 37 GBq 1 m. Ci = 37 MBq 1 u. Ci = 37 k. Bq � Multiply the number of becquerels by the equivalent number of curies. 1 Bq = 2. 7 x 10 -11 Ci 1 GBq = 0. 027 Ci or 27 m. Ci 1 MBq = 0. 0027 m. Ci or 27 u. Ci 1 k. Bq = 0. 027 u. Ci �

Converting Curie and Becquerel Examples: › Convert 100 u. Ci to k. Bq �If 1 u. Ci = 37 k. Bq, then (100 u. Ci)(37 k. Bq/u. Ci) = 3, 700 k. Bq or 3. 70 x 103 k. Bq › Convert 50 m. Ci to MBq �(50 m. Ci)(37 MBq/m. Ci) = 1, 850 or 1. 85 x 103 MBq

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 m. Gy � Gray to Rad › Multiply the number of Gy by the equivalent number of Rad. � 1 Gy = 100 rad � 1 m. Gy = 100 mrad

Converting between rad and gray Examples: › Convert 5 rad to grays �(5 rad)(0. 01 Gy/rad) = 0. 05 Gy or 50 m. Gy › Convert 0. 2 m. Gy to mrad �(0. 2 m. Gy)(100 mrad/m. Gy) = 20 mrad

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 m. Sv › Multiply the number of Sv by the equivalent number of rem � 1 Sv = 100 rem � 1 m. Sv = 100 mrem

Converting between Rem and Sievert Examples: › Convert 5 rem to sieverts �(5 mrem)(0. 01 Sv/rem) = 0. 05 Sv or 50 m. Sv › Convert 35 mrem to m. Sv �(35 mrem)(0. 01 m. Sv/mrem) = 0. 35 m. Sv › Convert 0. 3 Sv to rem �(0. 3 Sv)(100 rem/Sv) = 30 rem

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. 45 kg/lb) � Kilogram to pound › Lb = (kg)(2. 2 lb/kg)

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? �(12 kg)(2. 2 lb/kg) = 26 lb

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: 0 C = (o. F – 32)(5/9) � Centigrade 0 F= (0 C to fahrenhiet: x 9/5) + 32

Converting between Fahrenheit and centigrade Examples: › Convert 98. 8 0 F to 0 C �(98. 80 F – 32)(5/9) = 0 C (66. 80)(5/9) = 0 C 334/9= 37. 10 C › Convert 450 C to 0 F �(450 C x 9/5) + 32 = 0 F 81 + 32 = 1130 F

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 (10 x), and natural antilog (ex), functions on your instrument

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 10 x function key �Answer: 104 = 10, 000 › Find the natural antilog of 3. 219 (e 3. 219) using the ex function �Answer: e 3. 219 = 25

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 100 m. Ci there would be 50 m. Ci at the end of one half life, 25 m. Ci at the end of two half lives, 12. 5 and the end of three and so forth �

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

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 = Y 1 – Y 2 X 1 – X 2 � How to calculate the slope of a line: › Select two points on the line › Determine the coordinates for each point: X 1, Y 1 and X 2, Y 2 › Apply the coordinates to the equation: Y 1 – Y 2 X 1 – X 2

Slope Calculations Example: › Find the slope of a line having the following coordinates. �(250, 62) (350, 84) � 62 – 84 = -22 = 0. 22 250 - 350 -100