Basics Math Skills for Nuclear Medicine Technology :By: Jessenia Ramirez Basics Math Skills for Nuclear Medicine Technology


Significant Figures and Rounding Numbers :Significant Figures and Rounding Numbers Significant figures are units that reflect the accuracy of a number For example, if you have 15 tourniquets, the number 15 has two significant figures Now, lets say your NMT department has done 10 bone scans today, this number has only one significant figure Integers from 1 through 9 are always significant


Cont’d :Cont’d If a 0 is a place holder, as in 600 or 0.003, then it is NOT a significant figure When a 0 is in between integers, 305, then it is considered a significant figure When a 0 is the last digit of an accurate measurement such as the readout in the dose calibrator, 14.0 mCi, then it is a significant number


How to Determine How Many Significant Figures Should Be Used :How to Determine How Many Significant Figures Should Be Used There are three factors that need to be considered: How much accuracy can actually be achieved? How much accuracy is actually required, either by necessity or regulation? How much error can be introduced without affecting the quality of the results?


How to Round Numbers :How to Round Numbers Determine what your rounding digit is and look to the right side of it. If the digit is than 5, your rounding digit rounds up by one number. All digits that are on the right hand side of the requested rounding digit will become 0


Examples :Examples Round 9.351 to two significant figures The answer is 9.4 because the first digit dropped is a 5 and the retained number is odd. So therefore the retained number is increased by 1 Round 0.07685 The answer is 0.077. Since the 0 to the right of the decimal is a place holder, you drop the 8 and the 5 to produce a number with two significant figures. The retained number is rounded up because the first digit dropped, the 8, is greater than 5


Significant Figures and Mathematical Operations :Significant Figures and Mathematical Operations How to minimize error when rounding numbers during multiple mathematical steps Round each intermediate answer so it retains at least one more significant figure than the final answer Example: Always try and perform the operation without rounding until the final answer


How to Determine the Number of Significant Figures in a Product :How to Determine the Number of Significant Figures in a Product The product of an operation can only have as many significant figures as the least accurate number in the equation. If any number in the equation has 4 significant figures so must the answer


Example :Example Determine product of 5.22 x 3.123 to the appropriate significant figure 5.22 x 3.123= 16.30206 Since 5.22 is the least accurate number, with only 3 significant figures than the product of the operation must also have 3 significant figures, therefore the answer is 16.3


Powers and Exponents :Powers and Exponents When a number is multiplied by itself one or more times, it is said to be raised to the given power. Example: 3 x 3 x 3= 33 = 27 The power is expressed as an exponent or index Remember: 33 means 3 x 3 x 3 NOT 3 multiplied by 3


Roots :Roots The square root of a value is the number that was raised to the power of two to create the value. Example: x2 A scientific calculator will have a x and a x y function. The x function allows you to calculate the square root, while the x y function allows you to calculate any root


How to Calculate the Root of a Number :How to Calculate the Root of a Number Calculate the square root of 81 Answer: 81= 9, because 92=81 Convert 506 to a simple number Answer: 506= 22.5, because 22.5 2 =506


Scientific Notation :Scientific Notation Scientific notation uses exponents to make the recording and manipulation of very large and very small numbers more convenient Example: 5,110,000= 5.11 x 106


Converting a Whole Number to SN :Converting a Whole Number to SN Move the decimal point to the left until a single digit lies to the left of the decimal point Count the # of places the decimal was moved Use the # of places as the exponents of 10 Rewrite the #


Examples :Examples Convert 5,110,000 to scientific notation 1. 5 1 1 0 0 0 0. 2. The decimal is moved 6 places to the left 3. The exponential notation becomes 10 6 4. Rewrite the # as 5.11 x 10 6


Convert a Number in SN to a Whole Number :Convert a Number in SN to a Whole Number Move the decimal to the right the number of places to which 10 has been raised Add zeros if necessary Examples: 3.11 x 10 5 to a whole # 3 .1 1 0 0 0 The decimal is moved 5 places to the right, requiring the addition of 3 zeros, so 3.11 x 10 5 equals 311,000


Covert a Decimal to SN :Covert a Decimal to SN Move decimal point to the right until a single digit lies to the left of the decimal point Count the # of places the decimal was moved Use the # of places as the negative exponent of 10. The negative denotes that the # was created by moving the decimal to the right Rewrite the #


Examples :Examples Convert 0.00236 to SN 0 . 0 0 2 3 6 The decimal point is moved 3 places to the right, so 0.00236 equals 2.36 x 10 -3


Convert a Number in SN to a Decimal :Convert a Number in SN to a Decimal 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 Examples: Convert 5.9 x 10-6 to a decimal The decimal must be moved 6 places to the left. A sufficient number of zeros is added to the number 0 0 0 0 0 5 . 9= 5.9 x 10-6 equals 0.0000059


Mathematical Operations Using Exponential Numbers :Mathematical Operations Using Exponential Numbers Multiplication: (a m )(an)= a m+n Example: (2.5x10-1) x (5.0 x 10 -3) = (2.5 x 5.0) (10-1+ (-3))= 1.2 x 10 -3 Division: =am/an=am-n Example: 8.63 x 107 by 4.61 x 105= 8.63 x 107/ 4.61 x 105 = 8.63/4.61x 107/ 105= 1.872 x 107-5= 1.872 x 102


Cont’d :Cont’d Addition or Subtraction Reformat the numbers so the exponents are identical Add or subtract Round off to the appropriate significant figures Reformat the answer into standard scientific notation


Examples :Examples 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 Subtract 2.61 x 103 from 8.486 x 104 Reformat one of the numbers: 2.61 x 103= 0.261 x 104 Subtract: 8.486 x 104 - 0.261 x 103 = 8.225 x 104


Direct and Inverse Proportions :Direct and Inverse Proportions How to solve a direct proportion: Using: Cross multiply to give: x1y2= x2y1 Isolate the unknown(X), which can be any element in the equation Solve for X


Cont’d :Cont’d Using:  x1:y1:: x2:y2 Rearrange the equation so the product to the extremes( x1y2).  x1y2= x2y1 Isolate the unknown (X), which can be any element in the equation Solve for X


Inverse Proportion :Inverse Proportion How to solve inverse proportion: Using: x1y1= x2y2 Isolate the unknown (X), which can be any element in the equation Solve for X


Example :Example If a 200 uCi point source produces 6,000 cps, how many cps will be produced by a 500 uCi source? Cross multiply: (200 uci)(X cps)= (500 uci)(6000cps) Isolate and solve for X, X=


Example :Example (45 mCi)(0.8 ml)= (X mCi)(2.3ml)= (45 mCi)(0.8 ml)/ 2.3 ml= 16 mCi (65,000cpm)(2.0ml)/420 cpm= 310 ml


Converting Within the Metric System :Converting Within the Metric System


How to Convert Between Commonly used Units or Radioactivity :How to Convert Between Commonly used Units or Radioactivity 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 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= 0.500 mCi


Cont’d :Cont’d Convert 0.085 mCi to uCi When converting from a large unit to a smaller one, the number increase, so the decimal is moved to the right 3 places 0 . 0 8 5 mCi= 85 uCi


Converting between Ci and Bq :Converting between Ci and Bq Curies to Becquerels: Multiply the number of curies by the equivalent number of Becquerels 1 Ci= 37 GBq 1 mCi= 37 MBq I uCi= 37 kBq


Cont’d :Cont’d Multiply the number of Becquerel's by the equivalent number of curies 1 Bq= 2.7 x 10 -11 Ci 1 GBq= 0.027 Ci or 27 mCi 1 MBq= 0.027 mCi or 27 uCi I kBq= 0.027 uCi


Converting Between Rad and Gray :Converting Between Rad and Gray Rad: radiation absorbed dose Rad to Grays: 1 rad= 001 Gy 1 mrad= 0.01 mGy Grays to Rad: 1 Gy= 100 rad 1 mGy= 100 mrad


Converting Between Rem and Sievert :Converting Between Rem and Sievert Rem: roentgen equivalent man Rem to Sv 1 rem= 0.01 Sv 1 mrem= 0.01 mSv Sv to rem: 1 Sv= 100 rem 1 mSv=100 mrem


Converting Between Pound and Kilogram :Converting Between Pound and Kilogram 1 lb= 0.45 kg 1 kg= 2.2 lb Pounds to kilograms: Kg= (lb)(.45 kg/lb) Kilograms to pounds: lb= (kg)(2.2 lb/kg)


Converting between Fahrenheit and Centigrade :Converting between Fahrenheit and Centigrade In F scale, water freezes at 32 degrees and boils at 212 degrees On the C scale water freezes at 0 degrees and boils at 100 degrees F to C C= (F-32)(5/9) C to F F=(C x 9/5) + 32


Logs, Natural logs and Antilog :Logs, Natural logs and Antilog The log of a number is the power to which a base must be raised to produce that number. For example: 104= 10,000, therefore the logarithm to base 10 equals 4. In other words, log 10 10,00=4 An antilog or natural antilog is the inverse of the log or ln The natural antilog is expressed as e x


Examples :Examples Find the log of 100,000 Answer: log 100,000= 5 Find the ln of 25 Answer: ln 25= 3.219 Find the antilog of 4(10 4) using the 10 x function key Answer: 10 4= 10,000


Solving Equations with an Unknown in the Exponent :Solving Equations with an Unknown in the Exponent How to move an unknown out of the exponent: Isolate the component containing the exponent with the unknown (x) Simplify the side of the equation not containing x Take the log or ln of both sides of the equation. The exponent now becomes a whole number or a decmal Isolate and solve for x


Example :Example Solve for x in each of the following 74=10 2.37x 1.869=2.37x; x= 0.79


Slope Calculations :Slope Calculations 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/ x1- x2


Examples :Examples Find the slope of a line having the following coordinates (250,62) (350, 84)


References :References Abbott & Wardle, p.218-222 Wells, P. Practical mathematics. (1999). Society of nuclear medicine


Question 1 :Question 1 Convert 375 to a simple number


Answer 1 :Answer 1 19.4


Question 2 :Question 2 Convert 3564 to SN Convert 110.7 to SN Convert 27359.8 to SN


Answer 2 :Answer 2 3.564 X 103 1.107 x 102 2.73598 x 104


Question 3 :Question 3 Solve the following direct proportions


Answer 3 :Answer 3 0.2 mrem (0.6)(22)= 13.2/60


Question 4 :Question 4 Convert 15 uCi to kBq


Answer 4 :Answer 4 (15 uCi)(37 kBq/uCi)= 560 kBq


Question 5 :Question 5 Convert the following weight in pounds to kilograms 102lb 205 lb 236 lb 136 lb


Answer 5 :Answer 5 46 kg 92 kg 106 kg 61 kg