Chapter 2: Basic Mathematics Essentials of Radiologic Science, 2nd edition

MATHEMATICAL FORMULAS Mathematical formulas are the cornerstone for selecting or correcting technical factor settings This chapter’s purpose is to review the basic mathematic operations and conversions frequently used in the imaging sciences

BASIC MATHEMATICAL OPERATIONS Converting a fraction into a decimal: Divide the numerator into the denominator Numerator/denominator EX: 4/7 = 0.5714

BASIC MATHEMATICAL OPERATIONS Converting decimals to percentages Decimal to %: Move decimal two spaces to the right EX: 0.27 = 27% % to a decimal: Move decimal two spaces to the left EX: 89.2% = 0.892

ORDER OF OPERATION When multiple calculations are required in a single problem, it is important to follow the proper order of operation as outlined below or the correct solution will not be found: Perform calculations inside parentheses first Apply exponents Multiply and divide Perform addition and subtraction last

ORDER OF OPERATION EX: 2 + 3(3 + 5(2) + 32) – 10 ÷5 2 + 3(3 + 5(2) + 9) – 10 ÷5 2 + 3(3 + 10 + 9) – 10 ÷5 2 + 3(22) – 10 ÷5 2 + 66 – 10 ÷5 2 + 66 – 2 A: 66

EVALUATING EXPONENTS Rules for evaluating exponents: In multiplication, add the exponents EX: 102 X 105 = 1010 In division, subtract the exponents EX: 1 X 107/1 X 103 = 1 X 104 (or 10,000)

EVALUATING EXPONENTS Rules for evaluating exponents (cont’d): When raised to another power, multiply the exponents EX: (103)2 = 106 (or 1,000,000) A negative exponent indicates the value should be in the denominator EX: 10-6 = 1/106

POWERS OF TEN AND PREFIXES The use of exponents and the Powers of Ten allow scientists to compare units that are very large to microscopically small The metric system has assigned specific prefixes and symbols to each power of ten Using exponents for the powers of ten and their prefixes is a more economical method to solve problems involving very large or small values by eliminating the need to record all of the decimal places (EX: 1,000,000 meters = 106 meters = 1 megameter)

POWERS OF TEN AND PREFIXES

SCIENTIFIC NOTATION Def: Expressing a number using exponents starting with an integer (or a zero), the decimal point, and the remaining significant digits A related concept to the Powers of Ten EX: 1,082 may be expressed as 1.082 X 103

SCIENTIFIC NOTATION To convert a number to scientific notation: Find the first non-zero significant number EX: 1,060,000 (1) Next, place a decimal point directly to the right of the numeral, “1” (1.) Determine the remaining significant digits (06) In this case, the last four zeroes are not considered significant

SCIENTIFIC NOTATION To convert a number to scientific notation (cont’d): Count how many decimal places the decimal point will need to move to “restore” the number to its original value EX: 1.060000 (6 places) Ans: 1.06 X 106 Positive exponents indicate the number value is greater than 1 Negative exponents indicate the number value will be less than 1 EX: 3.421 X 10-4 = 0.0003421

SCIENTIFIC NOTATION Calculations using scientific notation: Multiply the like terms applying the appropriate rules for each operation EX: (3.0 X 107) (6.0 X 10-5) 3.0 X 6.0 = 18 107 X 10-5 = 102 Ans: 18 X 102 = 1.8 X 103 (scientific notation)

ALGEBRAIC EXPRESSIONS Algebraic expressions use letters to represent the unknown value; x is a commonly used letter Rules for solving: a(b + c): a must be multiplied by both b and c Ans: ab + ac When a number is added to x, the number must be subtracted from both sides to solve EX: x + 9 = 14 x + 9 – 9 = 14 – 9 x = 5

ALGEBRAIC EXPRESSIONS Rules for solving (cont’d): When x is multiplied by a number, the number must be divided by both sides to solve EX: 4x = 104 4x/4 = 104/4 x = 26

ALGEBRAIC EXPRESSIONS Rules for solving (cont’d): When both sides are ratios (or fractions), use cross-multiplication to solve EX: x/5 = 20/12 - 12x = 100 12x/12 = 100/12 - x = 8.33

RELATIONSHIPS BETWEEN VARIABLES Most radiographic formulas are used to demonstrate the relationship between two variables, or are used to compensate when one of the variables changes. Common relationships between variables: Direct Indirect (or related) Exponential Inverse

RELATIONSHIPS BETWEEN VARIABLES Direct (or directly proportional) Def: The change in variable A causes the same factor of change in variable B Radiographic example: mAs and density Using 2X the mAs will cause density to increase 2X Indirect Def: A change in variable A causes a change in variable B in the same direction, but not at the exact same rate or factor

RELATIONSHIPS BETWEEN VARIABLES Def: The change in variable A causes a change in the same direction on variable B, but at a greatly increased rate Radiographic example: kVp and density Increasing only 15% of the kVp will cause density to increase 2X

RELATIONSHIPS BETWEEN VARIABLES Def: The change in variable A causes a change in the opposite direction on variable B Radiographic example: Inverse Square Law I1/I2 = d22/d12 Increasing the distance will cause I (intensity) to decrease

OTHER RADIOGRAPHIC FORMULAS Calculating mAs: mAs = mA X time 15% Rule: kVp value X 1.15 or kVp value X 0.85 mAs-Distance Compensation Formula mAs1/mAs2 = d12/d22