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

2. 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

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

4. 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

5. 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

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

7. 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)

8. 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

9. 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)

10. POWERS OF TEN AND PREFIXES

11. 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 X 103

12. 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

13. 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: (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: X 10-4 =

14. 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)

15. 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

16. 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

17. 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 = x/12 = 100/12
- x = 8.33

18. 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

19. 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

20. 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

21. 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

22. 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