1.1 Arithmetical Terms, Signs, Fractions, and Decimals

Arithmetical Terms, Signs, and Operations

In aviation maintenance, mathematical precision is directly linked to flight safety. Whether calculating structural panel thickness, setting torque limits, or determining weight and balance, an aircraft technician must possess a flawless command of basic arithmetic. The European Aviation Safety Agency (EASA) Part-66 syllabus emphasizes these fundamentals under Module 1. This section covers arithmetical terms, positive and negative signs, order of operations, fractions, decimals, and their real-world applications in the hangar and on the flight line.

To communicate and calculate effectively, a technician must recognize and apply standard arithmetical terms. Every basic mathematical operation has distinct components:

• Addition (+): Combines numbers to find a sum. For example, if three hydraulic lines have volumes of 1.2 liters, 0.8 liters, and 1.5 liters, their sum is 3.5 liters.
• Subtraction (-): Finds the difference between two numbers. In subtraction, the number being subtracted from is the minuend, and the number being subtracted is the subtrahend. For instance, subtracting a wear measurement (subtrahend) from a nominal component thickness (minuend) yields the remaining material thickness (difference).
• Multiplication (x): Represents repeated addition of a number, the multiplicand, by another, the multiplier, to produce a product.
• Division (/): Splits a number, the dividend, by another, the divisor, to yield a quotient. Any remaining quantity that cannot be evenly divided is the remainder.

Positive and Negative Signs

In arithmetic, signs indicate both operations and directional values. A positive number (+) represents values greater than zero, while a negative number (-) represents values less than zero. In aviation, negative values are common:

1. Outside Air Temperature (OAT): At high altitudes, OAT drops well below zero (e.g., -40 degrees Celsius).
2. Altimeter Correction: Altimeter errors can be positive or negative, requiring additions or subtractions to find true altitude.
3. Center of Gravity (CG) Limits: A reference datum is established for weight and balance. Items located forward of the datum have a negative arm (distance), while items aft of the datum have a positive arm.

When manipulating signed numbers, specific rules apply:

• Addition: Adding two negative numbers yields a more negative number: (-5) + (-3) = -8. Adding numbers with different signs requires finding the difference between their absolute values and applying the sign of the larger absolute value: (-12) + 8 = -4.
• Subtraction: Subtracting a negative number is equivalent to adding its positive counterpart: 10 - (-5) = 10 + 5 = 15.
• Multiplication/Division: Multiplying or dividing two numbers with the same sign yields a positive result: (-4) x (-3) = 12. Multiplying or dividing numbers with opposite signs yields a negative result: (-15) / 3 = -5.

Order of Operations (BODMAS / PEMDAS)

Aviation maintenance manuals contain formulas for calculating structural repairs, electrical values, and engine performance. Without a standardized order of operations, calculations would yield inconsistent results. The international standard is governed by the acronym BODMAS (or PEMDAS in some regions):

1. Brackets / Parentheses: Solve terms inside brackets first, working from the innermost outward.
2. Orders / Exponents: Evaluate powers, roots, and indices (e.g., x squared, square root of y).
3. Division and Multiplication: Perform these operations from left to right as they appear.
4. Addition and Subtraction: Perform these operations from left to right as they appear.

Aviation Formula Example: Torque Extension

When a torque wrench requires an extension, the actual torque applied to the fastener (Ta) differs from the setting read on the wrench scale (Tw). The formula is:

Ta = (Tw x (L + E)) / L

Where L is the length of the torque wrench and E is the length of the extension.

Suppose a technician needs to apply a torque (Ta) of 120 ft-lb using a wrench of length L = 15 inches and an extension of length E = 3 inches. To find the correct wrench setting (Tw), the formula is rearranged:

Tw = (Ta x L) / (L + E)

Following the order of operations:

1. Evaluate the parentheses in the denominator first: L + E = 15 + 3 = 18.
2. Perform multiplication in the numerator: Ta x L = 120 x 15 = 1800.
3. Perform division: 1800 / 18 = 100 ft-lb.

Without parentheses around the denominator, a linear calculation like 120 x 15 / 15 + 3 would yield 120 x 1 + 3 = 123, leading to dangerous over-torqueing. This demonstrates the critical importance of following the established order of operations in aircraft maintenance.

Fractions in Aviation

Fractions represent parts of a whole, written as a numerator over a denominator (a/b). In aviation, fractions are extensively used in structural sheet metal work, fastener sizing, and tool selections. For instance, aerospace drill bits and Imperial aircraft bolts are sized in fractional inches (e.g., 3/32, 1/4, 5/16, or 13/64 inches).

Common Denominators

To add or subtract fractions, they must have a common denominator. The most efficient method is to find the lowest common denominator (LCD), which is the lowest common multiple of the individual denominators.

Worked Example: Sheet Metal Repair Stackup

An aviation technician is performing a flush repair on a fuselage skin. The repair patch requires stacking three sheets of aluminum alloy together with an adhesive layer. The thicknesses of the sheets are:

• Inner skin: 1/16 inch
• Doubler plate: 3/32 inch
• Outer skin: 5/64 inch

To calculate the total metal thickness, find the sum of these fractions:

Total Thickness = 1/16 + 3/32 + 5/64

1. Identify the denominators: 16, 32, and 64.
2. Find the LCD: 64 is a multiple of both 16 (16 x 4 = 64) and 32 (32 x 2 = 64).
3. Convert each fraction to have a denominator of 64:
   • 1/16 = (1 x 4) / (16 x 4) = 4/64
   • 3/32 = (3 x 2) / (32 x 2) = 6/64
   • 5/64 remains unchanged.
4. Add the numerators over the common denominator:
   • (4 + 6 + 5) / 64 = 15/64 inches.

Multiplying and Dividing Fractions

• Multiplication: Multiply the numerators together and the denominators together: (3/8) x (2/3) = 6/24 = 1/4
• Division: Multiply the first fraction by the reciprocal of the second (the divisor): (5/16) / (1/8) = (5/16) x (8/1) = 40/16 = 2.5 or 2 and 1/2.

Decimals and Precision Measurement

Decimals are another way of expressing fractions, based on powers of ten. In aviation, decimal measurements are the standard for high-precision components. Micrometers, dial indicators, and feeler gauges measure dimensions in thousandths or ten-thousandths of an inch (e.g., 0.001 in or 0.0001 in) or in hundredths of a millimeter (e.g., 0.01 mm).

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. This is vital when matching fractional specifications in a blueprint to decimal readouts on a digital micrometer:

• 3/8 inch = 3 / 8 = 0.375 inches
• 5/16 inch = 5 / 16 = 0.3125 inches

Converting Decimals to Fractions

To convert a decimal to a fraction, write the decimal as a fraction with a power of 10 in the denominator, then simplify.

For example, convert 0.0625 to a fraction:

0.0625 = 625 / 10000

Dividing both by their greatest common divisor (625):

(625 / 625) / (10000 / 625) = 1/16 inch.

Rounding Decimals

Aviation maintenance manuals specify limits to precise decimal places. Standard rounding rules apply:

1. Identify the target decimal place.
2. Examine the digit directly to the right.
3. If that digit is 5 or greater, round the target digit up by 1. If it is less than 5, leave the target digit unchanged.

For example, if a micrometer reads 0.21875 inches and the manual requires rounding to four decimal places, the value becomes 0.2188 inches (since the fifth digit is 5). If rounded to three decimal places, it becomes 0.219 inches (since the fourth digit is 7, which is greater than 5).