Module 1: Mathematics — B1.4 Category Syllabus Overview

This module provides the foundational mathematical knowledge required for aircraft maintenance engineering. It covers arithmetic operations, algebraic manipulation, geometry, trigonometry, and unit conversions — all applied to practical aeronautical contexts such as engine performance calculations, structural measurements, fuel computations, and maintenance tolerances. The syllabus is divided into four main areas: Arithmetic (1.1), Algebra (1.2), Geometry (1.3), and Trigonometry (1.4), with knowledge levels ranging from basic overview (Level 1) to detailed theoretical understanding (Level 3).

1.1 Arithmetic

Fractions, Decimals, and Percentages

Aircraft maintenance frequently requires working with fractional quantities, particularly when measuring fuel levels, calculating remaining service life, or interpreting inspection findings.

Key operations:

Addition/subtraction of fractions: Find a common denominator before adding or subtracting.
Multiplication of fractions: Multiply numerators together and denominators together.
Division of fractions: Multiply by the reciprocal of the divisor.
Fraction to decimal conversion: Divide the numerator by the denominator.
Percentage calculations: Percentage = (part / whole) × 100%

Example — Fuel quantity: A 200-litre tank is ¾ full. Current fuel = ¾ × 200 = 150 litres. Additional fuel required = 200 − 150 = 50 litres.

Example — TBO fraction: An engine has 1,500 hours TBO with 400 hours since overhaul. Remaining = 1,500 − 400 = 1,100 hours. Fraction remaining = 1,100 / 1,500 = 11/15 ≈ 0.733.

Ratios and Proportions

Ratios express the relationship between two quantities and are fundamental to engine parameters such as compression ratio and gearbox reduction ratios.

Compression ratio = (swept volume + clearance volume) / clearance volume

Example: Swept volume = 1.2 L, clearance volume = 0.15 L

Compression ratio = (1.2 + 0.15) / 0.15 = 1.35 / 0.15 = 9:1

Example: Swept volume = 500 cm³, clearance volume = 50 cm³

Compression ratio = (500 + 50) / 50 = 550 / 50 = 11:1

Gearbox reduction ratio: Engine RPM = Rotor RPM × Reduction Ratio

Example: Rotor RPM = 400, reduction ratio = 8:1

Engine RPM = 400 × 8 = 3,200 RPM

Averages and Statistical Measures

Maintenance inspections often involve multiple measurements of the same parameter. The arithmetic mean and range provide essential information about measurement consistency.

Arithmetic mean = sum of all values / number of values

Range = maximum value − minimum value

Example — Crankshaft journal measurements (mm): 49.98, 50.02, 50.00, 50.04

Mean = (49.98 + 50.02 + 50.00 + 50.04) / 4 = 200.04 / 4 = 50.01 mm
Range = 50.04 − 49.98 = 0.06 mm

Example — Runout measurements (mm): 0.02, 0.03, 0.01, 0.04, 0.02

Mean = 0.12 / 5 = 0.024 mm

Tolerances and Deviations

Manufacturing specifications include nominal dimensions with allowable tolerances. Deviation is calculated as:

Deviation = measured value − nominal value

Example: Nominal diameter = 45.80 mm, tolerance ±0.05 mm, measured = 45.72 mm

Deviation = 45.72 − 45.80 = −0.08 mm
Since |−0.08| > 0.05, the measurement is outside tolerance

1.2 Algebra

Solving Linear Equations

Algebraic manipulation is essential for rearranging formulas used in maintenance calculations.

Example: Solve 3x − 7 = 2x + 5

Subtract 2x from both sides: x − 7 = 5
Add 7 to both sides: x = 12

Example: Solve 3(x − 4) + 2x = 5x − 12

Expand: 3x − 12 + 2x = 5x − 12
Simplify: 5x − 12 = 5x − 12
Both sides identical → equation holds for all real values of x (identity)

Exponents and Indices

Exponent rules are fundamental for engineering notation and scientific calculations.

Multiplication rule: aᵐ × aⁿ = a^(m+n)

Example: 2³ × 2² = 2^(3+2) = 2⁵ = 32

Other key rules:

Division: aᵐ ÷ aⁿ = a^(m−n)
Power of a power: (aᵐ)ⁿ = a^(m×n)
Negative exponent: a⁻ⁿ = 1/aⁿ
Zero exponent: a⁰ = 1

Linear Relationships

Many aeronautical relationships are linear, such as temperature lapse rates with altitude.

Example — Temperature lapse: At 500 m, temperature = 15°C. Lapse rate = 2°C per 500 m.

Altitude difference to 2,500 m = 2,000 m
Number of 500-m intervals = 2,000 / 500 = 4
Temperature drop = 4 × 2°C = 8°C
Temperature at 2,500 m = 15 − 8 = 7°C

1.3 Geometry

Areas of Common Shapes

Rectangle: Area = length × width

Example — Rotor blade: Length = 3.5 m, chord = 0.25 m

Area = 3.5 × 0.25 = 0.875 m²

Circle: Area = π × r² = π × (d/2)²

Example — Hydraulic hose: Internal diameter = 12 mm

Area = 3.14 × (6)² = 3.14 × 36 = 113.04 mm²

Cylinder Volume and Engine Displacement

The swept volume of a cylinder is calculated using:

V = (π/4) × D² × L

Where:

V = swept volume
D = cylinder bore (diameter)
L = stroke length

Example: D = 10 cm, L = 8 cm, π = 3.14

V = (3.14/4) × (10)² × 8 = 0.785 × 100 × 8 = 628 cm³

Example — 4-cylinder engine: Bore = 100 mm, stroke = 80 mm

Per cylinder: V = 0.785 × (10 cm)² × 8 cm = 628 cm³
Total displacement = 4 × 628 = 2,512 cm³ (cc)

Example — Metric units: Bore = 120 mm, stroke = 140 mm, π = 3.142

V = (3.142/4) × (0.12 m)² × 0.14 m = 0.7855 × 0.0144 × 0.14 = 0.001584 m³
V = 1.584 litres

Pythagoras' Theorem

For right-angled triangles: c² = a² + b²

Example: Shorter sides = 3 cm and 4 cm

c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 cm

This theorem is widely used in airframe measurements, rigging checks, and structural alignment verification.

1.4 Trigonometry

Angular Measurement and Conversion

Angles can be expressed in degrees or radians. The relationship is:

2π radians = 360°

Conversion: radians = degrees × (π/180)

Angular Velocity

Angular velocity (ω) in radians per second is calculated from rotational speed:

ω = 2π × RPM / 60

Example: Engine at 2,400 RPM

ω = 2π × 2,400 / 60 = 2π × 40 = 251.3 rad/s

Example: Engine at 2,700 RPM

ω = 2π × 2,700 / 60 = 282.7 rad/s

Power, Torque, and Rotational Speed

The fundamental relationship for rotating machinery:

Power (W) = Torque (N·m) × Angular Velocity (rad/s)

Example: Engine produces 200 kW at 2,700 RPM

ω = 282.7 rad/s
Torque = 200,000 / 282.7 = 707 N·m

Rotor Tip Speed and RPM

Tip speed = ω × radius

Example: Tip speed = 220 m/s, rotor diameter = 12 m (radius = 6 m)

220 = (2π × RPM / 60) × 6
RPM = (220 × 60) / (2π × 6) = 13,200 / 37.7 = 350 RPM

Ignition Timing Calculations

Spark timing in degrees BTDC can be converted to time:

Time per revolution = 60 / RPM (seconds)

Time for angle θ = (θ / 360) × time per revolution

Example: Fires at 25° BTDC, engine at 3,000 RPM

Time per revolution = 60 / 3,000 = 0.02 s = 20 ms
Time for 25° = 20 ms × (25/360) = 1.39 ms

Units and Conversions

SI Base Units

Quantity	Unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Force	newton	N
Pressure	pascal	Pa
Energy	joule	J
Power	watt	W

Common Conversions

Torque:

1 ft-lb = 1.356 N·m
1 N·m = 0.7376 lb-ft

Example: 20 ft-lb = 20 × 1.356 = 27.12 N·m

Example: 25 N·m = 25 × 0.7376 = 18.44 lb-ft

Example: 25 ft-lb = 25 × 1.356 = 33.9 N·m

Pressure:

1 psi = 6.895 kPa
1 bar = 100 kPa

Example: 60 psi = 60 × 6.895 = 413.7 kPa

Volume:

1 US gallon = 3.785 litres

Example: 45 L/h ÷ 3.785 = 11.89 US gal/h

Temperature:

A difference of 1°C = 1 K
Absolute zero = −273.15°C

Example: Temperature drops from +15°C to −5°C

Drop = 15 − (−5) = 20°C = 20 K

Derived Quantities

Density: ρ = mass / volume

Example: Fuel density = 0.8 kg/L, tank capacity = 200 L

Mass = 200 × 0.8 = 160 kg

Example — Fuel consumption: 42 L/h, density = 0.72 kg/L

Mass flow = 42 × 0.72 = 30.24 kg/h
Per minute = 30.24 / 60 = 0.504 kg/min

Pressure and Force: Force = Pressure × Area

Example: Piston area = 80 cm² = 0.008 m², pressure = 50 bar = 5,000 kPa = 5,000,000 Pa

Force = 5,000,000 × 0.008 = 40,000 N = 40 kN

Speed: Speed = distance / time

Example: 150 nautical miles in 1 hour 15 minutes (1.25 hours)

Speed = 150 / 1.25 = 120 knots

Common Relationships Between Concepts

Engine performance: Power, torque, and RPM are interrelated through angular velocity. Increasing RPM at constant power reduces torque.
Engine geometry: Bore, stroke, and number of cylinders determine total displacement. Compression ratio links swept and clearance volumes.
Drive train: Engine RPM, gearbox reduction ratio, and rotor RPM are directly proportional.
Temperature and altitude: Lapse rates create linear temperature changes with altitude, affecting engine performance and density altitude calculations.
Measurement analysis: Mean values indicate central tendency while range indicates measurement spread — both essential for wear assessment and tolerance verification.
Unit systems: Many maintenance manuals mix SI and imperial units; accurate conversion is critical for correct torque application, pressure settings, and fuel calculations.

Typical Exam Focus Points

Torque/power/RPM calculations using the formula P = T × ω
Compression ratio calculations from swept and clearance volumes
Unit conversions between N·m and ft-lb, psi and kPa, litres and US gallons
Cylinder displacement calculations using V = (π/4) × D² × L
Angular velocity conversion from RPM to rad/s
Pythagoras' theorem applications in structural measurements
Algebraic manipulation of linear equations and identities
Exponent rules for engineering notation
Arithmetic mean and range calculations for inspection data
Ratio and proportion problems applied to gearboxes and fuel mixtures
Temperature difference conversions between Celsius and Kelvin
Fraction and percentage applications for fuel quantities and service life calculations
Speed/distance/time calculations for flight planning and performance verification

Regulatory References

Regulation (EU) No 1321/2014, Annex III (Part-66): Establishes the basic knowledge requirements for aircraft maintenance licences.
Appendix I to Part-66: Defines the Module 1 Mathematics syllabus with knowledge levels:
Level 1: Familiarisation with basic concepts
Level 2: General knowledge with practical application
Level 3: Detailed theory with in-depth understanding
AMC/GM to Part-66: Provides acceptable means of compliance and guidance material for examination standards and question formats.