Module 1: Mathematics — B2 Licence Category

1. Overview

Module 1 of the EASA Part-66 basic knowledge syllabus establishes the mathematical foundation required for the safe and effective execution of aircraft maintenance duties, particularly for B2 (avionics) licence holders. This module is not an abstract study of mathematics; rather, it is a focused, practical toolkit designed to support the calculations encountered in daily avionics maintenance, troubleshooting, and documentation.

The syllabus is structured around core arithmetic, algebra, geometry, and trigonometry, all applied to real-world aviation scenarios. The knowledge levels defined in Appendix I to Annex III (Part-66) guide the depth of study: Level 1 requires an overview, Level 2 a general understanding, and Level 3 a detailed theoretical grasp with the ability to apply the knowledge in practical situations. For the B2 category, many topics are studied at Level 3, reflecting the need for precise calculations in avionics systems.

The overarching goal is to enable the certifying staff to perform calculations with confidence and accuracy, whether converting units for a weight and balance report, determining the characteristics of an AC signal, or interpreting a wiring diagram. This module is the bedrock upon which all subsequent technical modules are built.

2. Key Concepts Explained in Detail

2.1 Arithmetic and Number Systems

This foundational area covers the manipulation of numbers, including fractions, decimals, and percentages, and the application of SI (Système International) prefixes.

SI Prefixes and Unit Conversion

The International System of Units (SI) is the global standard for measurement, and the aviation industry, particularly under EASA, operates predominantly in these units. However, legacy aircraft and documentation from other regions may use imperial units, making conversion a critical skill.

Prefix	Symbol	Factor	Example
Giga	G	10⁹	2.5 GHz = 2,500,000,000 Hz
Mega	M	10⁶	4.5 MHz = 4,500,000 Hz
Kilo	k	10³	0.025 kΩ = 25 Ω
(Base Unit)	-	10⁰	e.g., metre, ohm, hertz
Milli	m	10⁻³	250 mA = 0.25 A
Micro	µ	10⁻⁶	0.047 µF = 47,000 pF
Nano	n	10⁻⁹	-
Pico	p	10⁻¹²	1 µF = 1,000,000 pF

Example 1 (Unit Conversion): A capacitor has a value of 0.047 µF. Convert this to picofarads (pF).

Since 1 µF = 10⁻⁶ F and 1 pF = 10⁻¹² F, then 1 µF = 10⁶ pF.
Therefore, 0.047 µF × 10⁶ pF/µF = 47,000 pF.

Example 2 (Unit Conversion): An aircraft's altitude is 35,000 ft. Convert this to metres.

Using the conversion factor 1 ft = 0.3048 m.
35,000 ft × 0.3048 m/ft = 10,668 m.

Example 3 (Unit Conversion): A maintenance manual specifies a torque of 50 lbf·ft. Convert this to N·m.

Using the conversion factor 1 lbf·ft = 1.356 N·m.
50 lbf·ft × 1.356 N·m/lbf·ft = 67.8 N·m.

Example 4 (Unit Conversion): A temperature is 20°C. Convert to degrees Fahrenheit.

The conversion formula is °F = (°C × 9/5) + 32.
°F = (20 × 9/5) + 32 = 36 + 32 = 68°F.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is expressed as a product of a number (the mantissa) between 1 and 10, and a power of 10.

Multiplication: Multiply the mantissas and add the exponents.
Division: Divide the mantissas and subtract the exponents.

Example 5 (Scientific Notation): Calculate (3.2 × 10⁴) × (2.5 × 10⁻²).

Multiply mantissas: 3.2 × 2.5 = 8.0.
Add exponents: 4 + (-2) = 2.
Result: 8.0 × 10² = 800.

Example 6 (Scientific Notation): Express the result of the altitude conversion (10,668 m) in scientific notation.

10,668 m = 1.0668 × 10⁴ m.

Example 7 (Scientific Notation): A voltage of 12.5 V and a current of 250 mA are measured. Calculate the power.

First, convert current to amperes: 250 mA = 0.250 A.
Power (P) = V × I = 12.5 V × 0.250 A = 3.125 W.
In scientific notation: 3.125 × 10⁰ W.

Example 8 (Arithmetic Application): An aircraft battery supplies a current of 2.5 A for 4 hours. Calculate the total charge delivered.

Charge (Q) = Current (I) × Time (t) = 2.5 A × 4 h = 10 Ah.

Example 9 (Arithmetic Application): An aircraft battery has a capacity of 25 Ah and is discharged at 5 A for 3 hours. Calculate the remaining capacity.

Charge consumed = 5 A × 3 h = 15 Ah.
Remaining capacity = 25 Ah – 15 Ah = 10 Ah.

2.2 Algebra

Algebra introduces the use of symbols (variables) to represent unknown quantities, allowing for the formulation and solution of equations. This is essential for rearranging formulas and solving for specific parameters.

Solving Linear Equations

A linear equation is an equation where the highest power of the variable is 1. The goal is to isolate the variable on one side of the equation.

Example 10 (Linear Equation): Solve for x: 3(x + 2) = 15.

Divide both sides by 3: x + 2 = 5.
Subtract 2 from both sides: x = 3.

Transposition of Formulae

A key skill is the ability to rearrange a formula to make a different variable the subject. This is used extensively in electrical and electronic calculations.

Example 11 (Transposition): Ohm's law states V = I × R. Calculate the resistance (R) when V = 12 V and I = 0.5 A.

Rearrange the formula to make R the subject: R = V / I.
Substitute the values: R = 12 V / 0.5 A = 24 Ω.

Example 12 (Transposition): The formula for inductive reactance is XL = 2πfL. Calculate the inductance (L) given XL = 3 Ω and f = 400 Hz.

Rearrange to make L the subject: L = XL / (2πf).
Substitute values: L = 3 / (2 × 3.14 × 400) = 3 / 2512 = 0.001194 H ≈ 1.19 mH.

Example 13 (Transposition): The formula for wavelength is λ = c / f. Calculate the wavelength (λ) given c = 3 × 10⁸ m/s and f = 2.5 GHz.

First, convert frequency to Hz: 2.5 GHz = 2.5 × 10⁹ Hz.
Substitute values: λ = (3 × 10⁸) / (2.5 × 10⁹) = 0.12 m.

Simultaneous Equations and Quadratics

While not explicitly tested in the provided questions, a B2 technician may encounter situations requiring the solution of simultaneous equations (e.g., in network analysis) or quadratic equations (e.g., in certain resonance calculations). The syllabus requires a general knowledge of these methods.

2.3 Geometry

Geometry deals with the properties and relationships of points, lines, surfaces, and solids. For avionics, the most relevant areas are the calculation of areas and volumes for installation and packaging, and the use of trigonometric functions for resolving forces and signals.

Areas and Volumes

Rectangle: Area = length × width.
Triangle: Area = ½ × base × height.
Circle: Area = π × radius².
Rectangular Solid (Cuboid): Volume = length × width × height.
Cylinder: Volume = π × radius² × height.

Example 14 (Area): Calculate the area of a rectangular avionics rack that is 60 cm long and 40 cm wide.

Area = length × width = 60 cm × 40 cm = 2400 cm².

Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles. It is fundamental for resolving vectors, which is critical in navigation, weight and balance, and AC circuit analysis.

For a right-angled triangle with angle θ:

Sine (sin θ): Opposite / Hypotenuse.
Cosine (cos θ): Adjacent / Hypotenuse.
Tangent (tan θ): Opposite / Adjacent.

Special Triangles

Certain triangles have well-known properties that simplify calculations.

30-60-90 Triangle: The sides are in the ratio 1 : √3 : 2. The side opposite the 30° angle is half the hypotenuse.
45-45-90 Triangle: The sides are in the ratio 1 : 1 : √2.

Example 15 (Trigonometry): In a right-angled triangle, the side opposite the 30° angle measures 8 cm. Calculate the length of the hypotenuse.

In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse.
Therefore, hypotenuse = 2 × 8 cm = 16 cm.

Example 16 (Vector Application): An aircraft's true airspeed is 240 knots. The wind component along the track is a 25-knot headwind. Calculate the ground speed.

Ground speed = true airspeed – headwind component = 240 – 25 = 215 knots.

2.4 Graphs and Waveforms

The ability to interpret and analyse graphs is essential for understanding system performance, such as charging/discharging curves, frequency response, and signal waveforms.

Root Mean Square (RMS) Values

For alternating current (AC) and voltage, the RMS value is the equivalent DC value that would produce the same heating effect in a resistive load. The relationship between the peak value and the RMS value depends on the waveform shape.

Sine Wave: RMS = Peak / √2.
Triangular Wave: RMS = Peak / √3.
Square Wave: RMS = Peak.

Example 17 (RMS Value): A triangular waveform has a peak amplitude of 5 V and a frequency of 1 kHz. Calculate the RMS value.

For a triangular wave, RMS = Peak / √3 = 5 / 1.732 ≈ 2.89 V.

The Nyquist-Shannon Sampling Theorem

In digital avionics, analogue signals are sampled to be converted to digital form. The Nyquist-Shannon sampling theorem states that to accurately represent a signal without aliasing (a form of distortion), the sampling rate must be at least twice the highest frequency component of the signal. Therefore, the maximum frequency that can be accurately represented is half the sampling rate.

Example 18 (Nyquist Theorem): A digital avionics system samples an analogue signal at a rate of 8 kHz. What is the maximum frequency that can be accurately represented?

Maximum frequency = Sampling rate / 2 = 8 kHz / 2 = 4 kHz.

3. Important Formulas and Procedures

The following formulas are central to the calculations in this module and are frequently used in avionics maintenance.

Formula	Application
Ohm's Law: V = I × R	Relates voltage, current, and resistance.
Power: P = V × I	Calculates electrical power.
Charge: Q = I × t	Calculates electrical charge (e.g., battery capacity).
Inductive Reactance: XL = 2πfL	Calculates the opposition to AC current in an inductor.
Wavelength: λ = c / f	Relates wavelength, speed of light, and frequency.
RMS (Sine): VRMS = Vpeak / √2	Converts peak to RMS for sine waves.
RMS (Triangle): VRMS = Vpeak / √3	Converts peak to RMS for triangular waves.
Parallel Resistance: 1/R_total = 1/R1 + 1/R2 + ...	Calculates total resistance of parallel resistors.
Temperature Conversion: °F = (°C × 9/5) + 32	Converts Celsius to Fahrenheit.
Trigonometric Ratios: sin θ = O/H, cos θ = A/H, tan θ = O/A	Resolves vectors and triangle problems.

Procedures for Unit Conversion

Identify the given unit and the required unit.
Determine the conversion factor between the two units.
Multiply the given value by the conversion factor, ensuring the original unit cancels out.

Procedure for Solving Linear Equations

Simplify both sides of the equation by removing parentheses and combining like terms.
Use addition or subtraction to move all terms containing the variable to one side and all constant terms to the other.
Use multiplication or division to isolate the variable.

4. Common Relationships Between Concepts

The concepts in Module 1 are not isolated; they are interconnected and build upon each other.

Algebra is the tool for applying formulas: Ohm's Law (V = I × R) is a formula, but solving for R requires algebraic transposition.
Arithmetic is the tool for calculation: Once a formula is rearranged, arithmetic (including scientific notation and unit conversion) is used to compute the numerical answer.
Geometry and Trigonometry are tools for visualisation: They are used to understand physical layouts (areas, volumes) and to resolve forces and signals into components.
Graphs and Waveforms link mathematics to system behaviour: The RMS value is a mathematical calculation applied to a waveform, and the Nyquist theorem links the mathematical concept of sampling to the practical limitation of digital systems.
Unit conversion is a prerequisite for all calculations: A calculation is meaningless if the units are inconsistent. For example, in the wavelength calculation, the frequency must be in Hz (not GHz) to match the speed of light in m/s.

5. Typical Exam Focus Points

Based on the syllabus and typical examination patterns, the following areas are frequently tested:

SI Prefixes and Unit Conversions: Converting between prefixes (e.g., µF to pF, MHz to Hz, kΩ to Ω) and between imperial and metric units (e.g., ft to m, lbf·ft to N·m, °C to °F). This is a high-yield area.
Scientific Notation: Performing calculations (especially multiplication and division) with numbers in scientific notation and expressing final answers in this format.
Algebraic Manipulation: Solving linear equations and transposing formulas to find an unknown variable. Ohm's Law and power calculations are common contexts.
Basic Electrical Calculations: Applying formulas for charge (Q = I × t), power (P = V × I), and resistance (series and parallel).
Geometry and Trigonometry: Calculating areas of simple shapes and using trigonometric ratios (especially for 30-60-90 and 45-45-90 triangles) to solve for unknown sides.
Waveform Analysis: Calculating the RMS value of sine and triangular waveforms from their peak values. Understanding the Nyquist-Shannon sampling theorem.
Practical Applications: Interpreting resistor colour codes and applying vector addition/subtraction to navigation problems (e.g., ground speed).

A solid grasp of these core areas, with the ability to apply them to practical scenarios, is essential for success in the Module 1 examination and for a career as a B2 certifying staff member.