Module 1 — Mathematics
1.1 — Arithmetic
Arithmetic is the foundation of all mathematics in EASA Part 66. This section covers the essential number skills you will use throughout your career in aircraft maintenance — from calculating fuel loads and torque values to converting between measurement systems. Category A requires Level 1 (basic familiarisation); categories B1, B2, B2L, and B3 require Level 2 (general knowledge, able to apply).
Arithmetical Terms and Signs
You must know the standard mathematical symbols and the terms associated with each operation.
| Symbol | Meaning | Example |
|---|---|---|
| \( + \) | Addition (sum) | \( 8 + 5 = 13 \) |
| \( - \) | Subtraction (difference) | \( 15 - 7 = 8 \) |
| \( \times \) | Multiplication (product) | \( 6 \times 4 = 24 \) |
| \( \div \) | Division (quotient) | \( 20 \div 5 = 4 \) |
| \( = \) | Equals | \( 3 + 4 = 7 \) |
| \( \neq \) | Not equal to | \( 5 \neq 6 \) |
| \( > \) | Greater than | \( 10 > 7 \) |
| \( < \) | Less than | \( 3 < 8 \) |
| \( \geq \) | Greater than or equal to | \( x \geq 0 \) |
| \( \leq \) | Less than or equal to | \( T \leq 100 \) |
| \( \approx \) | Approximately equal to | \( \pi \approx 3.14 \) |
| \( \pm \) | Plus or minus (tolerance) | \( 50 \pm 2 \) mm |
Key terminology: In addition, the numbers being added are addends and the result is the sum. In subtraction, you subtract the subtrahend from the minuend to get the difference. In multiplication, the numbers are factors and the result is the product. In division, the dividend is divided by the divisor to give the quotient and possibly a remainder.
Aviation context: Tolerances (using ±) are critical in maintenance. A torque specification of \( 50 \pm 2 \) Nm means any value from 48 Nm to 52 Nm is acceptable.
Methods of Multiplication and Division
Long Multiplication
To multiply large numbers, break one factor into its place-value components and add the partial products.
Worked Example
Calculate \( 247 \times 36 \):
247 × 30 = 7,410
Total = 1,482 + 7,410 = 8,892
Long Division
Long division breaks a division problem into a sequence of smaller steps: divide, multiply, subtract, bring down.
Worked Example
Calculate \( 1,596 \div 12 \):
12 goes into 15 once (12), remainder 3. Bring down 9 → 39. 12 goes into 39 three times (36), remainder 3. Bring down 6 → 36. 12 goes into 36 exactly 3 times. Answer: 133
Estimation and Cross-checking
Always estimate before calculating. Round each number to one significant figure: \( 247 \times 36 \approx 250 \times 40 = 10{,}000 \). Your exact answer of 8,892 is reasonable. If you got 88,920, you would know something was wrong.
Common mistake: Misplacing the decimal point. A turbine oil capacity of 4.5 litres versus 45 litres is a critical difference. Always estimate first.
Fractions and Decimals
Types of Fractions
- Proper fraction: numerator < denominator, e.g. \( \frac{3}{4} \)
- Improper fraction: numerator ≥ denominator, e.g. \( \frac{7}{4} \)
- Mixed number: whole number + fraction, e.g. \( 1\frac{3}{4} \)
Operations with Fractions
Key Formulae
Addition/Subtraction (find a common denominator):
$$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$Multiplication:
$$ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} $$Division (invert and multiply):
$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} $$Worked Example
Calculate \( \frac{2}{3} + \frac{3}{5} \):
Common denominator = 15. So \( \frac{2}{3} = \frac{10}{15} \) and \( \frac{3}{5} = \frac{9}{15} \). Sum = \( \frac{10+9}{15} = \frac{19}{15} = 1\frac{4}{15} \)
Decimals
Decimals are another way of expressing fractions in base 10. The digit positions to the right of the decimal point represent tenths, hundredths, thousandths, etc.
Converting fractions to decimals: Divide the numerator by the denominator. For example, \( \frac{3}{8} = 3 \div 8 = 0.375 \).
Converting decimals to fractions: Write the decimal over the appropriate power of 10 and simplify. For example, \( 0.625 = \frac{625}{1000} = \frac{5}{8} \).
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| \( \frac{1}{2} \) | 0.5 | \( \frac{1}{8} \) | 0.125 |
| \( \frac{1}{3} \) | 0.333... | \( \frac{3}{8} \) | 0.375 |
| \( \frac{1}{4} \) | 0.25 | \( \frac{5}{8} \) | 0.625 |
| \( \frac{3}{4} \) | 0.75 | \( \frac{7}{8} \) | 0.875 |
| \( \frac{1}{5} \) | 0.2 | \( \frac{1}{16} \) | 0.0625 |
Decimal Places and Significant Figures
Decimal places (d.p.): the number of digits after the decimal point. For example, 3.142 has 3 d.p.
Significant figures (s.f.): count all digits starting from the first non-zero digit. For example, 0.00456 has 3 s.f.; 3,050 has 3 s.f. (the trailing zero may or may not be significant depending on context).
Factors and Multiples
A factor of a number divides into it exactly with no remainder. A multiple is the product of that number and any positive integer.
Examples
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Prime Numbers and Prime Factorisation
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... Note that 1 is not prime, and 2 is the only even prime.
Any integer can be written as a product of prime factors. For example: \( 60 = 2^2 \times 3 \times 5 \).
Highest Common Factor (HCF)
The HCF is the largest number that divides exactly into two or more numbers. To find it, identify the prime factors common to both numbers and multiply them together.
Worked Example
HCF of 36 and 48:
\( 36 = 2^2 \times 3^2 \), \( 48 = 2^4 \times 3 \). Common factors: \( 2^2 \times 3 = 12 \). HCF = 12
Lowest Common Multiple (LCM)
The LCM is the smallest number that is a multiple of two or more numbers. Take the highest power of each prime factor present in either number.
Worked Example
LCM of 36 and 48:
\( 36 = 2^2 \times 3^2 \), \( 48 = 2^4 \times 3 \). Highest powers: \( 2^4 \times 3^2 = 16 \times 9 = 144 \). LCM = 144
Weights, Measures and Conversion Factors
Aviation uses a mixture of SI (metric) and imperial/US customary units. You must be able to convert between them confidently.
Length
| Conversion | Factor |
|---|---|
| 1 inch (in) | 25.4 mm (exact) |
| 1 foot (ft) = 12 in | 0.3048 m |
| 1 yard (yd) = 3 ft | 0.9144 m |
| 1 mile (statute) | 1.609 km |
| 1 nautical mile (NM) | 1.852 km (exact) |
Mass
| Conversion | Factor |
|---|---|
| 1 pound (lb) | 0.4536 kg |
| 1 kilogram (kg) | 2.205 lb |
| 1 US ton (short ton) | 907.2 kg |
| 1 Imperial ton (long ton) | 1,016 kg |
| 1 metric tonne | 1,000 kg |
Volume
| Conversion | Factor |
|---|---|
| 1 US gallon | 3.785 litres |
| 1 Imperial gallon | 4.546 litres |
| 1 litre | 1,000 cm³ = 0.001 m³ |
Pressure
| Conversion | Factor |
|---|---|
| 1 atmosphere (atm) | 101.325 kPa = 14.696 psi |
| 1 bar | 100 kPa = 14.504 psi |
| 1 psi | 6.895 kPa |
| 1 atm | 29.92 inHg = 760 mmHg |
Temperature
Temperature Conversions
$$ {}^\circ\text{C} = \frac{5}{9}\,({}^\circ\text{F} - 32) \qquad\qquad {}^\circ\text{F} = \frac{9}{5}\,{}^\circ\text{C} + 32 $$ $$ \text{K} = {}^\circ\text{C} + 273.15 $$Aviation context: ISA (International Standard Atmosphere) sea-level temperature is 15 °C (59 °F, 288.15 K). Temperature decreases at approximately 2 °C per 1,000 ft of altitude in the troposphere.
Worked Example
Convert 68 °F to Celsius:
\( {}^\circ\text{C} = \frac{5}{9}(68 - 32) = \frac{5}{9} \times 36 = 20\,{}^\circ\text{C} \)
Ratio and Proportion
A ratio compares two quantities of the same kind. It is written as \( a : b \) and can also be expressed as the fraction \( \frac{a}{b} \).
Worked Example — Gear Ratio
A gear train has a driving gear with 20 teeth and a driven gear with 60 teeth. The gear ratio is:
\( 20 : 60 = 1 : 3 \)
This means the driven gear turns once for every 3 turns of the driving gear.
Direct Proportion
Two quantities are in direct proportion if when one increases, the other increases by the same factor. If \( y \) is directly proportional to \( x \), then \( y = kx \) for some constant \( k \).
Inverse Proportion
Two quantities are in inverse proportion if when one increases, the other decreases by the same factor. If \( y \) is inversely proportional to \( x \), then \( y = \frac{k}{x} \).
Aviation context: Boyle's Law states that gas pressure and volume are inversely proportional at constant temperature: \( P_1 V_1 = P_2 V_2 \). This is fundamental to understanding pneumatic and pressurisation systems.
Worked Example — Mixing Ratio
An oil-fuel mixture requires a ratio of 1 : 40. How much oil is needed for 20 litres of fuel?
Total parts = 1 + 40 = 41. Oil = \( \frac{1}{41} \times 20 = 0.488 \) litres ≈ 0.49 litres.
Averages and Percentages
Averages
- Mean (arithmetic average): sum of all values divided by the number of values. $$ \text{Mean} = \frac{\sum x}{n} $$
- Median: the middle value when data is arranged in order. For an even count of values, take the mean of the two middle values.
- Mode: the most frequently occurring value.
Worked Example
Five temperature readings (°C): 18, 22, 19, 22, 24
Mean = \( \frac{18 + 22 + 19 + 22 + 24}{5} = \frac{105}{5} = 21\,{}^\circ\text{C} \)
Ordered: 18, 19, 22, 22, 24 → Median = 22 °C
Mode = 22 °C (appears twice)
Percentages
A percentage is a fraction expressed out of 100. The symbol is %.
Percentage Formulae
Finding a percentage of a value:
$$ \text{Result} = \frac{\text{percentage}}{100} \times \text{value} $$Expressing one quantity as a percentage of another:
$$ \text{Percentage} = \frac{\text{part}}{\text{whole}} \times 100\% $$Percentage change:
$$ \text{Percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\% $$Worked Example — Efficiency
A generator produces 920 W of electrical power from 1,150 W of mechanical input. What is its efficiency?
\( \text{Efficiency} = \frac{920}{1150} \times 100\% = 80\% \)
Areas and Volumes
Area Formulae (2D Shapes)
| Shape | Formula |
|---|---|
| Rectangle | \( A = l \times w \) |
| Square | \( A = s^2 \) |
| Triangle | \( A = \frac{1}{2} \times b \times h \) |
| Parallelogram | \( A = b \times h \) |
| Trapezoid (Trapezium) | \( A = \frac{1}{2}(a + b) \times h \) |
| Circle | \( A = \pi r^2 \) |
Volume Formulae (3D Shapes)
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | \( V = s^3 \) | \( SA = 6s^2 \) |
| Rectangular prism | \( V = l \times w \times h \) | \( SA = 2(lw + lh + wh) \) |
| Cylinder | \( V = \pi r^2 h \) | \( SA = 2\pi r(r + h) \) |
| Cone | \( V = \frac{1}{3}\pi r^2 h \) | \( SA = \pi r(r + l) \), \( l \) = slant |
| Sphere | \( V = \frac{4}{3}\pi r^3 \) | \( SA = 4\pi r^2 \) |
Worked Example — Cylinder Volume
A piston engine cylinder has a bore (diameter) of 130 mm and a stroke of 150 mm. Find the swept volume of one cylinder.
Radius \( r = 65 \) mm = 0.065 m. Height \( h = 150 \) mm = 0.150 m.
\( V = \pi r^2 h = \pi \times 0.065^2 \times 0.150 = \pi \times 0.004225 \times 0.150 \approx 0.001991 \) m³ = 1,991 cm³ ≈ 1.99 litres
Squares, Cubes, Square and Cube Roots
Squaring a number means multiplying it by itself: \( n^2 = n \times n \). The square root reverses this: \( \sqrt{n^2} = n \).
Cubing means raising to the power of 3: \( n^3 = n \times n \times n \). The cube root reverses it: \( \sqrt[3]{n^3} = n \).
| \( n \) | \( n^2 \) | \( n^3 \) | \( \sqrt{n} \) | \( \sqrt[3]{n} \) |
|---|---|---|---|---|
| 1 | 1 | 1 | 1.000 | 1.000 |
| 2 | 4 | 8 | 1.414 | 1.260 |
| 3 | 9 | 27 | 1.732 | 1.442 |
| 4 | 16 | 64 | 2.000 | 1.587 |
| 5 | 25 | 125 | 2.236 | 1.710 |
| 10 | 100 | 1,000 | 3.162 | 2.154 |
Remember: The square root of a negative number is not a real number. However, squaring a negative number gives a positive result: \( (-5)^2 = 25 \).
Powers and Roots (Laws of Indices)
An index (plural: indices), also called an exponent or power, indicates how many times a base is multiplied by itself: \( a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}} \)
Laws of Indices
| Rule | Law | Example |
|---|---|---|
| Multiplication | \( a^m \times a^n = a^{m+n} \) | \( 2^3 \times 2^4 = 2^7 = 128 \) |
| Division | \( a^m \div a^n = a^{m-n} \) | \( 5^6 \div 5^2 = 5^4 = 625 \) |
| Power of a power | \( (a^m)^n = a^{mn} \) | \( (3^2)^3 = 3^6 = 729 \) |
| Zero index | \( a^0 = 1 \) | \( 7^0 = 1 \) |
| Negative index | \( a^{-n} = \frac{1}{a^n} \) | \( 2^{-3} = \frac{1}{8} \) |
| Fractional index | \( a^{1/n} = \sqrt[n]{a} \) | \( 27^{1/3} = 3 \) |
| General fractional | \( a^{m/n} = \sqrt[n]{a^m} \) | \( 8^{2/3} = (\sqrt[3]{8})^2 = 4 \) |
Worked Example
Simplify \( \frac{3^5 \times 3^{-2}}{3^2} \):
Numerator: \( 3^{5+(-2)} = 3^3 \). Then \( \frac{3^3}{3^2} = 3^{3-2} = 3^1 = \mathbf{3} \)
Aviation context: Scientific notation uses powers of 10 to express very large or small numbers. The speed of light is \( 3 \times 10^8 \) m/s. A micrometre is \( 1 \times 10^{-6} \) m. This is essential when working with electrical values, wavelengths, and material tolerances.
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