1.1 Numbering Systems: Decimal, Binary, Octal and Hexadecimal

1.1 Numbering Systems: Decimal, Binary, Octal and Hexadecimal

EASA Part-66 Appendix I topic 5.2, as set out in Commission Implementing Regulation (EU) 2023/989 (applicable from 12 June 2024), requires numbering-system knowledge at level 2 for category B2/B2L and level 1 for category B1. Categories A and B3 are not examined on 5.2. The current Appendix I gives the numbered heading and levels but no detailed content description; this guide uses the pre-12 June 2024 Appendix I description as historical study scope, not as current regulatory text. Module 5 is a multiple-choice paper only: B2 sits 72 questions in 90 minutes, B1 sits 40 questions in 50 minutes, and A/B3 sit 20 questions in 25 minutes, with a 75% pass mark, no negative marking, and no essay. The live paper uses three options; the practice items in this chapter use four options.

A numbering system is defined by a radix (base) r and a set of r digit symbols. Any integer N whose digits are d_n d_{n-1} … d_1 d_0 expands by place value:

N = d_n·r^n + d_{n-1}·r^{n-1} + … + d_1·r^1 + d_0·r^0

That expansion is the conversion method Module 5 actually tests. Decimal uses r = 10 and digits 0–9. Binary uses r = 2 and digits 0 and 1. Octal uses r = 8 and digits 0–7. Hexadecimal uses r = 16 and digits 0–9 plus A–F, where A through F stand for decimal 10 through 15.

Place values in the four bases used on the aircraft

A bit is a binary digit. Avionics words are packed from bits: an ARINC 429 word is 32 bits; a typical discrete status field is one byte (8 bits). Octal is a compact way to write groups of three bits. Hexadecimal is a compact way to write groups of four bits. That is why a bus analyser or a centralised maintenance computer almost never prints a 32-bit string of ones and zeros for a technician to read by eye.

Documentation prefixes vary. A trailing subscript (1101₂, 13₁₀, 15₈, F₁₆), a leading 0b / 0o / 0x in software listings, or an H suffix (3E8H) on a shop-tester dump all mean the same thing: the following (or preceding) digits belong to that radix. The single most examined hex fact is that digit F always equals decimal 15. Candidates who treat F as 16 (confusing the digit with the radix) or as 14 (mis-counting A as 1 instead of 10) lose a straightforward mark.

Worked conversion: binary 1101₂ equals decimal 13

Expand 1101₂ from the least significant bit on the right, where the weights are 2³, 2², 2¹ and 2⁰:

• 1 × 2³ = 1 × 8 = 8
• 1 × 2² = 1 × 4 = 4
• 0 × 2¹ = 0 × 2 = 0
• 1 × 2⁰ = 1 × 1 = 1

Sum = 8 + 4 + 0 + 1 = 13₁₀. Memorise the first powers of two (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) so the expansion is arithmetic rather than pattern matching.

Decimal 13 back to binary

Repeated division by 2 produces remainders that become bits from LSB to MSB:

• 13 ÷ 2 = 6 remainder 1
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Reading remainders from last to first gives 1101₂, which closes the round trip. If a question gives 1101 and asks for decimal, expand; if it gives 13 and asks for binary, divide. Do not mix the two procedures.

Hexadecimal F and mixed hex–decimal conversion

Hex digits A–F are decimal 10–15:

• A = 10₁₀ = 1010₂
• B = 11₁₀ = 1011₂
• C = 12₁₀ = 1100₂
• D = 13₁₀ = 1101₂
• E = 14₁₀ = 1110₂
• F = 15₁₀ = 1111₂

Convert hex 2F₁₆ to decimal:

2 × 16¹ + F × 16⁰ = 2 × 16 + 15 × 1 = 32 + 15 = 47₁₀.

Reverse: 47 ÷ 16 = 2 remainder 15, and remainder 15 is digit F, so 2F₁₆.

On the aircraft this mapping appears whenever a BITE page prints a byte such as 0F or FF. 0F is 0000 1111₂ = 15₁₀. FF is eight ones = 255₁₀ unsigned, or all bits set on a discrete byte. That is a status meaning, not a second copy of decimal 15.

Grouping: binary to octal in threes, binary to hex in fours

Because 8 = 2³, three binary bits map onto one octal digit. Because 16 = 2⁴, four binary bits map onto one hexadecimal digit. Always group from the LSB (right-hand end). If the leftover high-order group is short, pad with leading zeros. Never pad on the right: that would change the value.

Worked binary-to-octal grouping

Convert 110101₂ to octal. Group in threes from the right:

110 101

• 110₂ = 4 + 2 + 0 = 6, so octal digit 6
• 101₂ = 4 + 0 + 1 = 5, so octal digit 5

Result: 65₈. Check via decimal: 6 × 8 + 5 = 53₁₀, and 110101₂ = 32 + 16 + 4 + 1 = 53₁₀.

If the bit string is 1101 (four bits), pad on the left to 001 101, which groups as 15₈, not as 13₈. Grouping from the MSB without padding is the classic trap and yields a different octal number.

Worked binary-to-hex grouping

Convert 11011110₂ to hexadecimal. Group in fours from the right:

1101 1110

• 1101₂ = 13₁₀ = D₁₆
• 1110₂ = 14₁₀ = E₁₆

Result: DE₁₆. A second check: 13 × 16 + 14 = 208 + 14 = 222₁₀, and 11011110₂ = 128 + 64 + 16 + 8 + 4 + 2 = 222₁₀.

Never group octal in fours or hexadecimal in threes. Octal digit 8 does not exist; if a conversion produces an 8 or 9 as an octal digit, the grouping or the original bits are wrong.

Aircraft context: ARINC 429 octal labels

An ARINC 429 word is 32 bits. Bits 1–8 are the label, conventionally written as three octal digits covering 000₈ to 377₈ (decimal 0–255, every eight-bit pattern). Octal notation exists because eight bits split as 2 + 3 + 3: the most significant octal digit is only 0–3, which is why no legal label begins with 4–7. Examples a B2 technician reads on a bus analyser include labels in the 200₈ and 300₈ bands for air-data and position parameters, depending on the installation.

To recover the logical eight-bit value of a printed label, expand the octal digits and discard the leading pad bit. Label 203₈ → 010 000 011 as three padded octal groups, hence the actual eight-bit field 10000011. ARINC transmits the label most-significant bit first. Some interface ICs expose a bit-reversed stored label byte, so distinguish the serial label from the component register convention. The rest of the same word is SDI (bits 9–10), data (bits 11–29), SSM (bits 30–31) and odd parity (bit 32). For topic 5.2 preparation, the useful connection is why the label is octal: it is an eight-bit binary field grouped by threes, not a decimal parameter identifier and not a hexadecimal dump.

Maximum label 377₈ = 3 × 64 + 7 × 8 + 7 = 192 + 56 + 7 = 255₁₀ = 11111111₂, which confirms that 377₈ is eight bits filled with ones, not a decimal 377.

Aircraft context: hexadecimal dumps on BITE

When Built-In Test Equipment, a data-loader log or a shop tester dumps memory or a discrete status word, the display is almost always hexadecimal. Each pair of hex digits is one byte:

• 0A = 0000 1010₂ = 10₁₀
• 0F = 0000 1111₂ = 15₁₀ (F = 15 again)
• A5 = 1010 0101₂ = 165₁₀
• FF = 1111 1111₂ = 255₁₀ unsigned

A technician comparing a fault isolation manual with a CMC hex dump is performing exactly the conversions in 5.2: hex to binary to see which bit is set, then binary to decimal if the AMM quotes a decimal discrete number. Mixing the radix — treating 0F as if F were a decimal digit, or reading octal label 17 as decimal 17 — selects the wrong pin or the wrong ARINC label.

Conversion strategy under exam timing

B2 candidates have about 75 seconds per question on a 72-question, 90-minute paper. Use a fixed recipe:

1. Radix to decimal: multiply each digit by its place weight and add.
2. Decimal to radix: divide by the target radix; remainders are the new digits from LSB to MSB.
3. Binary to octal or hex: group from the LSB in threes (octal) or fours (hex); pad only on the left.
4. Sanity bounds: a hex digit never exceeds 15; an octal digit never exceeds 7; a binary digit is only 0 or 1. A written 8 inside an octal label, or a G inside a hex dump, is not a legal digit.

Fractional place values (2⁻¹ = 0.5, 16⁻¹ = 0.0625) appear in some textbooks. Module 5 questions almost always stay with integers. If a fractional binary such as 0.11₂ appears, expand it as 0.5 + 0.25 = 0.75₁₀ — same rule, negative powers.

Keep the four radices chained to the job: decimal for quantities a human reads (degrees, knots, litres), binary for the hardware, octal for ARINC 429 labels, hexadecimal for BITE and memory dumps. That operational mapping explains why the pre-12 June 2024 detailed Appendix I description listed binary, octal and hexadecimal together. It remains useful study scope, while the current consolidated text supplies only the 5.2 heading and category levels.