Binary, octal and hexadecimal numbering systems; conversion between them and decimal; binary coded decimal and two's complement for signed values.
Notes
Numbering systems — summary notes
Free
Main ideas
A positional system gives each digit a weight of baseᵏ, counting k from 0 at the least-significant digit. Binary (base 2) weights are …16, 8, 4, 2, 1; octal is base 8; hexadecimal is base 16.
Hexadecimal digits A, B, C, D, E, F represent decimal 10, 11, 12, 13, 14, 15 — not 11–16.
Binary converts to octal by grouping bits in THREES from the right, and to hexadecimal by grouping in FOURS from the right, padding the left-hand group with zeros.
BCD (binary coded decimal) encodes each decimal digit as its own 4-bit group: 29 → 0010 1001. It is not the same as the plain binary value of 29 (1 1101). BCD wastes the codes 1010–1111 but makes decimal display driving trivial.
Signed binary uses two's complement: invert every bit and add 1. The MSB is then the sign bit (1 = negative), and ordinary binary addition works for signed values with no special case.
Octal is the natural shorthand for ARINC 429 labels; hexadecimal is the natural shorthand for memory addresses and byte data.
⚠ Exam trap: mis-grouping is the commonest error — octal groups in 3s, hexadecimal in 4s. Reading a hex answer as if it were BCD (or vice versa) is the second commonest.
Key formulas
Positional value (base b)
value = Σ (dₖ · bᵏ), k = 0 at the LSD
Binary place values
… 128 · 64 · 32 · 16 · 8 · 4 · 2 · 1
Binary → octal
group bits in 3s from the right
Binary → hexadecimal
group bits in 4s from the right
Two's complement (n-bit)
invert all bits, then add 1; MSB = sign
Largest value in n bits
2ⁿ − 1 (unsigned)
Solved examples
Convert 1011 0110₂ to hexadecimal and to decimal.
Group in 4s: 1011 = B and 0110 = 6, so the value is B6₁₆. In decimal, add the place values where a 1 appears: 128 + 32 + 16 + 4 + 2 = 182₁₀.
Encode 45 in BCD, then represent −45 in 8-bit two's complement.
BCD takes each decimal digit separately: 4 → 0100 and 5 → 0101, giving 0100 0101. For two's complement, +45 = 0010 1101; invert to 1101 0010; add 1 to get 1101 0011. The leading 1 confirms a negative value.