Binary, octal, and hexadecimal number systems

Binary, octal, and hexadecimal number systems

1. What Number Systems Are (for BEEE Digital)

Digital electronics represents data in binary but uses octal/hex as shorthand.

System	Base	Digits	Place Weight	Example
Binary	2	0,1	$2^n$	$1011_2 =11_{10}$
Octal	8	0–7	$8^n$	$17_8 =15_{10}$
Decimal	10	0–9	$10^n$	$25_{10}$
Hexadecimal	16	0–9, A(10)–F(15)	$16^n$	$1A_{16}=26_{10}$

2. Why Binary + Hex/Octal for Digital

• Binary matches two voltage levels ($0V$ low, $Vcc$ high) → noise immune, transistor switch.
• Hex compact: 1 hex digit =4 bits (e.g., $1111_2=F$), 1 octal =3 bits — easier to read than long binary for 8-/16-bit words.

Binary→Decimal: 1011₂=1·8+0·4+1·2+1·1=11₁₀
Decimal→Binary (repeated divide by 2): 11÷2=5 rem1, 5÷2=2 r1, 2÷2=1 r0,1÷2=0 r1 ⇒ read rem reverse 1011 ✓
Binary→Hex (group 4 bits from right): 10111100₂=1011 1100=BC₁₆
Hex→Binary: A₍₁₆₎=1010₂ (each hex →4 bits)
Binary→Octal (group 3 bits): 101111₂=101 111=57₈
Octal↔Hex via binary intermediate.

5. Exam Points

• Define base, digits for binary/octal/hex — 3-5 marks, frequent.
• Convert binary↔decimal, binary↔octal/hex — 5-7 marks, very frequent (almost certain to ask one conversion).
• Show 1 hex=4 bits, 1 octal=3 bits — 3 marks, frequent.
• Mention why binary for digital, hex as shorthand — 3-5 marks, frequent.