Data Representation and Number Systems
Domain focus
Number-system questions test conversion between binary, octal, hexadecimal and denary, and the representation of characters and data units.
Core rules
- A bit is one binary digit; 8 bits = 1 byte; a byte has 256 possible values (0–255).
- Binary place values double: 128, 64, 32, 16, 8, 4, 2, 1.
- To convert denary to binary, divide repeatedly by 2 and read the remainders upwards.
- Octal uses base 8; hexadecimal uses base 16 with digits 0–9 and A–F.
- Group binary bits from the right: 3 bits make one octal digit; 4 bits make one hex digit.
- Character codes: ASCII uses 7/8 bits; Unicode supports many languages.
- Signed numbers may use two’s complement; overflow occurs when a result needs more bits than the register allows.
Worked example
Convert denary 45 to binary and then to hexadecimal. Reasoning: 45 ÷ 2 = 22 remainder 1; 22 ÷ 2 = 11 r 0; 11 ÷ 2 = 5 r 1; 5 ÷ 2 = 2 r 1; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1. Reading upwards gives 101101. Pad to 8 bits: 00101101. Group into nibbles: 0010 1101 = 2D, so the hexadecimal value is 2D. Check: (2 × 16) + 13 = 45.
Traps
- Reading remainders downwards, giving the reversed binary string.
- Grouping hex digits from the left instead of the right.
- Writing 1101 as “one thousand one hundred and one” in denary.
- Confusing bit and byte.
- Assuming ASCII can represent every world script; it cannot.