Topic 1 of 8 · Form 5
Data representation
Binary, denary, hexadecimal and binary-coded decimal (BCD). 1 kibibyte (KiB) = 1 024 bytes; 1 kilobyte (kB) = 1 000 bytes.
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Number systems
Binary, denary, hexadecimal and binary-coded decimal (BCD). 1 kibibyte (KiB) = 1 024 bytes; 1 kilobyte (kB) = 1 000 bytes.
Negative numbers: two's complement
To find −x in 8 bits, write x in binary, invert all bits and add 1. The most significant bit has place value −128. Range in 8 bits: −128 to +127.
Worked example: Represent −45 in 8-bit two's complement.
45 = 00101101 → invert 11010010 → add 1 → 11010011
Binary arithmetic and overflow
Overflow occurs when a result is too large for the bits available (e.g. two positive numbers giving a negative two's complement result).
Floating point
A number = mantissa × 2^exponent, both in two's complement. Normalised positive mantissas start 01, negative 10, giving maximum precision. More mantissa bits give precision; more exponent bits give range.
Worked example: Mantissa 0.1010000 and exponent 0011 (8-bit mantissa, 4-bit exponent). Find the denary value.
Mantissa = 0.5 + 0.125 = 0.625; exponent = 3 → 0.625 × 2³ = 5
Characters, images and sound
ASCII (7/8 bits) and Unicode; bitmap file size = width × height × colour depth; sound file size = sample rate × bit depth × duration × channels.
Compression
Lossless (run-length encoding, Huffman coding) and lossy (JPEG, MP3).
Check yourself
- Convert 11110110 (two's complement) to denary.
- A 1 024 × 768 image uses 24-bit colour. Find its size in bytes.
- Why are floating-point numbers normalised?
Show answers
- −128 + 64 + 32 + 16 + 4 + 2 = −10
- 1 024 × 768 × 3 = 2 359 296 bytes (2.25 MiB)
- To give the maximum precision for a given number of bits and a unique representation.
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