Data representation

Computer Science AS · Information representation. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Everything in a computer is stored as binary. You must move easily between denary, binary, hexadecimal and Binary Coded Decimal (BCD), where each denary digit gets its own 4-bit code.

Negative integers are stored in two's complement: the left-most bit has a negative place value, so in 8 bits it is worth −128. Subtraction is done by adding the two's complement of the number. If a result is too big or too small for the number of bits, overflow occurs and the answer is wrong.

Characters are stored as codes from a character set: ASCII, extended ASCII or Unicode. File sizes use binary prefixes (kibi, mebi) or decimal prefixes (kilo, mega), and these are not the same.

Rules to remember

Binary prefixes: 1 KiB = 2¹⁰ = 1024 bytes, 1 MiB = 2²⁰, 1 GiB = 2³⁰, 1 TiB = 2⁴⁰ bytes
Decimal prefixes: 1 kB = 10³ = 1000 bytes, 1 MB = 10⁶, 1 GB = 10⁹, 1 TB = 10¹² bytes
One hexadecimal digit = 4 bits; A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
BCD: each denary digit is coded separately in 4 bits, e.g. 59 = 0101 1001
One's complement: invert every bit. Two's complement: invert every bit, then add 1. The 8-bit two's complement range is −128 to +127
Overflow: the result is outside the range, e.g. adding two positive numbers gives a result with a sign bit of 1
ASCII uses 7 bits (128 characters), extended ASCII 8 bits (256 characters); Unicode uses more bits so it can code the characters of all languages

Common mistake

Students treat a BCD code as one pure binary number. In BCD each group of 4 bits is a separate denary digit, so 0001 0010 is 12, not 18.

Worked example

Convert the denary number 181 to hexadecimal and to 8-bit binary.

Divide by 16: 181 ÷ 16 = 11 remainder 5 (11 × 16 = 176, 181 − 176 = 5).
11 is the hex digit B and 5 is the hex digit 5, so the hex value is B5.
Write each hex digit as 4 bits: B = 1011, 5 = 0101, giving 1011 0101.
Check: 128 + 32 + 16 + 4 + 1 = 181.

Answer: 181 in denary is B5 in hexadecimal and 10110101 in binary.

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