Data Representation & Number Systems

1. Why Data Representation is Needed

A digital computer is built from two-state devices — a transistor is either on (1) or off (0). It therefore understands only binary, and every kind of information must be encoded as a pattern of bits: numbers, characters, images, audio, video and even the instructions of a program. Data representation is the set of rules for that encoding.

Data type	How it is represented	Unit of measurement
Integers	Positional binary, signed or unsigned	bit / byte
Real numbers	Floating point (IEEE-754: sign, exponent, mantissa)	bit
Characters	ASCII (8-bit), Unicode (8/16/32-bit), BCD for digits	byte
Images	Grid of pixels, each with a colour code (colour depth)	pixels × bits per pixel
Audio	Amplitude sampled at fixed intervals	samples/s (Hz) × bit depth
Video	Sequence of frames + compression	frames/s + bit rate
Instructions	Opcode + operand fields	bits per instruction word

2. Positional Number Systems

In a positional system, the value of a digit depends on its position. With radix (base) $r$ and digits $d$ :

Value = d_{n-1} × r^{n-1} + ... + d_1 × r^{1} + d_0 × r^{0} + d_{-1} × r^{-1} + ...

Two ideas follow directly from this: any base works, and the base tells you how many symbols are needed.

3. The Four Systems Used in Computing

System	Base (radix)	Digits allowed	Suffix examples	Why used
Binary	2	0, 1	$1011_2$ or 1011B	Native machine language; simple, reliable two-state circuits
Octal	8	0-7	$653_8$ or 653O	Compact shorthand for binary — 1 octal digit = 3 bits
Decimal	10	0-9	987 or 987D	Human everyday arithmetic
Hexadecimal	16	0-9, A-F (A=10 … F=15)	$2AF_{16}$ or 2AFH	Compact shorthand — 1 hex digit = 4 bits; used for addresses, memory dumps, MAC addresses
BCD	10 (as 4-bit groups)	0000-1001	25 → 0010 0101	Exact decimal arithmetic (calculators, banking); wastes 6 of 16 codes
Gray code	2 (reflected)	0, 1	3 → 0010	Only one bit changes between consecutive codes; used in encoders, K-maps

4. Counting in All Four Systems

Decimal	Binary (4-bit)	Octal	Hex
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

5. Why Binary is Used in Computers

Only two stable states (saturated / cut-off transistor, charged / discharged capacitor) — highly reliable and noise-resistant.
Simple circuits — one circuit (adder) handles all arithmetic; Boolean logic maps directly onto gates.
Cheap and easy to build with switches, relays, magnetic and optical media.
Arithmetic and error-control (parity, checksums) are simplest in base 2.
Octal and hexadecimal exist purely as human-friendly shorthands for binary: one digit condenses 3 or 4 bits, so long addresses become readable.

6. How Many Bits Do You Need?

To represent $n$ distinct values you need $k$ bits where

2^{k} ≥ n, that is, k = ⌈log₂ n⌉

Values to represent	Bits needed	Example
10 digits	4	BCD
26 letters	5 ( $2^{5}=32$ )	one alphabet
128 ASCII characters	7 (stored in 8)	ASCII
65,536 characters	16	Unicode BMP
256 colours	1 byte per pixel	GIF palette image

7. Exam Points

Why are octal and hexadecimal used in computer work? — 3-5 marks, very frequent.
Explain the positional number system with the general formula — 5 marks.
Compare binary, octal, decimal and hexadecimal with digits and base — 5 marks.
What is BCD? Why is it used in calculators? — 3 marks.
How many bits are needed to represent n characters/values? — 3 marks numerical.
How are pictures, sound and characters represented in a computer? — 5 marks.

Data Representation & Number Systems