Understand how data storage is measured

Data Storage, Transmission & Hardware – Cambridge IGCSE 0478

1. Data Representation

1.1 Binary, Decimal & Hexadecimal

• Binary (base‑2): digits 0 and 1. Used internally by computers.
• Decimal (base‑10): digits 0‑9. The number system we use in everyday life.
• Hexadecimal (base‑16): digits 0‑9 and A‑F. One hex digit = 4 bits.

Example conversion (8‑bit number):

Binary	Decimal	Hexadecimal
1101 0110	214	D6

1.2 Two’s‑Complement (negative integers)

To represent a negative integer in an n‑bit word:

1. Write the absolute value in binary.
2. Invert every bit (0→1, 1→0).
3. Add 1 to the result.

Example – –18 in 8‑bit two’s‑complement

1. +18 = 0001 0010
2. Invert → 1110 1101
3. Add 1 → 1110 1110

Result: 1110 1110₂ (‑18). The most‑significant bit (MSB) indicates sign (0 = positive, 1 = negative).

1.3 Logical Shifts

• Logical left shift (<<): moves all bits left, inserts 0 on the right. Equivalent to multiplication by 2 (ignoring overflow).
• Logical right shift (>>): moves all bits right, inserts 0 on the left. Equivalent to integer division by 2 (ignoring remainder).

Example (8‑bit unsigned)

Original	Binary	Shift left ×2	Result
25	0001 1001	0011 0010	50

If the shift moves a ‘1’ out of the MSB the bit is lost – this is **overflow**.

1.4 Text Representation

Encoding	Bits per character	Typical range
ASCII	7 bits (stored as 8 bits)	0‑127 (English letters, digits, basic symbols)
Unicode (UTF‑8)	1‑4 bytes per character	All world scripts; most common characters use 1 byte

Example – “IGCSE” in ASCII

5 characters × 1 byte = 5 bytes.

1.5 Image Representation (Bitmap)

• Resolution: width × height (pixels).
• Colour depth: bits per pixel (bpp). Common depths: 1, 8, 24, 32.
• Uncompressed size: size (bits) = width × height × colour‑depth
• Convert bits → bytes ÷ 8, then to KiB/MiB using 1 024.

1.6 Sound Representation (PCM audio)

• Sample‑rate (samples s⁻¹, e.g., 44 100 Hz).
• Bit‑depth (bits per sample, e.g., 16‑bit).
• Channels: 1 = mono, 2 = stereo.
• Uncompressed size: size (bits) = sample‑rate × bit‑depth × channels × duration (s)

1.7 Compression

• Lossless: original data can be recovered exactly (e.g., ZIP, RLE).
• Lossy: some information is permanently discarded for higher reduction (e.g., JPEG, MP3).
• Compression ratio R = S_original ÷ S_compressed.

2. Measuring Data Storage

2.1 Units

Unit (symbol)	Decimal (base 10)	Binary (IEC, base 2)
kilobyte (kB)	1 000 bytes	‑
kibibyte (KiB)	‑	1 024 bytes
megabyte (MB)	1 000 000 bytes	‑
mebibyte (MiB)	‑	1 048 576 bytes
gigabyte (GB)	1 000 000 000 bytes	‑
gibibyte (GiB)	‑	1 073 741 824 bytes
terabyte (TB)	1 000 000 000 000 bytes	‑
tebibyte (TiB)	‑	1 099 511 627 776 bytes

2.2 Converting between units

To convert upwards (bytes → KiB → MiB …) divide by 1 024 repeatedly.

To convert downwards (MiB → KiB → bytes) multiply by 1 024.

Example: 4 000 000 bytes ÷ 1 024 = 3 906.25 KiB ÷ 1 024 ≈ 3.81 MiB.

3. Calculating File Sizes

3.1 Integer data

32‑bit integer = 4 bytes.

For 1 000 000 integers:

size = 4 bytes × 1 000 000 = 4 000 000 bytes ≈ 3.81 MiB

3.2 Bitmap image (uncompressed)

Formula: size (bits) = width × height × colour‑depth

Example: 800 × 600 px, 24‑bit colour

• Bits = 800 × 600 × 24 = 11 520 000 bits
• Bytes = 11 520 000 ÷ 8 = 1 440 000 bytes
• MiB = 1 440 000 ÷ 1 024 ÷ 1 024 ≈ 1.37 MiB

3.3 PCM audio (uncompressed)

Formula: size (bits) = sample‑rate × bit‑depth × channels × duration (s)

Example: 3 min (180 s) mono, 44.1 kHz, 16‑bit

• Bits = 44 100 × 16 × 1 × 180 = 127 008 000 bits
• Bytes = 127 008 000 ÷ 8 = 15 876 000 bytes
• MiB ≈ 15.14 MiB

3.4 Text file (ASCII)

Each character = 1 byte.

Example: “Hello, world!” (13 characters)

• Size = 13 bytes ≈ 0.013 kB

4. Compression

4.1 Why compress?

• Save storage space – more files on a disk, USB stick, or cloud.
• Reduce transmission time and bandwidth usage.
• Lower cost of media and network resources.
• Enable real‑time services (video calls, streaming).

4.2 Types of compression

Method	Lossless?	Typical use	Typical ratio
Run‑Length Encoding (RLE)	Yes	Simple graphics, fax	5 : 1 – 10 : 1
ZIP / GZIP	Yes	Text, source code, spreadsheets	2 : 1 – 5 : 1
JPEG	No	Photographic images	≈ 10 : 1
MP3 / AAC	No	Music & speech	≈ 12 : 1 (music)

4.3 Compression ratio

R = S_original ÷ S_compressed

Lossless example: 1 440 000 bytes bitmap → 180 000 bytes after RLE

R = 1 440 000 ÷ 180 000 = 8 (file is 8 times smaller).

Lossy example: 50 MB PCM song → 4 MB MP3

R = 50 ÷ 4 = 12.5 (≈ 12½ times reduction, with some quality loss).

5. Data Transmission

5.1 Packet structure

• Header: source/destination address, control bits.
• Payload: the actual data.
• Trailer: error‑checking information (e.g., CRC).

5.2 Switching methods

• Circuit switching: dedicated path for the whole communication (e.g., traditional telephone).
• Packet switching: data broken into packets that travel independently (used on the Internet).

5.3 USB (Universal Serial Bus)

• Common wired transmission standard for peripherals.
• Supports simplex, half‑duplex and full‑duplex modes.
• Typical speeds: USB 2.0 = 480 Mbit s⁻¹, USB 3.0 = 5 Gbit s⁻¹.

5.4 Error‑detection & correction

Technique	How it works	Typical use
Even / odd parity bit	Add one bit so total number of 1’s is even (or odd).	Simple serial links, early memory modules.
Checksum	Sum of data bytes; receiver recomputes and compares.	Internet Protocol (IP) headers.
CRC (Cyclic Redundancy Check)	Polynomial division; detects burst errors.	Ethernet, USB, storage media.
ARQ (Automatic Repeat reQuest)	Receiver asks sender to retransmit corrupted packets.	TCP protocol.

5.5 Simple encryption (conceptual)

• Data is transformed using an algorithm and a key.
• Only someone with the correct key can reverse the process.
• In the syllabus you need only know that encryption protects data during transmission.

6. Hardware Overview

6.1 CPU architecture & the fetch‑decode‑execute (FDE) cycle

1. Fetch: read the next instruction from memory (address held in the Program Counter).
2. Decode: interpret the opcode; identify required operands.
3. Execute: perform the operation (ALU, register transfer, memory access).

Diagram placeholder: labelled CPU block showing ALU, Control Unit, Registers, and the three FDE stages.

6.2 Cores, cache & clock speed

• Core: an independent processing unit; modern CPUs have multiple cores.
• Cache: small, fast memory (L1, L2, L3) that stores frequently used data/instructions.
• Clock speed (MHz or GHz): how many cycles per second; higher speed → more instructions per second, all else equal.

6.3 Instruction set

Defines the operations a CPU can perform (e.g., ADD, SUB, LOAD, STORE, JUMP).

6.4 Embedded systems & I/O devices

• Embedded system: a computer built into another device (e.g., microwave, car engine controller).
• I/O devices: input (keyboard, mouse, sensor) and output (monitor, speaker, actuator).

6.5 Storage hierarchy

Level	Typical device	Typical capacity	Typical speed
Primary	RAM (DRAM)	4 GB – 32 GB	nanoseconds
Secondary	HDD / SSD / Optical disc	500 GB – 4 TB	micro‑ to milliseconds
Tertiary	Magnetic tape, Cloud storage	TB‑PB	seconds‑minutes (network latency)

6.6 Virtual memory

• Technique that uses part of secondary storage (often called a “page file” or “swap space”) to extend the apparent size of RAM.
• Pages are moved between RAM and disk as needed.

6.7 Cloud storage

• Data kept on remote servers accessed via the Internet.
• Advantages: accessibility, redundancy, scalability.
• Disadvantage: dependence on network connection and trust in provider.

6.8 Network hardware

• NIC (Network Interface Card): provides a physical connection to a network; has a unique MAC address (48‑bit hexadecimal).
• IP addressing: IPv4 (32‑bit) vs. IPv6 (128‑bit); used for logical routing.
• Router: forwards packets between networks, uses IP addresses to determine the best path.

7. Software Overview

7.1 System software vs. application software

• System software: operating system (OS) and utility programs that manage hardware resources.
• Application software: programs that perform specific tasks for the user (e.g., word processor, web browser, games).

7.2 Core OS functions (relevant to the syllabus)

• File management: create, store, retrieve, delete files; maintain directories.
• Memory management: allocate RAM to processes, implement virtual memory.
• Process management: start, schedule, and terminate programs.
• Device control: drivers translate OS commands to hardware actions.
• Security: user authentication, permissions, basic encryption support.

8. Summary

• Data is represented in binary; hexadecimal provides a convenient shorthand.
• Two’s‑complement and logical shifts are essential for integer arithmetic.
• Text, images, and sound each have specific storage formulas; use the binary (IEC) units required by the syllabus.
• Compression (lossless vs. lossy) reduces storage and transmission requirements; the compression ratio quantifies the reduction.
• Data transmission relies on packets, switching, USB, and error‑detection methods such as parity, checksum, CRC, and ARQ.
• Understanding CPU operation, cores, cache, instruction sets, and the storage hierarchy links hardware to the data concepts above.
• System software manages hardware resources, while application software provides user‑oriented functionality.
• All these ideas together explain how modern computers store, process, and move information efficiently.