Understand how and why hexadecimal is used as a beneficial method of data representation

Data Representation – Number Systems

1. Why is hexadecimal used?

Hexadecimal (base‑16) is a compact, human‑readable way of representing binary data. One hexadecimal digit (a “hex digit”) corresponds exactly to four binary bits (a “nibble”). This relationship means that large binary numbers can be written with far fewer characters, making it easier to:

• Read and write memory addresses.
• Interpret machine code and colour values in graphics.
• Debug programs by comparing binary patterns with a short notation.

2. Converting between number systems

2.1 Binary ↔ Decimal (Denary)

Binary → Decimal

Example: 101101₂

1. Write the powers of 2 under each bit (right‑most bit = 2⁰).
2. Multiply each bit by its power of 2 and add the results.

Bit	2⁵	2⁴	2³	2²	2¹	2⁰
Value	1	0	1	1	0	1

Calculation: 1·32 + 0·16 + 1·8 + 1·4 + 0·2 + 1·1 = 45₁₀

Decimal → Binary

Example: 156₁₀

1. Divide the number by 2, record the remainder.
2. Repeat with the quotient until it becomes 0.
3. Read the remainders in reverse order.

Division	Quotient	Remainder
156 ÷ 2	78	0
78 ÷ 2	39	0
39 ÷ 2	19	1
19 ÷ 2	9	1
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1

Reading the remainders bottom‑up gives 10011100₂.

2.2 Decimal ↔ Hexadecimal

Decimal → Hexadecimal

Example: 156₁₀ → ?₁₆

1. Divide the decimal number by 16, keep the remainder.
2. Repeat with the quotient until it becomes 0.
3. Write the remainders in reverse order, using A‑F for values 10‑15.

Division	Quotient	Remainder
156 ÷ 16	9	12 (C)
9 ÷ 16	0	9

Result: 9C₁₆.

Hexadecimal → Decimal

Example: 0x7F → ?₁₀

• 7 × 16¹ = 112
• F (15) × 16⁰ = 15

Sum = 112 + 15 = 127₁₀.

2.3 Binary ↔ Hexadecimal (quick conversion)

Group binary digits into nibbles (4 bits) starting from the right‑most side, then replace each nibble with its hex equivalent.

Example: 101101110010₂ → ?₁₆

Binary nibble	Hex digit
1011	B
0111	7
0010	2

Result: B72₁₆.

2.4 Quick‑reference tables (syllabus‑friendly)

These tables show the maximum values for 8‑bit and 16‑bit unsigned numbers and give the direct conversion triples.

Bits	Maximum unsigned value (Dec)	Binary (full)	Hex
8 bits	255	1111 1111₂	FF₁₆
16 bits	65 535	1111 1111 1111 1111₂	FFFF₁₆

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

3. Binary arithmetic, overflow and logical shifts

3.1 Overflow in an 8‑bit register

Computers usually work with fixed‑size bytes (8 bits). The largest unsigned value that can be stored in one byte is 255₁₀ (11111111₂). Adding two numbers that exceed this limit produces an overflow – the carry out of the most‑significant bit is discarded.

Example: 150₁₀ + 130₁₀

• 150₁₀ = 10010110₂
• 130₁₀ = 10000010₂
• Binary addition: 10010110
+10000010
-----------
1 00011000 (9‑bit result)
• Discard the left‑most carry → 00011000₂ = 24₁₀.

The mathematically correct sum is 280₁₀, but an 8‑bit register can only hold 24₁₀; the discarded carry sets the overflow flag.

3.2 Logical shifts

A logical shift moves all bits left or right, inserting 0s into the vacated positions. It is used for fast multiplication/division by powers of two.

• Logical left shift (<< 1): 00110101₂ → 01101010₂ (value doubles).
• Logical right shift (>> 1): 11001010₂ → 01100101₂ (value halves, highest bit becomes 0).

Logical shifts treat the number as unsigned; the sign bit is not preserved.

3.3 Two’s‑complement representation of negative numbers

Two’s‑complement allows a single binary pattern to represent both positive and negative integers.

1. Write the absolute value in binary (using the required number of bits).
2. Invert every bit (0 → 1, 1 → 0).
3. Add 1 to the inverted result.

Example: –45₁₀ in an 8‑bit system

• 45₁₀ = 00101101₂
• Invert → 11010010₂
• Add 1 → 11010011₂

Thus, –45₁₀ is stored as 11010011₂. To verify, add the positive and negative forms:

 00101101
+11010011
-----------
100000000   (extra carry discarded → 00000000)

4. Hexadecimal in memory addressing

Modern computers address memory in bytes. Since a byte consists of 8 bits (two nibbles), a memory address is naturally expressed in hexadecimal.

Example address: 0x3F2A → binary 0011 1111 0010 1010. Using hexadecimal reduces a 16‑bit address to just four readable characters.

5. Common uses of hexadecimal

• Representing colour values in HTML/CSS (e.g., #FF5733).
• Displaying machine code and instruction sets.
• Debugging memory dumps and stack traces.
• Specifying Unicode code points (e.g., U+00E9 for “é”).

6. Character sets – ASCII and Unicode (hex view)

Both ASCII and Unicode assign a hexadecimal code point to each character. The table below shows a small sample relevant to the IGCSE syllabus.

Char	ASCII (Dec)	ASCII (Hex)	Unicode (U+)
A	65	41	U+0041
a	97	61	U+0061
0	48	30	U+0030
Space	32	20	U+0020
€	—	—	U+20AC
é	—	—	U+00E9

7. File‑size calculation example (image data)

File size (in bytes) = (width × height × colour‑depth) ÷ 8.

Example: A 1920 × 1080 pixel image with 24‑bit colour (true colour).

• Pixels = 1920 × 1080 = 2 073 600.
• Total bits = 2 073 600 × 24 = 49 766 400 bits.
• Bytes = 49 766 400 ÷ 8 = 6 220 800 bytes ≈ 5.93 MiB.

8. Quick reference diagram

Data Transmission

1. Packet structure

A data packet is the basic unit of communication in most networks. The typical layout is:

Field	Purpose
Header	Contains source/destination addresses, packet type, length, and error‑checking information (e.g., CRC).
Payload (Data)	The actual user data being transferred.
Trailer	Often a frame‑check sequence or other end‑of‑packet marker.

2. Packet‑switching vs. circuit‑switching

• Packet‑switching: Data is broken into packets that travel independently; each packet may follow a different route. Used in the Internet, Ethernet, Wi‑Fi.
• Circuit‑switching: A dedicated path is established for the duration of a communication session (e.g., traditional telephone networks).

3. Transmission modes

Mode	Direction	Typical use
Simplex	One‑way only	Keyboard to computer, TV broadcast.
Half‑duplex	Two‑way, but not simultaneously	Walkie‑talkies, RS‑485.
Full‑duplex	Two‑way simultaneously	Ethernet, USB, telephone.

4. Serial vs. parallel transmission

Characteristic	Serial	Parallel
Bits sent per clock cycle	1	Multiple (usually 8‑16)
Cabling	Single pair (or few pairs)	Many wires, bulkier
Distance	Longer (up to kilometres)	Short (centimetres‑metres)
Common examples	USB, RS‑232, Ethernet	Older printer ports, internal CPU‑bus

5. Mini‑activity – USB as a transmission interface

1. Identify the three USB transfer types: Control, Bulk, Isochronous.
2. Explain why USB is considered a full‑duplex, serial, packet‑switched interface.
3. Discuss the role of the USB “endpoint” address (shown in hexadecimal, e.g., 0x81 for an IN endpoint).

Computer Hardware – Core Components

1. CPU architecture (simplified)

• Control Unit (CU) – fetches, decodes and controls execution of instructions.
• Arithmetic‑Logic Unit (ALU) – performs arithmetic and logical operations on binary data.
• Registers – small, fast storage locations (e.g., accumulator, program counter) that hold binary values, often displayed in hexadecimal during debugging.
• System bus – carries data, addresses and control signals between CPU, memory and peripherals.

2. Network Interface Card (NIC) and MAC address

A NIC provides the hardware link to a network. Each NIC has a unique 48‑bit Media Access Control (MAC) address, usually written in hexadecimal:

Example: 00:1A:2B:3C:4D:5E

The MAC address is used for link‑layer addressing on Ethernet and Wi‑Fi.

3. IP addressing – IPv4 vs. IPv6

Feature	IPv4	IPv6
Length	32 bits (4 octets)	128 bits (16 octets)
Notation	Decimal dotted‑quad (e.g., 192.168.0.1)	Hexadecimal groups (e.g., 2001:0db8:85a3:0000:0000:8a2e:0370:7334)
Number of addresses	≈ 4.3 × 10⁹	≈ 3.4 × 10³⁸
Header size	20 bytes (minimum)	40 bytes (fixed)

4. Primary vs. secondary storage

Aspect	Primary (volatile)	Secondary (non‑volatile)
Typical devices	RAM (DRAM, SRAM)	HDD, SSD, optical discs, USB flash drives
Speed	Nanoseconds‑microseconds	Milliseconds (HDD) to microseconds (SSD)
Power dependence	Data lost when power is removed	Data retained without power
Typical capacity (2025)	4 GB – 64 GB	500 GB – 8 TB (HDD) / 256 GB – 4 TB (SSD)

Key Take‑aways

• One hex digit = four binary bits (a nibble); this makes hex a concise representation of binary data.
• All required conversions (binary ↔ decimal, decimal ↔ hex, binary ↔ hex) are systematic and reversible.
• Understanding overflow, logical shifts and two’s‑complement is essential for working with memory addresses, machine code and signed arithmetic.
• Hexadecimal links naturally to other data‑representation topics:
  • Text – ASCII/Unicode code points are given in hex.
  • Images – colour values are three hex bytes (R,G,B).
  • Sound – sample values are binary numbers that can be examined in hex.
• Data transmission concepts (packet structure, switching, transmission modes) and core hardware components (CPU, NIC, storage) complete the IGCSE 0478 syllabus requirements.