Number Systems & Base Conversions

In Cambridge IGCSE Computer Science, all data inside a digital computer is processed and stored in binary format (base 2) using discrete electrical voltage states: $0$ (low voltage / off) and $1$ (high voltage / on).

Learning Objectives

• Understand why computers use binary representation for instructions and data.
• Convert positive integers between Denary (base 10), Binary (base 2), and Hexadecimal (base 16) up to 16-bit values.
• Explain the practical benefits and applications of hexadecimal representation in computer science.

Core Syllabus Theory

1. The Three Number Bases

2. Conversion Methods

A. Binary to Denary

Construct an 8-bit binary positional place value table:

B. Denary to Binary (Successive Subtraction or Division by 2)

To convert $109_{10}$ to 8-bit binary:

1. $109 \ge 64 \rightarrow \text{place 1 at 64, remainder } 45$
2. $45 \ge 32 \rightarrow \text{place 1 at 32, remainder } 13$
3. $13 \ge 8 \rightarrow \text{place 1 at 8, remainder } 5$
4. $5 \ge 4 \rightarrow \text{place 1 at 4, remainder } 1$
5. $1 \ge 1 \rightarrow \text{place 1 at 1, remainder } 0$

C. Binary to Hexadecimal (4-bit Nibble Grouping)

Group the binary string into 4-bit nibbles from right to left:

3. Why Hexadecimal is Used

1. Human Readability: Hexadecimal strings are significantly shorter than binary (e.g. 8 binary bits condense into 2 hex digits), making them much easier for humans to read, write, and remember.
2. Error Reduction: Shorter strings drastically reduce transcription errors when programmers copy memory addresses or byte values.
3. Direct Nibble Mapping: Each hexadecimal digit corresponds exactly to one 4-bit binary nibble ($2^4 = 16$), making conversion between binary and hex instantaneous without complex calculations.

4. Common Hexadecimal Applications

• HTML/CSS Colour Codes: #FF0000 (Red), #00FF00 (Green), #0000FF (Blue) — 24-bit true colour represented as 6 hex digits.
• MAC Addresses: Physical hardware identifiers formatted as 12 hex digits (e.g. 00:1A:2B:3C:4D:5E).
• Memory Dumps & Debugging: Displaying contents of RAM when troubleshooting system crashes or buffer overflows.
• IPv6 Addresses: 128-bit network addresses formatted in 8 groups of 4 hexadecimal digits.

Exam Tips & Common Pitfalls

Note: Pad to 8 Bits: In exam questions, always write binary answers as full 8-bit bytes (e.g. write 00101101 rather than 101101) unless instructed otherwise.

Key Terminology Glossary

• Bit: A binary digit, the smallest unit of digital data ($0$ or $1$).
• Nibble: A group of 4 binary bits (equivalent to 1 hexadecimal digit).
• Byte: A group of 8 binary bits (capable of representing $2^8 = 256$ distinct values).

Written practice and self-review

Identify two distinct reasons why computer scientists use hexadecimal notation instead of binary when displaying MAC addresses or memory addresses. Explain why this representation does NOT reduce the amount of physical storage required by the computer.

Write an explanation before checking the theory. Check that each claim answers the question, uses the correct concept and explains why it applies. Discuss uncertain points with a teacher. The structured exercise below checks specific facts; it does not automatically mark this explanation.