Number bases — AQA GCSE Computer Science

Number bases are foundational mathematical systems used in Computer Science to represent numerical data and memory states. While humans traditionally use denary (base-10), digital computer architectures rely entirely on binary (base-2) because their microprocessors are constructed from billions of microscopic transistors that exist in one of two physical states: on (voltage present) or off (voltage absent).

To make raw binary strings manageable for developers, computer scientists use hexadecimal (base-16) as a concise shorthand. Mastery of base conversions between denary, binary, and hexadecimal is critical for understanding low-level data representation, addressing modes, colour codes, and network protocols across the AQA specification.

• →Base definition: The base or radix indicates the total count of unique symbols (digits) available in that system: Base-2 (0-1), Base-10 (0-9), and Base-16 (0-9 followed by A-F).
• →Binary place values: Columns increase exponentially from right to left by powers of two (1, 2, 4, 8, 16, 32, 64, 128 for a standard byte).
• →Nibble mapping: Exactly 4 binary bits (one nibble) correspond to exactly 1 hexadecimal character, because 2 to the power of 4 equals 16.
• →Human utility of hexadecimal: Hexadecimal reduces the length of binary strings by a factor of four, reducing transcription mistakes in MAC addresses, IPv6 addresses, and HTML colour codes.

• Decimal is base 10 and uses the ten digits 0–9.
• Binary is base 2 and uses only the digits 0 and 1.
• Hexadecimal is base 16 and uses 0–9 plus A–F, where A represents 10 and F represents 15.
• Each position in a number has a place value that is a power of the base: in base 10 the columns are 10⁰, 10¹, 10² …; in base 2 they are 2⁰, 2¹, 2² …; in base 16 they are 16⁰, 16¹, 16² …
• The same quantity can be represented in different bases, for example decimal 13 equals binary 1101 and hexadecimal D.
• The base tells you how many distinct digit symbols the system uses before a new column is needed.

• 💡Learn the digit set for each base so you can spot an invalid representation quickly.
• 💡Remember the place-value columns are powers of the base, which explains why 10 in any base equals that base in decimal.
• 💡Practise converting a small decimal number such as 13 into binary and hexadecimal to confirm you understand the relationship.
• 💡Always sketch out an 8-column place value grid (128, 64, 32, 16, 8, 4, 2, 1) across the top of your exam page for binary conversion questions to prevent arithmetic slips.
• 💡When asked why hexadecimal is preferred over binary, use phrases like 'shorter representation', 'easier for humans to read and understand', and 'less prone to typing errors' to hit mark scheme keywords.
• 💡Double-check your final bit string contains exactly 8 bits if the question asks for a full byte, padding with leading zeroes if necessary.

• Assuming every number written without a label is decimal: correction — check the context, because 10 in binary means two and 10 in hexadecimal means sixteen.
• Thinking hexadecimal uses only the letters A–F with no digits: correction — hexadecimal uses 0–9 as well as A–F, sixteen symbols in total.
• Believing the digit 2 is valid in binary: correction — binary uses only 0 and 1, so any digit of 2 or above signals a different base.
• Believing that hexadecimal saves memory inside computer chips. In reality, computers store and process every piece of data purely in binary; hex is purely an interface notation for human convenience.
• Forgetting that 10 in hexadecimal is equivalent to denary 16, not denary 10, because the first column represents 16s and the zero represents units.
• Confusing the letters in hexadecimal by starting 'A' at 9 instead of 10, which shifts all subsequent letters by one.

1. 1Day 1: Master 8-bit binary to denary and denary to binary conversions using the column addition and subtraction methods.
2. 2Day 2: Practice hexadecimal to denary conversions, memorising the values of letters A through F.
3. 3Day 3: Practice the split-nibble technique to convert directly between 8-bit binary and 2-digit hexadecimal.
4. 4Day 4: Work through timed AQA past-paper questions focusing on contextual reasons why hexadecimal is used in real computing systems.

• 📋Direct conversion calculations: 1-mark or 2-mark procedural questions requiring conversions between denary, binary, and hexadecimal.
• 📋Short explanation questions: 2-mark questions asking why programmers prefer hexadecimal over binary for representing memory addresses or colour codes.
• 📋Data analysis and state identification: Questions asking students to identify valid hexadecimal values or determine the maximum value expressible in a given number of bits.