Converting between number bases — AQA GCSE Computer Science

Number base conversion is the process of translating values across binary (base 2), denary (base 10), and hexadecimal (base 16) systems. Computers natively operate using binary switches (transistors representing 0 or 1), whereas humans naturally calculate using denary. Hexadecimal bridges this gap by offering a concise, human-readable shorthand for binary data.

Mastering these conversions is fundamental to GCSE Computer Science as it underpins how computers store data types, execute instructions, and represent colours or MAC addresses. Being fluent in conversions allows students to understand low-level architecture, memory management, and debugging without error.

• →Base systems define the number of unique digits available: base 2 uses 0 and 1; base 10 uses 0 to 9; base 16 uses 0 to 9 and A to F.
• →Place values increase as powers of the base moving right-to-left: base 2 powers (1, 2, 4, 8, 16, 32, 64, 128) and base 16 powers (1, 16, 256).
• →A nibble (4 binary bits) maps directly to exactly one hexadecimal digit (e.g. 1111 in binary is F in hexadecimal), enabling straightforward conversions.
• →To convert between denary and hexadecimal easily, convert through binary as an intermediate stepping stone.

• Recognises the eight place values 128, 64, 32, 16, 8, 4, 2 and 1 and that eight bits span 0 to 255.
• Converts a given decimal correctly by subtracting place values from left to right and recording 1 or 0 in each column.
• Verifies an answer by summing the place values of the 1 bits to recover the original decimal.
• Identifies boundary cases: 0 as 00000000 and 255 as 11111111, and explains why 256 needs a ninth bit.
• Keeps all eight positions, including leading zeros, so the result is a full 8-bit pattern.

• 💡Write the place values 128 to 1 above your working before converting, so each bit is placed deliberately.
• 💡Always check by adding the place values of the 1s; this catches most slips quickly.
• 💡For multiple-choice items, eliminate options with more than eight bits or with a value-sum that does not match the target decimal.
• 💡Always pad binary answers with leading zeros if the question specifies an 8-bit byte (e.g. write 00101101 rather than 101101).
• 💡Double-check your addition when converting binary to denary by adding the active place values twice to eliminate simple arithmetic errors.
• 💡When converting binary to hex, group bits into sets of 4 starting from the least significant bit (rightmost) towards the left, adding leading zeros if needed.

• Writing bits right to left in the wrong order: the correction is to fix the place values above the columns first, largest on the left, then fill each column.
• Dropping leading zeros so 5 becomes 101 rather than 00000101: the correction is to always show all eight positions for a 0 to 255 value.
• Adding place values instead of subtracting as each bit is set: the correction is to track the running remainder and only set a bit when its place value fits.
• Thinking hexadecimal ends at 16 or uses 16 digits up to '16': Hexadecimal uses digits 0-9 and letters A-F, representing values 0 to 15. The value 16 is written as '10' in hex.
• Dividing by 2 incorrectly or reading the remainders top-to-bottom instead of bottom-to-top when using repeated division to convert denary to binary.
• Splitting an 8-bit byte into two nibbles unevenly (e.g. 5 bits and 3 bits) instead of two equal groups of 4 bits from right to left.

1. 1Day 1-2: Master binary-to-denary and denary-to-binary conversions using 8-bit place-value grids.
2. 2Day 3-4: Learn hexadecimal digits (0-9, A-F) and practice 4-bit nibble to single hex character mapping.
3. 3Day 5-6: Practise full two-digit hex to denary and denary to hex conversions, using binary as an intermediate step.
4. 4Day 7: Complete timed AQA past-paper questions focusing on precision, showing all workings, and padding to 8 bits.

• 📋Direct short-answer conversions (e.g. 'Convert the binary byte 11001010 to hexadecimal'): 1-2 marks requiring accuracy or nibble splitting.
• 📋Working-out structured questions (e.g. 'Convert 77 to an 8-bit binary number, showing all your working'): 2 marks assessing both method and final byte.
• 📋Contextual application questions (e.g. explaining why HTML colour codes or MAC addresses use hexadecimal instead of binary): 2-3 marks focusing on readability and error reduction.