COMPUTER SCIENCE / UNSIGNED BINARY / INDEPENDENT SUPPLEMENT
METHOD SUMMARY + ORIGINAL PRACTICE
Binary numbers, without lost carries
GCSE Computer Science
24 original tasks with worked solutions: conversion, addition, logical shifts and error repair. Learn to preserve
the value, track each carry and recognise when a fixed width loses information.
Route Pages
Place values and conversion; complete worked example 2
Carry ledger; complete worked addition 3
Logical shifts and width limits 4
24 practice tasks in four sets 5-8
Separate explained solutions 9-12
Scope, references and repair checklist 13
Start here
Use a pencil and spare paper. Try each set before opening the solutions. Keep eight positions whenever an
eight-bit answer is requested; label decimal, binary and hex clearly. Prerequisites: whole-number addition,
division and place value.
Scope and limits
Supports selected AQA GCSE Computer Science 8525 number-base and binary-arithmetic content for the
specification first taught in September 2025. It is a narrow supplement, not full-course notes. Tasks 22-24 are
clearly marked extensions. No negative numbers, two's complement, fractions, coding tasks, official exam
questions or grade predictions.
AI-assisted original composition, with separately coded numerical checks and page-by-page visual review. No independent
expert review, attendance, achieved grade or examination-board endorsement is claimed.
Version 1.0 | September 2026 | Original methods and practice 1
, COMPUTER SCIENCE / UNSIGNED BINARY / INDEPENDENT SUPPLEMENT
01 / CONVERT
Keep the value; change the notation
Decimal uses ten digits; binary uses two; hexadecimal uses sixteen. Here all bit patterns represent unsigned
whole numbers. A leading 1 is a place value, not a minus sign.
Binary weight 128 64 32 16 8 4 2 1
Example bits 1 0 1 0 0 1 1 1
Worked example A: convert decimal 167
Choose the largest weight that fits: 167 - 128 = 39. Skip 64; use 32, leaving 7. Skip 16 and 8; use 4, 2 and 1.
Reading all eight columns gives 10100111. Check by adding the chosen weights: 128 + 32 + 4 + 2 + 1 = 167.
Split into four-bit groups: 1010 | 0111. The first group is 10 (hex A); the second is 7. Therefore 167 decimal =
10100111 binary = A7 hexadecimal. A separate check is 10 x 16 + 7 = 167.
Decimal 0-9 10 11 12 13 14 15
Hex digit 0-9 A B C D E F
Choose a route
Binary to decimal: add the weights whose bits are 1.
Decimal to binary: test weights from 128 down to 1; write 0 for skipped weights.
Binary to hex: group from the right in fours; pad the left with zeros if needed.
Hex to binary: write exactly four bits per hex digit.
Decimal to two-digit hex: divide by 16; the quotient is the left digit and the remainder is the right digit.
Watch the zeros
Decimal 7 becomes 00000111 in eight bits. Its hex form is 07 when two digits are requested. Padding on the
left does not change value. Appending a zero on the right of a binary whole number multiplies its value by 2 if
width is allowed to grow.
Version 1.0 | September 2026 | Original methods and practice 2
METHOD SUMMARY + ORIGINAL PRACTICE
Binary numbers, without lost carries
GCSE Computer Science
24 original tasks with worked solutions: conversion, addition, logical shifts and error repair. Learn to preserve
the value, track each carry and recognise when a fixed width loses information.
Route Pages
Place values and conversion; complete worked example 2
Carry ledger; complete worked addition 3
Logical shifts and width limits 4
24 practice tasks in four sets 5-8
Separate explained solutions 9-12
Scope, references and repair checklist 13
Start here
Use a pencil and spare paper. Try each set before opening the solutions. Keep eight positions whenever an
eight-bit answer is requested; label decimal, binary and hex clearly. Prerequisites: whole-number addition,
division and place value.
Scope and limits
Supports selected AQA GCSE Computer Science 8525 number-base and binary-arithmetic content for the
specification first taught in September 2025. It is a narrow supplement, not full-course notes. Tasks 22-24 are
clearly marked extensions. No negative numbers, two's complement, fractions, coding tasks, official exam
questions or grade predictions.
AI-assisted original composition, with separately coded numerical checks and page-by-page visual review. No independent
expert review, attendance, achieved grade or examination-board endorsement is claimed.
Version 1.0 | September 2026 | Original methods and practice 1
, COMPUTER SCIENCE / UNSIGNED BINARY / INDEPENDENT SUPPLEMENT
01 / CONVERT
Keep the value; change the notation
Decimal uses ten digits; binary uses two; hexadecimal uses sixteen. Here all bit patterns represent unsigned
whole numbers. A leading 1 is a place value, not a minus sign.
Binary weight 128 64 32 16 8 4 2 1
Example bits 1 0 1 0 0 1 1 1
Worked example A: convert decimal 167
Choose the largest weight that fits: 167 - 128 = 39. Skip 64; use 32, leaving 7. Skip 16 and 8; use 4, 2 and 1.
Reading all eight columns gives 10100111. Check by adding the chosen weights: 128 + 32 + 4 + 2 + 1 = 167.
Split into four-bit groups: 1010 | 0111. The first group is 10 (hex A); the second is 7. Therefore 167 decimal =
10100111 binary = A7 hexadecimal. A separate check is 10 x 16 + 7 = 167.
Decimal 0-9 10 11 12 13 14 15
Hex digit 0-9 A B C D E F
Choose a route
Binary to decimal: add the weights whose bits are 1.
Decimal to binary: test weights from 128 down to 1; write 0 for skipped weights.
Binary to hex: group from the right in fours; pad the left with zeros if needed.
Hex to binary: write exactly four bits per hex digit.
Decimal to two-digit hex: divide by 16; the quotient is the left digit and the remainder is the right digit.
Watch the zeros
Decimal 7 becomes 00000111 in eight bits. Its hex form is 07 when two digits are requested. Padding on the
left does not change value. Appending a zero on the right of a binary whole number multiplies its value by 2 if
width is allowed to grow.
Version 1.0 | September 2026 | Original methods and practice 2
