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Convert between binary, decimal, hexadecimal, and octal with step-by-step working. Binary arithmetic, two's complement, and ASCII conversion. Perfect for GCSE Computer Science.
Binary to Decimal
Sum each bit × 2^position (rightmost = position 0)
1011 = 1×8 + 0×4 + 1×2 + 1×1 = 11
Decimal to Binary
Divide by 2 repeatedly, record remainders, read bottom-up
13 ÷ 2 = 6r1, 6 ÷ 2 = 3r0, 3 ÷ 2 = 1r1, 1 ÷ 2 = 0r1 → 1101
Binary to Hex
Group bits in 4s (from right), convert each group
11010110 → 1101 0110 → D6
Reading binary remainders in wrong order
Always read remainders from BOTTOM to TOP when converting decimal to binary.
Forgetting to pad hex-to-binary
Each hex digit must become exactly 4 binary bits. Pad with leading zeros if needed.
A number base (or radix) determines how many different digits can be used in a counting system. We normally use base 10 (decimal) with digits 0-9, but computers use base 2 (binary) with just 0 and 1.
In GCSE Computer Science, you need to understand binary, decimal, hexadecimal (base 16), and sometimes octal (base 8).
Multiply each digit by its place value (power of 2) and add
1011₂ = 8+0+2+1 = 11₁₀
Repeatedly divide by 2 and read remainders bottom-up
13 → 6r1 → 3r0 → 1r1 → 0r1 = 1101₂
Each hex digit = 4 binary bits (a nibble)
1010 1111₂ = AF₁₆
Memorise the first 16 place values: 1, 2, 4, 8, 16, 32, 64, 128, 256... This makes binary-decimal conversion much faster!
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1)
Example: 1011 + 0110 = 10001
Step 1: Start with +5 = 00000101
Step 2: Flip all bits = 11111010
Step 3: Add 1 = 11111011
-5 in two's complement = 11111011
Write the place values (1, 2, 4, 8, 16...) above each binary digit from right to left. Add up the place values where there's a 1. Example: 1011 = 8+2+1 = 11.
Divide by 2 repeatedly and note the remainders. Read the remainders from bottom to top for the binary number. Example: 13÷2=6r1, 6÷2=3r0, 3÷2=1r1, 1÷2=0r1 → 1101.
Hexadecimal is used in computing because each hex digit represents exactly 4 binary bits. This makes it easier to read and write binary data. It's used for colours (e.g., #FF0000), memory addresses, and MAC addresses.
Add column by column from right to left using: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1). If three 1s meet: 1+1+1=11 (write 1, carry 1).
Two's complement is how computers store negative numbers. To find it: flip all bits (0→1, 1→0), then add 1. The leftmost bit is the sign bit (0=positive, 1=negative).
Yes! It covers the entire GCSE number systems topic for OCR, AQA, and Edexcel, including binary/hex conversion, binary arithmetic, two's complement, and ASCII.
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