Binary / Hex / Decimal Converter

Binary (base-2) uses only 0 and 1 to represent numbers. Each position represents a power of 2, starting from the right: 1, 2, 4, 8, 16 and so on. This converter translates between binary, decimal (base-10), octal (base-8) and hexadecimal (base-16).

Binary is the foundation of all digital computing — every piece of data in a computer is ultimately stored as a sequence of binary digits (bits). Hexadecimal is commonly used in programming as a shorthand for binary, where each hex digit represents exactly four bits.

Why binary matters. All digital computing is binary at the lowest level: 0 and 1 represent OFF/ON, false/true, low/high voltage. CPUs have billions of transistors switching between these two states. Storage units are powers of 2: byte=8 bits (256 values, 2⁸); short=16 bits (65,536); int=32 bits (4.3 billion); long=64 bits (18 quintillion). UK A-Level Computer Science covers binary, hexadecimal, octal, two's complement representation.

Decimal to binary conversion. Method 1 (division by 2): divide by 2, write remainders, read bottom-up. Sample: 42 ÷ 2 = 21 r0; 21 ÷ 2 = 10 r1; 10 ÷ 2 = 5 r0; 5 ÷ 2 = 2 r1; 2 ÷ 2 = 1 r0; 1 ÷ 2 = 0 r1. Read upwards: 101010. Method 2 (subtraction): subtract largest power of 2 fitting, mark 1; repeat. 42 = 32+8+2 = 101010. Verify: 32+8+2 = 42 ✓.

Binary to decimal — positional notation. Each binary digit (bit) represents a power of 2. Reading right to left: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128. Sample 10110: (1×16) + (0×8) + (1×4) + (1×2) + (0×1) = 16+4+2 = 22. Memorise powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.

Binary arithmetic — addition and subtraction. Addition rules: 0+0=0; 0+1=1; 1+0=1; 1+1=10 (carry 1). Sample: 1011 + 0110 = 10001 (add bit by bit with carry). Subtraction: same as decimal but borrow 2 instead of 10. Negative numbers: 'two's complement' — flip all bits, add 1. Sample 8-bit −5: 5 = 00000101; flip = 11111010; +1 = 11111011. Two's complement allows addition/subtraction with same circuit.

Hexadecimal — the human-friendly binary. Base-16 representation. Digits 0-9 + A-F (A=10, B=11, C=12, D=13, E=14, F=15). Each hex digit = 4 binary digits (nibble). Sample: binary 11010110 → 1101=D, 0110=6 → 0xD6. Decimal 214 → D6 hex. UK A-Level CS uses extensively. Hex used in: HTML colors (#FFFFFF = white = 16,777,215 = decimal); MAC addresses; memory addresses (0x7FFE...); file dumps. Easier to read than long binary strings.