Binary to Decimal Converter

Each binary digit (bit) represents a power of 2, starting from the rightmost bit (2⁰). To convert, multiply each bit by its positional value and sum the results. Example: 1011₂ = (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11₁₀. This calculator shows the full step-by-step breakdown.

Repeatedly divide the decimal number by 16 and record the remainders. Read the remainders bottom to top. Remainders 10–15 become A–F. Example: 255 ÷ 16 = 15 remainder 15 (F), then 15 ÷ 16 = 0 remainder 15 (F). So 255₁₀ = FF₁₆. Hex is used in web colors (#FF5733), memory addresses, and CSS.

Hexadecimal (base 16) uses digits 0–9 and letters A–F. Developers use it because each hex digit exactly represents 4 binary bits (a nibble), so one byte (8 bits) is two hex digits. This makes hex compact and easy to read: FF₁₆ = 11111111₂ = 255₁₀. Hex is used for memory addresses, color codes, file signatures, and encoding.

Octal (base 8) uses digits 0–7. Each octal digit represents exactly 3 binary bits. Unix/Linux file permissions use octal — chmod 755 means rwxr-xr-x in binary (111 101 101). Octal was common in older computing systems before hex became dominant. It's still used in Unix permissions and some assembly languages.

Two's complement is the standard way computers represent negative integers in binary. To negate a number: invert all bits (one's complement) then add 1. Example: +5 = 0101₂, -5 = 1010 + 1 = 1011₂. This system has one representation of zero and makes arithmetic circuits simpler. Most modern CPUs use two's complement for signed integers.

A bit is a single binary digit (0 or 1). A byte is 8 bits. A nibble is 4 bits (half a byte). Modern computers process data in bytes. Memory is measured in bytes (KB, MB, GB). Network speed is measured in bits per second (Mbps, Gbps). So 100 Mbps internet = 100/8 ≈ 12.5 MB/s actual download speed.