Base Converter

Multiply each binary digit by 2 raised to its position (starting from 0 on the right) and sum the results. For example, binary 1010 = (1×8) + (0×4) + (1×2) + (0×1) = 10. The calculator does this automatically — just enter your binary number and select 'All Bases' as the output to see the decimal equivalent alongside hex, octal, and padded binary.

Two's complement is how computers represent signed (negative and positive) integers in binary. The leftmost bit indicates sign, and the value is derived by inverting all bits and adding 1. Toggle 'Show Two's Complement' to see the signed representation of your number at your selected bit width. This is useful for low-level programming, digital logic design, and understanding how integer overflow works.

Hexadecimal compactly represents binary data. Each hex digit corresponds to exactly 4 bits. Common uses include: color codes in web design (#FF5733), memory addresses in debugging, Unicode code points (U+0041 for 'A'), MAC addresses (00:1A:2B:3C:4D:5E), and machine code representation. The calculator shows the hex equivalent for any number you enter.

Leading zeros appear when binary padding is enabled and the number doesn't fill the selected bit width. For example, decimal 15 in 8-bit padded binary is 00001111 — the four leading zeros ensure exactly 8 bits are displayed. This is standard in digital systems where every value occupies a fixed number of bits. Disable padding or change the bit width to adjust the output format.

Octal (base 8) uses digits 0-7, with each digit representing 3 bits. Hexadecimal (base 16) uses 0-9 and A-F, with each digit representing 4 bits. Octal was popular in early computing when 12, 24, and 36-bit word sizes were common — since 3 divides evenly into those. Today, hexadecimal is far more common because modern architectures use 8/16/32/64-bit sizes where 4-bit nibbles are the natural grouping. Octal persists mainly in Unix file permissions (chmod 755).