Convert binary numbers to decimal instantly with full step-by-step positional breakdown. Supports signed two's complement, fractional binary, and batch conversion of multiple values at once.
Type or paste one or more binary numbers (0s and 1s). For batch mode, enter one binary number per line. For signed two's complement, enable the signed toggle. Fractional binary like 101.11 is also supported.
The decimal equivalent appears instantly as you type. The step-by-step panel shows the positional breakdown — each bit multiplied by its power of 2, summed to give the final result.
The result panel also shows the equivalent in hex and octal. Click Copy to copy the decimal result. In batch mode, all results are displayed in a table you can copy or download as CSV.
Each bit in a binary number represents a power of 2, starting from 2⁰ on the right. To convert, multiply each bit by its corresponding power of 2 and sum all the results. For example, binary 1011 = (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11 in decimal. This tool shows every step of this calculation.
Binary is base-2, using only two digits: 0 and 1. Computers use binary because transistors — the fundamental building blocks of processors — naturally exist in two states: on (1) and off (0). All data in a computer, whether text, images, or instructions, is ultimately stored and processed as sequences of these 0s and 1s at the hardware level.
Two's complement is the standard method computers use to represent negative binary integers. In an 8-bit signed number, the leftmost bit is the sign bit — 0 means positive, 1 means negative. To convert a negative two's complement binary number, invert all bits and add 1. For example, 11110110 in 8-bit two's complement = -(00001010) = -10. Enable the Signed toggle in this tool to handle two's complement correctly.
A fractional binary number has a binary point (like a decimal point) separating the integer part from the fractional part. Bits to the right of the binary point represent negative powers of 2: 2⁻¹ (0.5), 2⁻² (0.25), 2⁻³ (0.125), and so on. For example, 101.11 = 4 + 1 + 0.5 + 0.25 = 5.75. This is the basis for how floating-point numbers are represented in computer hardware.
This converter handles binary numbers up to 64 bits, which corresponds to decimal values up to 18,446,744,073,709,551,615 (unsigned) or ±9,223,372,036,854,775,807 (signed 64-bit). For educational purposes, most conversions involve 4, 8, 16, or 32-bit numbers corresponding to nibbles, bytes, words, and double-words in computer architecture.
Binary, hexadecimal (base-16), and octal (base-8) are closely related because 16 = 2⁴ and 8 = 2³. Every single hex digit represents exactly 4 binary bits (e.g. hex F = binary 1111). Every octal digit represents exactly 3 binary bits (octal 7 = binary 111). This makes hex and octal convenient shorthand for reading and writing binary values in programming.
