Binary to Decimal Converter

Binary is base-2 positional notation: every digit is worth 2 to the power of its position counted from the right, starting at 0. To convert, you add up the place values of every 1 bit. For 1010 that is 2^3 + 2^1 = 8 + 2 = 10. For 11111111 it is 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. The same rule scales to any length: the leftmost bit of an N-bit string is worth 2^(N-1), the rightmost is always worth 1.

Two's complement is how almost every CPU stores signed integers. In an N-bit signed field, if the leading bit is 1, the value is parseInt(s, 2) - 2^N. So 11111111 read as unsigned 8-bit is 255, but read as signed 8-bit it is 255 - 256 = -1. 10000000 is the most negative value, -128. The trick is that addition and subtraction work the same way for negative numbers as for positive ones with no special hardware: -1 + 1 wraps to 0 inside the 8 bits, exactly as the math demands. Choose the right width on this converter to see what your bit pattern actually encodes.

The bit length sets two things: the maximum representable value and where the sign bit sits. An unsigned 8-bit field holds 0 to 255, signed 8-bit holds -128 to 127. Double the width and the unsigned ceiling jumps to 65,535 (16-bit) or 4,294,967,295 (32-bit), while the signed range becomes -32,768 to 32,767 or -2,147,483,648 to 2,147,483,647. The exact same bit pattern 11111111 means 255 in an 8-bit unsigned register, 255 in a 16-bit register (zero-extended on the left), and -1 only when you tell the CPU to interpret it as signed 8-bit. Width is part of the type, not part of the bits.

The integer side uses positive powers of 2 and the fractional side uses negative powers. In 101.011, the fractional places are 2^-1, 2^-2, and 2^-3, so the value is 4 + 1 + 0.25 + 0.125 = 5.375. Binary floating-point formats use a significand and exponent; decimals such as 0.1 require rounding because their binary expansion repeats.

Hex is shorthand for binary in groups of 4 bits, and octal is shorthand in groups of 3 bits, because 16 and 8 are both powers of 2. One hex digit always equals exactly 4 binary digits: 0xF is 1111, 0xA is 1010, so 0xFF is 11111111 = 255. One octal digit equals exactly 3 binary digits: 0o7 is 111. This is why hex is the standard way to write byte values in memory dumps, color codes (#ff5733), MAC addresses, and SHA hashes: every two hex digits is exactly one byte. The converter shows all three on every result so you can drop straight into whichever notation your tool expects.