Binary to Decimal Converter

The positional method

A binary number is a string of bits, each worth a power of two. Reading right to left, the positions carry the weights 1, 2, 4, 8, 16, 32, 64, 128 and so on — each one double the previous. To convert, multiply every bit by its position weight and add the results together. Bits that are 0 contribute nothing, so in practice you only sum the weights where a 1 appears.

Take the binary value 1011. From the right, the bits sit at positions 0, 1, 2 and 3, with weights 1, 2, 4 and 8:

1 0 1 1
8 4 2 1   ← position weights
8+0+2+1 = 11

So 1011 in binary equals 11 in decimal. The same routine scales to any length — an 8-bit byte simply has eight columns, with the leftmost worth 128.

The doubling shortcut

For longer numbers there is a faster mental trick that avoids large powers. Start with 0, then walk the bits left to right: double your running total and add the current bit. For 1011 the steps are: start 0, then 0×2+1 = 1, then 1×2+0 = 2, then 2×2+1 = 5, then 5×2+1 = 11. You reach the answer with nothing harder than doubling and adding one.

This is exactly how a computer's hardware evaluates a binary string, and it is the algorithm our converter uses internally with BigInt so that even hundred-digit binary numbers stay exact.

Reference table

The last row is worth memorising: eight set bits make 255, the largest value a single byte can hold without a sign. That is why color channels, IP address octets and many hardware registers top out at 255.

Where you'll use it

• Reading the output of bitwise operations and permission masks (e.g. Unix chmod bits).
• Decoding network subnet masks and converting IP octets between binary and dotted-decimal.
• Interpreting microcontroller register values and sensor flags in embedded projects.
• Checking answers in computer-architecture and digital-logic coursework.

Signed binary

Plain positional conversion assumes an unsigned value. If your binary represents a signed integer using two's complement, the most significant bit is a sign bit: when it is 1 the number is negative. To read it, flip every bit, add one, convert the result normally and prepend a minus sign — or just paste it into the converter with an 8/16/32/64-bit width selected and let it handle the sign for you.