What is a population count?

The population count, also called the Hamming weight, is the number of bits set to 1 in a number's binary representation. For example, the byte 10110110 has a population count of 5.

How is it computed efficiently?

This tool uses Kernighan's algorithm: repeatedly clearing the lowest set bit with x AND (x minus 1) and counting how many iterations it takes. That loops once per set bit rather than once per total bit.

What is popcount used for?

It appears in error-correcting codes, cryptography, bitmap database indexes, similarity hashing, and game engines that store board state in bitboards. Most modern CPUs have a dedicated POPCNT instruction for it.

Can it handle very large numbers?

Yes. The calculator uses JavaScript BigInt, so there is no 32-bit or 64-bit limit. You can count the set bits in numbers hundreds of digits long.

Why does the zero count depend on width?

Leading zeros are infinite in principle, so the zero count shown is relative to the minimal bit width needed to write the number. A value of 6 is 110 in binary, giving one zero bit, not infinitely many.

Bit Population Count (popcount)

Population count, also known as the Hamming weight or simply popcount, answers a deceptively useful question: how many bits in this number are set to 1? This tool computes it for any non-negative integer entered in decimal, hex, or binary, and shows the binary form so you can verify the count by eye.

How it works

The tool counts set bits using Kernighan’s bit-clearing trick rather than checking every bit position:

count = 0
while x != 0:
    x = x & (x - 1)   // clears the lowest set bit
    count += 1

Subtracting 1 flips the lowest set bit to 0 and turns the zeros below it into ones; ANDing with the original then clears that bit and everything below. Because the loop runs exactly once per set bit, it is faster than scanning all bits when set bits are sparse. The zero count is the total bit width of the number minus the population count.

Example and notes

The decimal value 182 is 10110110 in binary, which has five 1 bits, so its population count is 5 and it has three zero bits across its eight-bit width. Real CPUs expose this as the POPCNT (x86) or CNT (ARM) instruction, and it underpins bitboard chess engines, Bloom-filter saturation checks, and Hamming distance between hashes. This calculator uses BigInt, so there is no upper limit on the size of the integer you can analyse.