Exponents and Bit Manipulation

Every power of $2$ maps to a single bit. $2^0 = 1$ is bit $0$. $2^3 = 8$ is bit $3$. In code, $2^k$ is just 1 << k (left shift by $k$).

This connection is everywhere. Checking if bit $k$ is set in a number $n$: n & (1 << k). Setting bit $k$: n | (1 << k). Clearing bit $k$: n & ~(1 << k).

A number with all $k$ lower bits set is $2^k - 1$. In binary, $2^4 - 1 = 15 = 1111_2$. You get this with (1 << k) - 1, and it's useful as a bitmask.

Checking if $n$ is a power of $2$: n & (n - 1) == 0. A power of $2$ has exactly one bit set. Subtracting $1$ flips that bit and sets all lower bits, so the AND gives zero.