Bitwise Calculator

The Bitwise Operators

A bitwise operator ignores the numeric value of its operands and works on their individual binary digits, comparing bit 0 with bit 0, bit 1 with bit 1, and so on down the word. The behaviour of each operator is fully described by a truth table covering the four possible input pairs.

AND, OR, XOR

AND produces a 1 only when both input bits are 1, which makes it the natural tool for masking. OR produces a 1 when either bit is 1, which makes it the tool for setting bits. XOR (exclusive or) produces a 1 only when the two bits differ, which makes it the tool for toggling and for cheap reversible transformations.

NAND, NOR, XNOR

These are the negated forms, the outputs of AND, OR, and XOR with every bit flipped. NAND and NOR matter in hardware because either one alone is functionally complete: any logic circuit whatsoever can be built from nothing but NAND gates, which is why they are the cheapest primitive to fabricate. XNOR outputs 1 when the bits match, so it is an equality test at the bit level.

NOT

NOT is the only unary operator here. It inverts every bit of its single operand, turning each 1 into a 0 and each 0 into a 1.

Left Shift and Right Shift

Shifting slides every bit sideways by a given number of positions. Shifting left by n positions multiplies the value by 2 to the power n, filling the vacated low bits with zeros. Shifting right by n positions divides by 2 to the power n and discards whatever falls off the right hand end, so it always rounds toward zero rather than rounding to nearest.

Worked Examples

Write both operands in binary, line them up, and apply the operator column by column. Using 12 (binary 1100) and 10 (binary 1010):

What Bitwise Operations Are Used For

Bit flags and permission masks

A single integer can carry dozens of independent booleans, one per bit. Assign each option a power of two, then combine them with OR and interrogate them with AND. This keeps configuration compact in storage and makes checks a single CPU instruction.

Unix file permissions

The familiar chmod 755 is exactly this scheme written in octal. Each octal digit holds three permission bits for one class of user: 7 is 111 (read, write, execute), 5 is 101 (read and execute, no write). Owner, group, and others get one digit each.

Packed colour values

An RGB colour is often stored as a single integer with red in the high byte, green in the middle, and blue in the low byte. Shifting and masking pulls the channels back out.

Hashing and fast arithmetic

Hash functions lean heavily on XOR and shifts because they mix bits quickly and reversibly. Compilers also turn multiplication and division by powers of two into shifts, since a shift costs a single cycle where a division can cost twenty or more.

Signed vs Unsigned Right Shift

JavaScript has two right shift operators and the difference matters. The arithmetic shift >> preserves the sign bit, copying it into the vacated high positions, so a negative number stays negative. The logical shift >>> always fills with zeros, which turns a negative value into a large positive one. It is also the idiomatic way to coerce a value to an unsigned 32-bit integer, using x >>> 0.

One more limitation worth knowing: JavaScript converts operands to 32-bit integers before applying any bitwise operator, so values above 2 to the power 31 behave unexpectedly and bit operations silently truncate anything larger. For wider values use BigInt, which supports the same operators without the 32-bit ceiling. The calculator above works on 32-bit unsigned values.