Bitwise Operators on Integers

A bitset is a fixed number of bits, and bitwise operators let you act on the whole set in a single expression. Reading a line like that and knowing the answer is a useful skill, because it lets you understand bitset code without tracing each bit by hand. This article teaches the two families of operators behind those expressions. The first family combines or inverts bits, one position at a time. The second family, the shift operators, moves every bit a given number of positions to the left or to the right.

Other bitwise operators, such as OR and XOR, follow the same position-by-position pattern with their own rules. OR gives a 1 when at least one input bit is 1, and XOR gives a 1 when the two input bits differ. Once you know that the operation is applied to each position separately, you only need the per-bit rule for each operator to predict the whole result.

The clearest way to read a bitwise result is to go through the positions from left to right and apply the per-bit rule at each one. The diagram below does this for an AND of two four-bit values, 1011 and 1110. Each position is treated separately, and the input bits at that position are shown beneath the result bit.

Reading the diagram from left to right, the first position has 1 and 1, so the result is 1. The second position has 0 and 1, so the result is 0, because at least one input is 0. The third position has 1 and 1, giving 1. The last position has 1 and 0, giving 0. The full result is 1010.

Consider what happens when a bitset is combined with its own complement. At every position, the two bits are opposites, so one of them is always 0. An AND of a bit with its own opposite is therefore always 0, because the two bits can never both be 1. This means every position of the result is 0, and the whole result is all zeros.

The shift operators move every bit of a bitset a given number of positions in one direction. The number written after the operator sets how far the bits travel. The two directions are handled by two operators, and both follow the same rule about what fills the gap they leave behind. The subsections below take each direction in turn.

A bitset has a fixed width, meaning it holds a set number of positions. That width decides which bits exist at all. A complement is applied across every position the bitset has, so the same value can produce different bit patterns at different widths. The diagram below shows the complement of the four-bit value 1011 written out in an eight-bit bitset, where the value sits in the lower four positions and the upper four are 0 before the complement.

In a four-bit bitset, the complement of 1011 is 0100. In an eight-bit bitset holding the same value, the complement is 11110100. The two results are not the same bit pattern, and that difference is one of the most common reasons a bitset result looks wrong. Shifts follow the same width rule: a bit shifted past the left edge is discarded rather than wrapping around to the right end.

Different operators combine the same bits in different ways. The table applies several operators to the same two four-bit values, 1011 and 1110. The complement is applied to 1011 alone, and the shifts are applied to 1011 alone as well. Comparing the rows shows how the operator decides the outcome at each position.

When a bitset result differs from what you expected, the cause is almost always one of three things: the width, the direction of the shift, or the operator. Checking them in a fixed order helps you find the mismatch quickly. Start with the width, because it decides which bits exist. Then check the shift direction, because it decides which end the bits leave from and which end the zeros enter. Finally, check the operator, because it decides the rule applied at each position.