Lesson 34.1 · Bit Manipulation

Binary, Two's Complement and the Operators

Java ints are 32-bit two's complement. & | ^ ~ work bit by bit; << shifts left; >> shifts right keeping the sign; >>> shifts right filling with zeros.

14 min

Think of it like this

A row of 32 light switches. Each operator is a rule for combining two rows of switches: & keeps a light on only if both are on, | if either is on, ^ if exactly one is on, ~ flips every switch.

1.Binary numbers

Bit i is worth 2ⁱ: 12 = 8 + 4 = 1100. Two's complement stores negatives as ~x + 1, so −1 is all 32 bits set, the top bit is the sign, and the range is −2³¹ to 2³¹ − 1.

x & y (AND), x | y (OR), x ^ y (XOR), ~x (NOT). x << k multiplies by 2ᵏ (bits fall off the top). x >> k divides by 2ᵏ rounding down, copying the sign bit. x >>> k fills with zeros, treating x as unsigned.

Precedence trap: == binds tighter than &, so x & 1 == 0 doesn't compile; write (x & 1) == 0.

Main.java

public class Main {
    public static void main(String[] args) {
        int a = 12, b = 10;                         // 1100 and 1010
        System.out.println("a & b = " + (a & b));
        System.out.println("a | b = " + (a | b));
        System.out.println("a ^ b = " + (a ^ b));
        System.out.println("~a = " + (~a));
        System.out.println("a << 1 = " + (a << 1));
        System.out.println("a >> 2 = " + (a >> 2));
        System.out.println("-8 >> 1 = " + (-8 >> 1));
        System.out.println("-8 >>> 28 = " + (-8 >>> 28));
        System.out.println("-8 in binary = " + Integer.toBinaryString(-8));
    }
}

Output

a & b = 8
a | b = 14
a ^ b = 6
~a = -13
a << 1 = 24
a >> 2 = 3
-8 >> 1 = -4
-8 >>> 28 = 15
-8 in binary = 11111111111111111111111111111000

Remember

Two's complement: −x = ~x + 1.
>> keeps the sign, >>> doesn't.
Parenthesise bit expressions in comparisons.

Common mistakes

Using >> on negative numbers when an unsigned shift was intended.
Shifting by 32 or more (Java uses only the low 5 bits of the shift count).

Words used in this lesson

Bit: A single binary digit, 0 or 1.
Two's complement: The standard way computers store negative integers: flip all bits and add 1.
Mask: An int whose set bits select the positions you care about.

Command Palette

Binary, Two's Complement and the Operators

1.Binary numbers