How is System.out.println(4*2147483647) equal to -4 in java?

Question

I am a Java beginner and pretty confused with this.

How is System.out.println(4*2147483647) equal to -4 in java?

Answer 1

It is due to integer silent overflow.

2147483647 == Integer.MAX_VALUE is the maximum value for an integer.

Silent overflow means that 2147483647 + 1 == Integer.MIN_VALUE = -2147483648
You can then see that 2147483647 + 2147483647 == 2147483647 + (-2147483648 + - 1) == -2

In other words, 2147483647 * 2 == -2, and you can now see why 2147483647 * 4 == -4.

More technically, the result is defined by the Java Language Specification #15.17.1:

If an integer multiplication overflows, then the result is the low-order bits of the mathematical product as represented in some sufficiently large two's-complement format. As a result, if overflow occurs, then the sign of the result may not be the same as the sign of the mathematical product of the two operand values.

Answer 2

It has to do with the way processors perform a binary mutiplication. The number you have in there is the maximum signed integer, which is represented as 0111111111111111111111111 (I haven't checked the number of ones there, but I assume you get the idea).

When you mutiply by 4, is like doing a left shift by 2, which results in 11111111111111111111100 (which represents -4). You might want to read how mutiplication is done in binary.