Why is absolute of Integer.MIN_VALUE equivalent to Integer.MIN_VALUE

Question

In java when I say Integer i = Math.abs(Integer.MIN_VALUE). I get the same value as the answer,which means i contains Integer.MIN_VALUE. I have verified the same in C++ as well.

Why this behavior?

Answer 1

Read this in Effective Java by Joshua Bloch.

I found the answer for this question and here is the explanation: Computers work with binary arithmentic, the logic of Math.abs in java or the absolute function in any language is like below:

if(num >= 0)
    return num;
else
    return (2's complement of the num);

Note : How to find 2's complement

For a given number, we find it's 1's complement first and then add 1 to it. For e.g. Consider our number to be 10101 1's complement= 01010 2's complement= 01011 (added 1 to the 1`s complement)

Now, for the sake of making it easy and clear, let us say that our Integer(signed) size is 3 bit, then here is the possible list of numbers which can be produced using the four bits:

000 --> 0 (0)
001 --> 1 (1)
010 --> 2 (2)
011 --> 3 (3) 
100 --> 4 (-4)
101 --> 5 (-3)
110 --> 6 (-2)
111 --> 7 (-1)

Now that this is signed, it means half of the numbers are negative and the other half are positive(The negative numbers are the ones with the first bit 1). Let us start from 000 and try to find its negative number, it would be the two's complement of 000.

2's complement of `000` = 1 + `111` = `000`
2's complement of `001` = 1 + `110` = `111`
2's complement of `010` = 1 + `101` = `110`
2's complement of `011` = 1 + `100` = `101`
2's complement of `100` = 1 + `011` = `100`
2's complement of `101` = 1 + `010` = `011`
2's complement of `110` = 1 + `001` = `010`  
2's complement of `111` = 1 + `000` = `001`

From the above demonstration, we find that 2's complement of 111(-1) is 001(1), similarly 2's complement of 110(-2) is 010(2), 2's complement of 101(-3) is 011(3) and 2's complement of 100(-4) is 100(-4) and as we can see that -4 is the smallest negative number possible using 3 bits.

This is the reason why absolute of Integer.MIN_VALUE is Integer.MIN_VALUE.

Answer 2

There is an inherent asymmetry that is the root cause of this effect. The number of 32-bit bit patterns is even. One of those patterns is used for zero. That leaves an odd number of non-zero values. The number of positive values and the number of negative values cannot be equal because their sum is odd.

In the 2's complement representation used for Java integers, the number of negative numbers is one greater than the number of positive numbers. Each negative number other than Integer.MIN_VALUE corresponds to a positive number that is both its negation and its absolute value. Integer.MIN_VALUE is left over, with no corresponding positive int. Math.abs and negation map it to itself.