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

Answer 1

Why this behavior?

It is a mathematical consequence of the choice of representation used in all modern computers.

Here's an informal proof of why it has to be that way.

1. A signed binary number representation with N bits has 2^N possible values; i.e. a set integers with an even number of elements.

2. Remove zero from the set. The set is now has an odd number of elements.

3. Now remove all pairs of numbers of the form {n, -n}. Each time we remove a pair of numbers, the set still contains an odd number of elements.

4. We are now left with a set that contains an odd number of integers for which n is in the set, but -n is not in the set. Since the set size is odd, it cannot be empty; i.e. there is at least one number with this property.

In the representations specified by Java (and indeed used all other practical languages), integer types have 2^N-1 negative values and 2^N-1 - 1 values greater than zero. The value that has the strange property is MIN_VALUE. This representation is called two's complement.

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Strictly speaking, an integer representation doesn't have to have this anomaly:

• It is possible to have two zeros (-0 and +0); e.g. signed magnitude or one's complement representations. (This invalidates step 2 of the proof: there are now 2 zeros to remove.)

• It is possible to exclude -2^N-1; i.e. make it an illegal value. (This invalidates step 1 of the proof: the initial set now has an odd number of values.)

• It is possible to specify integer arithmetic so that (for example) negating MIN_VALUE raises an exception. For instance Math.abs(Integer.MIN_VALUE) would throw an exception.

However, all of these things have significant performance implications, especially since modern computer hardware only supports twos-complement arithmetic natively. They also have issues in relation to writing reliable integer codes ...