Why is the max integer in java 2^31 - 1 and not 2^31 [duplicate]

Question

Sorry if this is a really basic question, but why is there a minus one for the positive side?

Does it have to do with the zero being stored or something? I thought computing the highest possible decimal number for binary would just be to add the powers of two up, like for a 3 bit unsigned it would be

1*2^0 + 1*2^1 + 1*2^2 = 7

Shouldn't the same rule apply for java integers? Thanks

Answer 1

Because Java can support max signed int as 0x7fffffff which is 2^31-1.

2^31 = 0x80000000 is negative so Positive is 2^31-1

Binary level comparasion would be:

10000000000000000000000000000000  --> 2147483648 --> 2^31
01111111111111111111111111111111  --> 2147483647 --> 2^31 -1
^ Sign bit

Answer 2

The same rule does apply... 7 is 2^3 - 1. And yes, it's because of the 0. :)

In contrast, negatives go to -(2^31)

So there's 2^31 negative numbers, one 0, and 2^31-1 strict positives, which add to...

2^31 + 1 + 2^31 - 1 = 2 * 2^31 = 2^32