Why has the int32 type a maximum value of 2³¹ − 1?

Question

Possible Duplicate:
What is "2's Complement"?

I know int32 has a length of 32 bits (4 bytes). I assume it has 2³² values but as half of them must be under zero, I guess it has something to do with this. I would like to know why exactly int32 has max. positive number 2³¹ − 1.

Solution 1

This most significant bit is used to code the sign (1 meaning negative), so only 31 bits are available for the actual value.

Int32.MaxValue =  2^31 - 1 = 01111111111111111111111111111111
                  1        = 00000000000000000000000000000001
                  0        = 00000000000000000000000000000000
                 -1        = 11111111111111111111111111111111
Int32.MinValue = -2^31     = 10000000000000000000000000000000

Solution 2

2³² possible values

− 2³¹ values used for negative integers

− 1 value used for zero

= 2³¹ − 1 values available for positive integers

Solution 3

2³² is about 4.2 billion. This is the maximum number of VALUES that a binary number with 32 digits (a 32-bit number) can represent.

Those values can be any values in any range. In an UNSIGNED 32-bit number, the valid values are from 0 to 2³² − 1 (instead of 1 to 2³², but the same number of VALUES, about 4.2 billion).

In a SIGNED 32-bit number, one of the 32 bits is used to indicate whether the number is negative or not. This reduces the number of values by a factor of 2¹, or by half. This leaves 2³¹, which is about 2.1 billion. This means the range is now about −2.1 billion to 2.1 billion. Same number of values, different range.

Solution 4

You have 2^31 values below zero (minimum value = -2^31), 2^31-1 values above zero and zero itself. That makes 2^31 + 2^31-1 + 1 = 2*2^31 = 2^32 values :) ...

The other explanation involves the way how negative numbers are represented (using the two-complement): Shortly, the most-significant bit indicates a negative number, so you have 2^31 positive numbers (including zero) left, which gives us the range 0..2^31-1

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Comments

[Shruti Saagar]

If we are going to consider 0 as positive number(hypothetically), then we have equal number of values besides the imaginary line between -1 and 0 .

[ShadowRanger]

@HashimAziz: The flaw being that, without referencing two's complement representation, it doesn't explain why the maximum value is 2³¹ − 1, while the minimum is -2³¹.