Why is the maximum value of an unsigned n-bit integer 2^n-1 and not 2^n?

Question

The maximum value of an n-bit integer is 2ⁿ-1. Why do we have the \"minus 1\"? Why isn\'t the maximum just 2ⁿ?

Answer 1

In most programming languages integer is a signed value (see two's complement).

For example, in Java and .NET integer most left byte is reserved for sign:

• 0 => positive or zero number
• 1 => negative number

Then the maximum value for 32-bit number is limited by 2^31. And adding -1 we get 2^31 - 1.

Why does -1 appear?

Look at more simple example with unsigned Byte (8-bits):

  1  1  1  1  1  1  1  1
128 64 32 16  8  4  2  1  <-- the most right bit cannot represent 2
--- --------------------
128 + 127 = 255 

As other guys pointed out the most right bit can have a maximum value of 1, not 2, because of 0/1 values.

Int32.MaxValue = 2147483647 (.NET)