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

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?

Solution 1:

The -1 is because integers start at 0, but our counting starts at 1.

So, 2^32-1 is the maximum value for a 32-bit unsigned integer (32 binary digits). 2^32 is the number of possible values.

To simplify why, look at decimal. 10^2-1 is the maximum value of a 2-digit decimal number (99). Because our intuitive human counting starts at 1, but integers are 0-based, 10^2 is the number of values (100).

Solution 2:

2^32 in binary:

1 00000000 00000000 00000000 00000000

2^32 - 1 in binary:

11111111 11111111 11111111 11111111

As you can see, 2^32 takes 33 bits, whereas 2^32 - 1 is the maximum value of a 32 bit integer.

The reason for the seemingly "off-by-one" error here is that the lowest bit represents a one, not a two. So the first bit is actually 2^0, the second bit is 2^1, etc...

Solution 3:

2³² in binary is one followed by 32 zeroes, for a total of 33 bits. That doesn't fit in a 32-bit int value.