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

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.

Answer 2

In most programming languages, 0 is a number too.

Answer 3

If you're just starting out with programming, I suggest you take a look at this wiki article on signed number representations

As Vicente has stated, the reason you subtract 1 is because 0 is also an included number. As a simple example, with 3 bits, you can represent the following non-negative integers

0 : 000
1 : 001
2 : 010
3 : 011
4 : 100
5 : 101
6 : 110
7 : 111

Anything beyond that requires more than 3 digits. Hence, the max number you can represent is 2^3-1=7. Thus, you can extend this to any n and say that you can express integers in the range [0,2^n -1]. Now you can go read that article and understand the different forms, and representing negative integers, etc.

Answer 4

Because 0 is also represented. The amount of numbers you can represent is indeed 2^n with n bits, but the maximum number is 2^n-1 because you have to start the count in 0, that is, every bit set to 0.

For 1 bit: 0, 1
For 2 bits: 0, 1, 2, 3
For 3 bits: 0, 1, 2, 3, 4, 5, 6, 7

And so on.

Answer 5

In the field of computing we start counting from 0.

Answer 6

Why do we have the "minus 1"?

Just answer the question: What is the maximum value of an 1-bit integer?

One bit integer can store only two (2¹) values: 0 and 1. Last value is 1₂ = 1₁₀

Two bit integer can store only four (2²) values: 00, 01, 10 and 11. Last value is 11₂ = 3₁₀

Thus, when integer can stores N, values last value will be N-1 because counting starts from zero.

n bit integer can store 2n values. Where last will be 2ⁿ-1

Example: One byte can store 2⁸ (256) values. Where first is 0 and last is 255

Why isn't the maximum just 2n?

Because counting starts from zero. Look at first value for any n bit integer.
For example byte: 00000000

This would be very confusing if:
00000001 means 2
00000000 means 1

would not? ;-)