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

If I ask you what is the biggest value you can fit into a 2-digit number, would you say it's 10² (100) or 10²-1 (99)? Obviously the latter. It follows that if I ask you what the biggest n-digit number is, it would be 10ⁿ-1. But why is there the "-1"? Quite simply, because we can represent 0 in a 2-digit number also as 00 (but everyone just writes 0).

Let's replace 10 with an arbitrary base, b. It follows that for a given base, b, the biggest n-digit number you can represent is bⁿ-1. Using a 32-bit (n = 32) base-2 (b = 2) number, we see that the biggest value we can represent 2³²-1.

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Another way of thinking about it is to use smaller numbers. Say we have a 1-bit number. Would you tell me the biggest value it can represent is 2¹ or 2¹-1?