binary - How many values can be represented with n bits?

Question

For example, if n=9, then how many different values can be represented in 9 binary digits (bits)?

My thinking is that if I set each of those 9 bits to 1, I will make the highest number possible that those 9 digits are able to represent. Therefore, the highest value is 1 1111 1111 which equals 511 in decimal. I conclude that, therefore, 9 digits of binary can represent 511 different values.

Is my thought process correct? If not, could someone kindly explain what I'm missing? How can I generalize it to n bits?

Answer 1

2⁹ = 512 values, because that's how many combinations of zeroes and ones you can have.

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What those values represent however will depend on the system you are using. If it's an unsigned integer, you will have:

000000000 = 0 (min)
000000001 = 1
...
111111110 = 510
111111111 = 511 (max)

In two's complement, which is commonly used to represent integers in binary, you'll have:

000000000 = 0
000000001 = 1
...
011111110 = 254
011111111 = 255 (max)
100000000 = -256 (min) <- yay integer overflow
100000001 = -255
...
111111110 = -2
111111111 = -1

In general, with k bits you can represent 2^k values. Their range will depend on the system you are using:

Unsigned: 0 to 2^k-1
Signed: -2^k-1 to 2^k-1-1