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

User · Answer

A better way to solve it is to start small.

Let's start with 1 bit. Which can either be 1 or 0. That's 2 values, or 10 in binary.

Now 2 bits, which can either be 00, 01, 10 or 11 That's 4 values, or 100 in binary... See the pattern?

User · Answer

What you re missing  Zero is a value

User · Answer

There s an easier way to think about this  Start with 1 bit  This can obviously represent 2 values  0 or 1   What happens when we add a bit  We can now represent twice as many values  the values we could represent before with a 0 appended and the values we could represent before with a 1 appended   So the the number of values we can represent with n bits is just 2 n  2 to the power n

User · Answer

Without wanting to give you the answer here is the logic   You have 2 possible values in each digit  you have 9 of them   like in base 10 where you have 10 different values by digit say  you have 2 of them  which makes from 0 to 99    0 to 99 makes 100 numbers  if you do the calcul you have an exponential function  base numberOfDigits  10 2   100   2 9   512

User · Answer

29   512 values  because that s how many combinations of zeroes and ones you can have     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   lt - yay integer overflow 100000001   -255     111111110   -2 111111111   -1   In general  with k bits you can represent 2k values  Their range will depend on the system you are using      Unsigned  0 to 2k-1   Signed  -2k-1 to 2k-1-1

User · Answer

The thing you are missing is which encoding scheme is being used   There are different ways to encode binary numbers   Look into signed number representations   For 9 bits  the ranges and the amount of numbers that can be represented will differ depending on the system used

User · Answer

Okay  since it already  leaked   You re missing zero  so the correct answer is 512  511 is the greatest one  but it s 0 to 511  not 1 to 511    By the way  an good followup exercise would be to generalize this   How many different values can be represented in n binary digits  bits

[binary] How many values can be represented with n bits?

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