The range of numbers of an n-bit cell in general?

Apr 2013
69
1
USA
With unsigned binary representation, what is the range of numbers as written in binary
and in decimal for the following cells?

(e)an n-bit cell in general

This problem comes from the Computer Systems by Warford book if that helps. I don't know how to write it. I know for example a three bit cell the answer is:
in binary 000-111 and decimal 0 to 7.
 

romsek

MHF Helper
Nov 2013
6,838
3,079
California
With unsigned binary representation, what is the range of numbers as written in binary
and in decimal for the following cells?

(e)an n-bit cell in general

This problem comes from the Computer Systems by Warford book if that helps. I don't know how to write it. I know for example a three bit cell the answer is:
in binary 000-111 and decimal 0 to 7.
in general, in base k, n digits can represent kn numbers so if 0 is to be your first number you have the range of [0,kn-1].

for binary k=2
for decimal k=10
 
Apr 2013
69
1
USA
in general, in base k, n digits can represent kn numbers so if 0 is to be your first number you have the range of [0,kn-1].

for binary k=2
for decimal k=10
And for two's complement binary representation, with a 7-bit cell, would the range be [-7,6]? I'm guessing the formula could be [-n, to n-1], where n is the number of bits in the cell. but that's decimal . I don't know how to write it for binary.
 

romsek

MHF Helper
Nov 2013
6,838
3,079
California
And for two's complement binary representation, with a 7-bit cell, would the range be [-7,6]? I'm guessing the formula could be [-n, to n-1], where n is the number of bits in the cell. but that's decimal . I don't know how to write it for binary.
\(\displaystyle [-2^{n-1}, 2^{n-1}-1]\) for an n bit, binary string, so for n=7 the range is \(\displaystyle [-2^6,2^6-1]=[-64,63]\)