GATE CSE 1994 | Question: 2.7

Question

Consider $n$-bit (including sign bit) $2's$ complement representation of integer numbers. The range of integer values, $N$, that can be represented is ______ $\leq N \leq $______ .

Comments

https://chortle.ccsu.edu/AssemblyTutorial/Chapter-08/ass08_20.html

Answer 1

$-2^{n-1} \leq N \leq 2^{n-1} -1$

Example : Let us have $3$ bit binary numbers (unsigned )

$000\; (0_{10})$ to $111(7_{10})$  total of $8 (2^3)$  numbers.

But when we have one sign bit then we have half the number of negatives $-4$ to $-1,$ $0$ and $1$ to $3.$

  bit pattern:    100   101  110  111  000  001  010  011

  1's comp:       -3     -2   -1    0   0    1    2    3

  2's comp.:      -4     -3   -2   -1   0    1    2    3

Comments

An n-bit 2s complement number, $x_{n-1}x_{n-2}...x_1x_0$ can be written as $-x_{n-1}2^{n-1} + \sum_{i=0}^{n-2}x_i2^i$, where $x_{n-1}$ is the sign bit.

To get the smallest number, set sign bit to 1 and all other bits to 0. That gives you $-2^{n-1}$.

To get the largest, set sign bit to 0 and all other bits to 1. Through GP sum you get $2^{n-1} - 1$.

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an except from carl HAMACHER

same to same.

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can you please explain gp of that series

Answer 2

( - 2 )^n-1   to  ( 2^n-1 - 1)