Negative Numbers in Binary

Date: 08/19/99 at 17:06:35
From: Ramesh Akella
Subject: Binary Number

How do I represent -53 as a binary number? Is it 110101? Wouldn't that 
be +53?

Can you help me?

Date: 08/19/99 at 22:58:21
From: Doctor Peterson
Subject: Re: Binary Number

Hi, Ramesh.

Mathematically, there's nothing wrong with writing -53 as -110101, 
using a separate negative sign as we do in decimal. In computers, the 
usual method for storing signed numbers is called "twos-complement": 
to negate a number, you take the complement (changing 0 to 1 and 
1 to 0) and add 1:

      53 --> 00110101
     -53 --> 11001010 + 1 = 11001011

Since any negative number has the leftmost bit set, the effect of this 
is to take the upper half of the range of numbers that can be stored 
in a fixed number of bits (eight bits in my example) and use them for 
negative numbers. The numbers 0 to 127 are positive; numbers 128 to 
255 represent -128 to -1. You can still add two numbers just as you do 
with unsigned numbers:

       -53 =    11001011
     +  63 =    00111111
     -----   -----------
        10 = (1)00001010

In this case an extra carry digit is generated, but it is ignored and 
the result is the correct answer.

You can learn more about this by using a searcher like Google 
(http://www.google.com/) to look for the phrase  "twos complement."

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/