Complement of a Number

Date: 7/10/96 at 10:51:3
From: Anonymous
Subject: Complement of a Number

What is the formula to find the complement of a number? 
What about the binary complement?

                               -Steve

Date: 7/11/96 at 18:15:51
From: Doctor Anthony
Subject: Re: Complement of a Number

To complement a number in base 10, you subtract it from a row of 9's. 
E.g. the complement of 5097 is 4912.  Likewise, in base 2, the 
complement of a number is obtained by subtraction from a row of 1's.  
All you have to do in fact is to interchange 0's and 1's. 
E.g. the complement of 10011101011 is
                       01100010100

In binary, a convenient way of subtraction is to add the complement.  
There is a slight complication when there is a carry 1 at the extreme 
left after the addition. In this case the carry 1 is moved to the 
extreme right and added. When making up the complement it is also 
necessary to make up the second number (with leading zeros if 
necessary) to the same number of digits as the first number before 
complementing.

Example. Find 101001101    Write this    101001101 
             -011100101                 +100011010
                                        -----------
                                        1001100111

Here we have a carry 1, so remove it and add to the extreme right hand 
column.  
                                         001100111
                                                 1
                                         ----------
                                           1101000 

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    

Date: 06/19/2008 at 14:44:07
From: Bob
Subject: 55778: Complement of a Number

I was searching for terminology regarding what the term for the 
original number is.  The complement is the "complement" but what
do you call the number you started with (i.e. the complement's 
complement)?
 
I came across your answer for the complement of a number being 
the number subtracted from a series of 9's.  I believe this is 
actually the diminished complement which is equal to the complement 
less one, so the complement of 9 is 1, 8 is 2, etc. and the 
complement of 0.25 is 0.75. 

I think generally the complement of x is thought of as b^n - x, 
where n is an integer such that base^n >= x >= base^(n-1).

Just wanted to bring this to your attention to review.  Thanks.

Date: 06/22/2008 at 23:15:57
From: Doctor Peterson
Subject: Re: 55778: Complement of a Number

Hi, Bob.

Dr. Anthony evidently assumed that "complement" meant "nines
complement" or "ones complement", which you are calling "diminished
[radix] complement".  I don't think it's a good idea to use the word
"complement" without any modifier at all, unless the context makes it
very clear (e.g. in the C programming language the word is used of the
"~" operator to mean the ones complement).  To clarify this, the
archived answer could be modified by adding something like this at the
start:

  You have not said what kind of complements you are asking about,
  but I will assume you mean the "tens complement" in base 10 and
  the "ones complement" in base 2.

In my experience, it is the tens (radix) complement that is used for 
subtraction, though as Dr. Anthony showed, it can be done.  So with
this one addition to prevent people from misinterpreting it, it's
probably good enough.  Note that this is one of our earlier answers;
there are others that go into more depth on different kinds of 
complements.


- Doctor Peterson, The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/