Converting Negative Decimals to Hexadecimal

Date: 05/30/2001 at 08:27:06
From: Josh Goodpaster
Subject: Converting negative decimal numbers to hexadecimal numbers

I'm trying to find out the formula for converting a negative decimal 
number to a hexadecimal number.

Using calculators I can get any of the conversions like - 16 = F0, but 
what I really need to know is HOW to get to F0 from - .16

Date: 06/05/2001 at 15:47:09
From: Doctor Twe
Subject: Re: Converting negative decimal numbers to hexidecimal 
numbers

Hi Josh - thanks for writing to Dr. Math.

That way of representing signed number is called "sixteen's 
complement" (the hexadecimal equivalent of two's complement in 
binary). Generally, sixteen's complement hexadecimal is used as a 
"shorthand notation" for two's complement signed binary numbers, 
therefore I think it is easier to understand when taken in two steps.

To find the sixteen's complement hexadecimal value, you would first 
convert the number to two's complement signed binary, then convert 
that binary value to hexadecimal.

To learn about two's complement, check out:

   Two's Complement
   http://mathforum.org/dr.math/problems/corbin.07.13.99.html   

   Negative Numbers in Binary
   http://mathforum.org/dr.math/problems/akella.8.19.99.html   

To learn about converting binary to hexadecimal, check out:

   Binary to Hexadecimal
   http://mathforum.org/dr.math/problems/hamilton12.8.98.html   

   Binary Conversion
   http://mathforum.org/dr.math/problems/stirling1.7.98.html   

   Converting Bases (bottom of the page)
   http://mathforum.org/dr.math/problems/reinhardt12.21.97.html   

all from our Ask Dr. Math archives.

Let's do an example. Suppose we want to find the hexadecimal 
representation of -53 (base 10). First, we convert 53 to binary:

                       1 1 0 1 0 1
                      / / / / / /
          0  r 1  ---+ / / / / /
        ----          / / / / /
      2 ) 1  r 1  ---+ / / / /
        ----          / / / /
      2 ) 3  r 0  ---+ / / /
        ----          / / /
      2 ) 6  r 1  ---+ / /
       -----          / /
     2 ) 13  r 0  ---+ /
       -----          /
     2 ) 26  r 1  ---+
       -----
     2 ) 53
        
So 53 (base 10) = 110101 (base 2). Next, we take the two's complement. 
We'll use 8-bit values (as in your example), so we have to fill the 
value to 8 bits by adding 2 leading zeroes on the left. Then invert 
the bits and add 1. We then have:

       00110101   (positive value)
           |
           v
       11001010   (invert)
     +        1   (add 1)
      ----------
       11001011   (two's complement negative result)

This is how -53 is represented in two's complement signed binary. The 
final step is to convert the binary to hexadecimal. We simply group 
the bits into groups of four, then convert each group to its 
hexadecimal equivalent, like so:

     1100 1011
     \__/ \__/
      12   11
       C    B

So -53 (base 10) = CB (signed hexadecimal using complements).

I hope this helps. If you have any more questions, write back.

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